Abstract

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the irreducible modules of the Temperley-Lieb algebra.

1. Introduction

A goal of the categorification theorist is to replace interesting endomorphisms of a vector space with interesting endofunctors of a category. The question is what makes these functors interesting? In the pivotal paper of Chuang and Rouquier [1], a fresh paradigm emerged. They noticed that by specifying structure on the natural transformations (morphisms) between these functors one obtains more useful categorifications (in this case, the added utility is a certain derived equivalence). The categorification of quantum groups by Rouquier [2], Lauda [3], and Khovanov and Lauda [4] has shown that categorifying an algebra itself (with a category ) will specify what this additional structure should be for a categorification of any representation of that algebra: a functor from to an endofunctor category. That their categorifications provide the “correct” extra structure is confirmed by the facts that existing geometric categorifications conform to it (see [5]) and that irreducible representations of can be categorified in this framework (see [6, 7]). The salient feature of these categorifications is that, instead of being defined abstractly, the morphisms are presented by generators and relations, making it straightforward to define functors out of .

In the case of the Hecke algebra , categorifications have existed for some time, in the guise of category or perverse sheaves on the flag variety. In [8] Soergel rephrased these categorifications in a more combinatorial way, constructing an additive categorification of by a certain full monoidal subcategory of graded -bimodules, where is a polynomial ring. Objects in this full subcategory are called Soergel bimodules. There are deep connections between Soergel bimodules, representation theory, and geometry, and we refer the reader to [811] for more details. Categorifications using category and variants thereof are common in the literature, and often Soergel bimodules are used to aid calculations (see, e.g., [1214]).

In [15], Elias and Khovanov provides (in type ) a presentation of by generators and relations, where morphisms can be viewed diagrammatically as decorated graphs in a plane. To be more precise, the diagrammatics are for a smaller category , the (ungraded) category of Bott-Samelson bimodules, described in Section 2.1. Soergel bimodules are obtained from by taking the graded Karoubi envelope. This is in exact analogy with the procedures of Khovanov and Lauda in [4] and related papers.

The Temperley-Lieb algebra is a well-known quotient of , and it can be categorified by a quotient of , as this paper endeavors to show. Thus, we have a naturally arising categorification by generators and relations, and we expect it to be a useful one. Objects in can no longer be viewed as -bimodules (though their Hom spaces will be -bimodules), so that diagrammatics provide the simplest way to define the category.

The most complicated generator of is killed in the quotient to , making easy to describe diagrammatically in its own right. Take a category where objects are sequences of indices between and (denoted ). Morphisms will be given by (linear combinations of) collections of graphs embedded in , one for each index , such that the graphs have only trivalent or univalent vertices, and such that and are disjoint. Each graph will have a degree, making Hom spaces into a graded vector space. The intersection of the graphs with and determines the source and target objects, respectively. Finally, some local graphical relations are imposed on these morphisms. This defines , and we take the graded Karoubi envelope to obtain .

The proof that categorifies uses a method similar to that in [15]. We show first that is a potential categorification of , in the sense described in Section 2.2. Categorifications and potential categorifications define a pairing on given by , the graded dimension which takes values in . Equivalently, it defines a trace on via where is the monoidal identity (see Section 2.1). The difficult part is to prove the following lemma.

Lemma 1. The trace induced on from is the map defined in Section 2.2.

Given this lemma, it is surprisingly easy (see Section 3.3) to show the main theorem.

Theorem 2. Let be the graded additive closure of , and let be the graded Karoubi envelope of . Then is Krull-Schmidt and idempotent closed, so , and categorifies .

To prove the lemma, we note that there is a convenient set of elements in , the nonrepeating monomials, whose values determine any pairing; hence, there is a convenient set of objects whose Hom spaces will determine all Hom spaces. If is a nonrepeating sequence, the Hom space we must calculate is (up to shift) a quotient of by a two-sided ideal . We use graphical methods to determine these rings explicitly, giving generators for the ideals in which define them. As an interesting side note, these ideals also occur elsewhere in nature.

Proposition 3. Let be the reflection representation of , and identify with its coordinate ring. Let be the union of all the lines in which are intersections of reflection-fixed hyperplanes, and let be the ideal which gives the reduced scheme structure on . Then Hom spaces in are -bimodules, and the ideals cut out subvarieties of given by lines with certain transverseness properties (see Section 3.7 for details).

Also in Section 3.7, we give a topological interpretation of the ideals , using a functor defined by Vaz [16].

Now, let be the parabolic subalgebra of given by ignoring the index , and let be the induced (right) representation from the sign representation of . Such an induced representation is useful because it is a quotient of and also contains an irreducible module of as a submodule. All irreducibles can be constructed this way.

We provide a diagrammatic categorification of as a quotient of , and a categorification of as a full subcategory of , in a fashion analogous to quantum group categorifications. Having found a diagrammatic categorification of the positive half of the quantum group, Khovanov and Lauda in [17] conjectured that the highest weight modules (naturally quotients of ) could be categorified by quotients of by the appearance of certain pictures on the left. This approach was proven correct by Lauda and Vazirani [6] (for the -module structure), and then used by Webster to categorify tensor products [18]. Similarly, to obtain , we mod out by diagrams where any index except appears on the left. The proof that this works is similar in style to the proof of Theorem 2: one calculates the dimension of all Hom spaces by calculating enough Hom spaces to specify a unique pairing on and then uses simple arguments to identify the Grothendieck group.

Theorem 4. The category is idempotent closed and Krull-Schmidt. Its Grothendieck group is isomorphic to . Letting be the full subcategory generated by indecomposables which decategorify to elements of , one has that is idempotent closed and Krull-Schmidt, with Grothendieck group isomorphic to .

A future paper will categorify all representations induced from the sign and trivial representations of parabolic subalgebras of and . Induced representations were categorified more generally in [13] in the context of category , although not diagrammatically. We believe that our categorification should describe what happens in [13] after applying Soergel’s functor.

Soergel bimodules are intrinsically linked with braids, as was shown by Rouquier in [19, 20], who used them to construct braid group actions (these braid group actions also appear in the category context, see [21]). As such, morphisms between Soergel bimodules should correspond roughly to movies, and the graphs appearing in the diagrammatic presentation of the category should be (heuristically) viewed as 2-dimensional holograms of braid cobordisms. This is studied in [22]. The Temperley-Lieb quotient is associated with the representation theory of , for which all braids degenerate into 1-manifolds, and braid cobordisms degenerate into surfaces with disorientations. There is a functor from to the category of disorientations constructed by Vaz [16]. The functor is faithful (though certainly not full), as we remark in Section 3.7. This in turn yields a topological motivation of the variety and its subvarieties . Because is not full, there might be actions of that do not extend to actions of disoriented cobordisms. Cobordisms have long been a reasonable candidate for morphisms in Temperley-Lieb categorifications, although we hope will provide a useful substitute, with more explicit and computable Hom spaces.

Categorification and the Temperley-Lieb algebra have a long history. Khovanov in [23] constructed a categorification of using a TQFT, which was slightly generalized by Bar-Natan in [24]. This was then used to categorify the Jones polynomial. Bernstein et al. in [25] provide a categorical action of the Temperley-Lieb algebra by Zuckerman and projective functors on category . Stroppel [14] showed that this categorical action extends to the full tangle algebroid, and also investigated the natural transformations between projective functors. Recent work of Brundan and Stroppel [26] connects these Temperley-Lieb categorifications to Khovanov-Lauda-Rouquier algebras, among other things. We hope that our diagrammatics will help to understand the morphisms in these categorifications.

The organization of this paper is as follows. Section 2 will provide a quick overview of the Hecke and Temperley-Lieb algebras, and the diagrammatic definition of the category . Section 3 begins by defining the quotient category diagrammatically in its own right (which makes a thorough understanding of the diagrammatic calculus for unnecessary). Section 3.3 proves Theorem 2, modulo Lemma 1 which requires all the work. The remaining sections of that chapter do all the work, and starting with Section 3.6 one will not miss any important ideas if one skips the proofs. Section 4 begins with a discussion of cell modules for and certain other modules, and then goes on to categorify these modules, requiring only very simple diagrammatic arguments.

This paper is reasonably self-contained. We do not require familiarity with [15] and do not use any results other than Corollary 28. We do quote some results for motivational reasons, but the difficult graphical arguments of that paper can often be drastically simplified for the Temperley-Lieb setting, so that we provide easier proofs for the results we need. Familiarity with diagrammatics for monoidal categories with adjunction would be useful, and [3] provides a good introduction. More details on preliminary topics can be found in [15].

2. Preliminaries

Notation 5. Fix , and let index the vertices of the Dynkin diagram . We use the word index for an element of , and the letters always represent indices. Indices are adjacent if , and distant if , and questions of adjacency always refer to the Dynkin diagram, not the position of indices in a word or picture.

Notation 6. Let with simple reflections . Let be a field of characteristic not dividing ; all vector spaces will be over this field. Let , where ; it is a graded ring, with . We will abuse notation and refer to elements of and their images in in the same way, and will refer to both as polynomials. Note that , where , since modulo . The ring arises as the coordinate ring of , the reflection representation of (the span of the root system), and are the simple coroots.
There is an obvious action of on , which permutes the generators . For each index we have a Demazure operator , a map of degree from to the invariant subring , which is -linear and sends to . Explicitly, .

Notation 7. Let be the -linear involution of switching and . Given a -linear map of modules, we call it antilinear if it is -linear after twisting by , or in other words if . We write .
Let be a -algebra. In this paper we always use the word trace to designate a -linear map satisfying . We use the word pairing or semilinear pairing to denote a -linear map which is -linear in the second factor and -antilinear in the first factor.

2.1. The Hecke Algebra and the Soergel Categorification

We state here without proof a number of basic facts about the Hecke algebra, its traces, and Soergel’s categorification. For more background, see Soergel’s original definition of his categorification [8], or an easier version [11]. A similar overview with more discussion can be found in [15]. A more in-depth introduction, connecting Soergel bimodules to other parts of representation theory, can be found in [13].

Definition 8. Denote by the Hecke algebra for . It is a -algebra, specified here by its Kazhdan-Lusztig presentation: it has generators and relations

Definition 9. Given two objects in a graded -linear (possibly additive) category , where denotes the grading shift, the graded hom space between them is the graded vector space . Given a class of objects in , we can define a category with morphisms enriched in graded vector spaces, whose objects are and whose morphisms are . Let us call this an enriched full subcategory, which we often shorten to the adjective enriched. While the enriched subcategory is neither additive nor graded, it has enough information to recover the hom spaces between grading shifts and direct sums of objects in .
Let -bim denote the category of finitely-generated graded (resp., ungraded) -bimodules. Then spaces in -bim will be graded -bimodules. For , let -bim be defined by , where is the invariant subring. A Bott-Samelson bimodule is a tensor product in -bim, where here and henceforth denotes the tensor product over . Let be the enriched full subcategory generated by the Bott-Samelson bimodules; it is a monoidal category, but is neither additive nor graded. Let denote the full subcategory of -bim given by all (finite) direct sums of grading shifts of Bott-Samelson bimodules; it is monoidal, additive, and graded. Finally, let denote the category of Soergel bimodules or special bimodules, the full subcategory of -bim given by all (finite) direct sums of grading shifts of summands of Bott-Samelson bimodules; it is monoidal, additive, graded, and idempotent closed.

One can observe that all bimodules in are free and finitely generated when viewed as either left -modules or right -modules, and therefore the same is true of any space. The following proposition parallels the Kazhdan-Lusztig presentation for .

Proposition 10. The category is generated (as an additive, monoidal category) by objects which satisfy

From this we might expect the next result.

Proposition 11. The Grothendieck ring of is isomorphic to , with being sent to , and being sent to . The Grothendieck ring of is isomorphic to as well.

Remark 12. The proof of this statement is not immediately obvious. There is clearly a surjective morphism from to . When one takes the idempotent closure of a category, one adds new indecomposables and can potentially enlarge the Grothendieck group. Soergel showed, via a support filtration, that all the new indecomposables in have symbols in which can be reached from certain symbols in by a unitriangular matrix (see [11]). Therefore, the Grothendieck rings of and are equal. Since is idempotent closed and is embedded in - , it has the Krull-Schmidt property and the Grothendieck group behaves as one would expect: it has a basis given by indecomposables. By classifying indecomposables and using the unitriangular matrix, Soergel showed that the map from to is actually an isomorphism.
It is important to note that one does not know what the image of the indecomposables of in is. The Soergel conjecture, still unproven in generality, proposes that the indecomposables of descend to the Kazhdan-Lusztig basis of (see [11]).

Notation 13. We write the monomial as , where is a finite sequence of indices; by abuse of notation, we sometimes refer to this monomial simply as . If is as above, we say the monomial has length . We call a monomial nonrepeating if for , and increasing if . The empty set is a sequence of length 0, and . Similarly, in , write as . Note that , the monoidal identity. For an arbitrary index and sequence , we write if appears in .

Given two objects -bim we say they are biadjoint if and are left and right adjoints of each other, and the same for and . If and are biadjoint, so are and . We often want to specify additional compatibility between various adjunction maps, but we pass over the details here (see [3] for more information on biadjunction).

Proposition 14. Each object in (resp., , ) has a biadjoint, and is self-biadjoint. Let be the -antilinear anti-involution on which fixes , that is, , where reverses the order of a sequence. There is a contravariant functor on sending an object to its biadjoint, and it descends on the Grothendieck ring to .

Definition 15. An adjoint pairing on is a pairing where each is self-adjoint, so that and for all and all . Equivalently, for any , and .

There is a bijection between adjoint pairings and traces , defined by letting , or conversely . Adjoint pairings appear often in the literature, for instance [27] (although they are usually -linear in both factors, unlike our current semilinear definition). Semilinear adjoint pairings will be crucially important, due to the following remark.

Remark 16. Let be a monoidal category with objects , such that are self-biadjoint. We assume that is additive and graded and has isomorphisms (2.4)–(2.6). We call such a category a potential categorification of . In this case, there is a map of rings from to sending to , and (under suitable finite-dimensionality conditions) we get an adjoint semilinear pairing on via , the graded dimension as a vector space. Denote the pairing and its associated trace map as and .
Instead, we may assume is an enriched monoidal subcategory, containing objects . The isomorphisms (2.4)–(2.6) typically have no meaning in this context, since there are no grading shifts or direct sums, but we can require that they Yoneda-hold; that is, they hold after the application of any functor (to graded vector spaces). There is no definition of a Grothendieck ring in this case, but we still get an induced adjoint semilinear pairing induced by Hom spaces. We call this an enriched potential categorification.

We may use pairings to distinguish between different potential categorifications. The next proposition allows us to specify the pairing induced by a categorification by only investigating certain spaces.

Proposition 17. Traces on are uniquely determined by their values on increasing monomials . Equivalently, adjoint pairings are determined by for increasing . If is nonrepeating and is a permutation of , then .

We quickly sketch the proof. Moving an index from the beginning of a monomial to the end, or vice versa, will be called cycling the monomial. It is clear, using biadjointness or the definition of trace, that the value of is invariant under cycling. It is not difficult to show that any monomial in (in the letters ) will reduce, using the Coxeter relations and cycling, to an increasing monomial. When the monomial is already nonrepeating, one needs only use cycling and for distant . Finally, using induction on the length of the monomial, the same principle shows that any monomial in reduces to a linear combination of increasing monomials, and therefore is determined by these.

The upshot is that, given a potential categorification, one knows the dimension of all so long as one knows the dimension of for increasing . Note that not every choice of for all increasing will yield a well-defined trace map.

Consider the adjoint pairing given by for nonrepeating of length . This is the semilinear version of the pairing found in [27] which picks out the coefficient of the identity in the standard basis of and is called the standard pairing. Soergel showed that is a free graded left (or right) -module of rank using this pairing. In particular, for increasing , is generated by a single element in degree . Since the graded dimension of is we have that is a rescaling of the standard pairing.

Now let be the quotient map by the ideal generated by all . It is a homomorphism to a commutative algebra, so it is a trace. The corresponding pairing satisfies and for monomials if either monomial is not . We call this the trivial pairing, .

2.2. The Temperley-Lieb Algebra

Here again we state without proof some basic facts about Temperley-Lieb algebras. They were originally defined by Temperley and Lieb in [28], and were given a topological interpretation by Kauffman [29]. There are many good expositions for the topic, such as [30, 31].

Definition 18. The Temperley-Lieb algebra is the -algebra generated by with relations

Proposition 19. For adjacent , consider the element of defined by , where the equality arises from relation (2.3). There is a surjective map sending to for all , and whose kernel is generated by for adjacent .

Once again, write for a monomial in the above generators, with all the same conventions as before. The map descends from to , and we define an adjoint pairing on in the same way, with replacing everywhere. The results of Proposition 17 apply equally to .

Definition 20. A category as in Remark 16 is a potential categorification of if it has objects satisfying We call it an enriched potential categorification if it is an enriched category with objects such that these isomorphisms Yoneda-hold.

A permutation is called 321-avoiding if it never happens that, for , . It turns out that, using the Temperley-Lieb relations, every monomial is equal to a scalar times some where is 321-avoiding; that is, if viewed as a word in the symmetric group, it represents a reduced expression for a 321-avoiding permutation. Moreover, between 321-avoiding monomials, the only further relations come from , and hence it is easy to pick out a basis from this spanning set. See [30] for more details.

The Temperley-Lieb algebra has a well-known topological interpretation where an element of is a linear combination of crossingless matchings (isotopy classes of embedded planar 1-manifolds) between bottom points and top points. Multiplication of crossingless matchings consists of vertical concatenation (where is above ), followed by removing any circles and replacing them with a factor of . In this picture, becomes the following:

The basis of 321-avoiding monomials agrees with the basis of crossingless matchings. Any increasing monomial is 321-avoiding. Increasing monomials are easy to visualize topologically, as they have only “right waves” and “simple cups and caps.” For example:

As an example of a monomial which is not increasing:

Given a crossingless matching, its closure is a configuration of circles in the punctured plane obtained by wrapping the top boundary around the puncture to close up with the bottom boundary, as in Figure 1. Circle configurations have two topological invariants: the number of circles and the nesting number which is the number of circles which surround the puncture and is equal to for some . Given a scaling factor for each possible nesting number, one constructs a trace by letting where is the number of circles in the closure of and is the scaling factor associated to its nesting number . To calculate , we place below an upside-down copy of (or vice versa), and then take the closure. All pairings/traces on can be constructed this way, so they are all topological in nature.

The Temperley-Lieb algebra has a standard pairing of its own for which for all nesting numbers : as above. One can check that for an increasing monomial. This is not related to the standard pairing on , which does not descend to . On the other hand, clearly does descend to a pairing trivial pairing on , which only evaluates to a nonzero number when the nesting number is .

It turns out that the pairing on arising from our categorification will satisfy and . We will call the associated trace . Clearly . In particular, on any monomial , our trace will agree with a rescaling of the standard trace. When , the algebras and are already isomorphic, and agrees with the rescaling of the standard trace on discussed at the end of Section 2.1.

2.3. Definition of Soergel Diagrammatics

We now give a diagrammatic description of the category , as discovered in [15]. Since the category to be defined will be equivalent to the category of Bott-Samelson bimodules, we will abuse notation temporarily and use the same names.

Definition 21. In this paper, a planar graph in the strip is a finite graph with boundary embedded in . In other words, all vertices of occur in the interior , and removing the vertices, we have a 1-manifold with boundary whose intersection with is precisely its boundary. This allows for edges which connect two vertices, edges which connect a vertex to the boundary, edges which connect two points on the boundary, and edges which form circles (closed 1-manifolds embedded in the plane).
We generally refer to as the boundary, which consists of two components, the top boundary , and the bottom boundary . We refer to a local segment of an edge which hits the boundary as a boundary edge; there is one boundary edge for each point on the boundary of the graph. We use the word component to mean a connected component of a graph with boundary.
This definition clearly extends to other subsets of the plane with boundary, so that we can speak of planar graphs in a disk or planar graphs in an annulus. The annulus has two boundary components, inner and outer. When we do not specify, we always mean a planar graph in the strip.

We will be drawing morphisms in as planar graphs with edges labelled in . Instead of putting labels everywhere, we color the edges, assigning a color to each index in . Henceforth, we use the term “color” and “index” interchangeably.

We now define anew. Let be the monoidal category, with hom spaces enriched over graded vector spaces, which is defined as follows.

Definition 22. An object in is given by a sequence of indices , which is visualized as points on the real line , labelled or “colored” by the indices in order from left to right. These objects are also called . The monoidal structure on objects is concatenation of sequences.

Definition 23. Consider the set of isotopy classes of planar graphs in the strip whose edges are colored by indices in such that only four types of vertices exist: univalent vertices or “dots”, trivalent vertices with all three adjoining edges of the same color, 4-valent vertices whose adjoining edges alternate in colors between distant and , and 6-valent vertices whose adjoining edges alternate between adjacent and . This set has a grading, where the degree of a graph is +1 for each dot and -1 for each trivalent vertex; -valent and -valent vertices are of degree . The allowable vertices, which we call “generators,” are pictured here:

The intersection of a graph with the boundary yields two sequences of colored points on , the top boundary and the bottom boundary . In this case, the graph is viewed as a morphism from to . For instance, if “blue" corresponds to the index and “red" to , then the lower right generator is a degree 0 morphism from to . Although this paper is easiest to read in color, it should be readable in black and white: the colors appearing are typically either blue, red, green, or miscellaneous and irrelevant. We throughout use the convention that blue (the darker color) is always adjacent to red (the middle color) and distant from green (the lighter color).

We let be the graded vector space with basis given by planar graphs as above which have the correct top and bottom boundary, modulo relations (2.13) through (2.27). As usual in a diagrammatic category, composition of morphisms is given by vertical concatenation (read from bottom to top), the monoidal structure is given by horizontal concatenation, and relations are to be interpreted monoidally (i.e., they may be applied locally inside any other planar diagram).

The relations are given in terms of colored graphs, but with no explicit assignment of indices to colors. They hold for any assignment of indices to colors, so long as certain adjacency conditions hold. We will specify adjacency for all pictures, although one can generally deduce it from the fact that 6-valent vertices only join adjacent colors, and 4-valent vertices only join distant colors.

For example, these first four relations hold, with blue representing a generic index.

We will repeatedly call a picture looking like by the name “needle.” Note that a needle is not necessarily zero if there is something in the interior. Note that a circle is just a needle with a dot attached, by , so that an empty circle evaluates to .

Remark 24. It is an immediate consequence of relations and that any tree (connected graph with boundary without cycles) of one color is equal to (i) if it has no boundary, two dots connected by an edge. Call the entire component a double dot. (ii) if it has one boundary edge, a single dot connected by the edge to the boundary. Call the component a boundary dot. (iii) if it has more boundary edges, a tree with no dots and the fewest possible number of trivalent vertices needed to connect the boundaries. Moreover, any two such trees are equal. Call the component a simple tree. We refer to this as tree reduction.

This applies only to components of a graph which are a single color. Even if the blue part of a graph looks like a tree, if other colors overlap, then we may not apply tree reduction in general.

In the following relations, the two colors are distant

In this relation, two colors are adjacent, and both distant to the third color.

In this relation, all three colors are mutually distant.

Remark 25. Relations through indicate that any part of the graph colored and any part of the graph colored “do not interact” for distant and , that is, one may visualize sliding the -colored part past the -colored part, and it will not change the morphism. We call this the distant sliding property.

In the following relations, the two colors are adjacent.

In this final relation, the colors have the same adjacency as

This concludes the list of relations defining .

Remark 26. We chose here to describe in terms of planar graphs with relations, with the notion of isotopy built-in, rather than in terms of generators and relations. Note, however, that using isotopy and , we get Therefore, all “cups" and “caps" can be expressed in terms of the generators. By adding new relations corresponding to isotopy, one could give a presentation of the category where the “generators” above (and their isotopy twists) are really generators. This is how the category is presented in [15].

We will occasionally use a shorthand to represent double dots. We identify a double dot colored with the polynomial , and for a linear combination of disjoint unions of double dots in the same region of a graph, we associate the appropriate linear combination of products of . For any polynomial , a square box with a polynomial in a region will represent the corresponding linear combination of graphs with double dots.

For instance,

Relations , , and are referred to as dot forcing rules, because they describe at what price one can “force" a double dot to the other side of a line. The three relations imply that, given a line and an arbitrary collection of double dots on the left side of that line, one can express the morphism as a sum of diagrams where all double dots are on the right side, or where the line is “broken" (as illustrated next). Rephrasing this, for any polynomial there exist polynomials and such that

The polynomials appearing can in fact be found using the Demazure operator , and in particular, . One particular implication is that whenever is a polynomial invariant under (and blue represents ). As an exercise, the reader can check that slides through a line colored . These polynomial relations are easy to deduce, or one can refer to [15] (see page 7, pages 16-17, and relation ).

We have an bimodule action of on morphisms by placing boxes (i.e., double dots) in the leftmost or rightmost regions of a graph. Now we can formulate the main result of [15].

Theorem 27. There is a functor from this diagrammatic category to the earlier definition in terms of Bott-Samelson bimodules. This functor sends to the bimodule and a planar graph to a map of bimodules, preserving the grading and the -bimodule action on morphisms. This functor is an equivalence of categories.

Corollary 28. The -bimodules are free as left (or right) -modules. In other words, placing double dots to the left of a graph is a torsion-free operation.

Now, we have justified our abuse of notation. In this paper, we will never need to know explicitly what map of -bimodules a planar graph corresponds to, so the interested reader can see [15] for details. In fact, we will not use Theorem 27 at all, preferring to work entirely with planar graphs. However, we do use Corollary 28, a fact which would be difficult to prove diagrammatically.

The proof of Theorem 27 can be quickly summarized: first, one explicitly constructs a functor from the diagrammatic category to the Bott-Samelson category. Then, using the observations of the next section, one shows that the diagrammatic category is a potential categorification of and that the diagrammatic category, the Bott-Samelson category, and the image of the former in the latter all induce the same adjoint pairing on . Therefore, the functor is fully faithful.

2.4. Understanding Soergel Diagrammatics

Let us explain diagrammatically why the category is a potential categorification of , and induces the aforementioned adjoint pairing.

Definition 29. Given a category whose morphism spaces are -modules, one may take its additive closure, which formally adds direct sums of objects and yields an additive category. Given whose morphism spaces are graded -modules, one may take its grading closure which formally adds shifts of objects, but restricts morphisms to be homogeneous of degree 0. Given an additive category, one may take the idempotent completion or Karoubi envelope, which formally adds direct summands. Recall that the Karoubi envelope has as objects pairs where is an object in and an idempotent endomorphism of . This object acts as though it were the “image” of this projection and behaves like a direct summand. When taking the Karoubi envelope of a graded category (or a category with graded morphisms) one restricts to homogeneous degree 0 idempotents. We refer in this paper to the entire process which takes a category , whose morphism spaces are graded -modules, and returns the Karoubi envelope of its additive and grading closure as taking the graded Karoubi envelope. All these transformations interact nicely with monoidal structures. For more information on Karoubi envelopes see [32].
We let be the graded additive closure of , and let be the graded Karoubi envelope of .

We wish to show that the isomorphisms through hold in . Relation immediately implies that for distant, with the isomorphism being given by the 4-valent vertex.

We have the following equality: To obtain this, use to stretch two dots from the two lines into the middle, and then use to connect them. The identity decomposes as a sum of two orthogonal idempotents, each of which is the composition of a “projection" and an “inclusion" map of degree , to and from (explicitly, where , , ). This implies that and is a typical example of how direct sum decompositions work in diagrammatic categories.

Similarly, the two color variants of relation together express the direct sum decompositions in the Karoubi envelope Again, the identity is decomposed into orthogonal idempotents. The second idempotent factors through , and the corresponding object in the Karoubi envelope will be isomorphic to . The first idempotent, which we call a “doubled 6-valent vertex,” corresponds to a new object in the idempotent completion. It turns out that the doubled 6-valent vertex for “blue red blue" is isomorphic in the Karoubi envelope to the doubled 6-valent vertex for “red blue red" (i.e., their images are isomorphic). We may abuse notation and call both of these new objects ; it is a summand of both and . The image of in the Grothendieck group is .

We can also understand the induced pairing on using diagrammatic arguments. The theorems below are proven in [15], and we will not use them in this paper (except motivationally), proving their analogs in the Temperley-Lieb case directly.

Theorem 30 (Color Reduction). Consider a morphism , and suppose that the index (blue) appears in zero times (resp.,: once). Then is in the -span of graphs which only contain blue in the form of double dots in the leftmost region of the graph (resp., as well as a single boundary dot). This result may be obtained simultaneously for multiple indices .

Corollary 2.23The space is precisely the graded ring . In other words, it is freely generated (over double dots) by the empty diagram. The space for nonrepeating is a free left (or right) -module of rank 1, generated by the following morphism of degree .

The proof of the theorem does not use any sophisticated technology, only convoluted pictorial arguments. It comprises the bulk of [15]. The corollary implies that for nonrepeating of length , as stated in Section 2.1.

2.5. Aside from Karoubi Envelopes and Quotients

Return to the setup of Definition 29. If is a full subcategory of (graded) -bimodules for some ring , then the transformations described above behave as one would expect them to. In particular, the Karoubi envelope agrees with the full subcategory which includes all summands of the previous objects. The Grothendieck group of the Karoubi envelope is in some sense “under control" if one understands indecomposable -bimodules already. On the other hand, the Karoubi envelope of an arbitrary additive category may be enormous, and to control the size of its Grothendieck group, one should understand and classify all idempotents in the category, a serious task. Also, arbitrary additive categories need not have the Krull-Schmidt property, making their Grothendieck groups even more complicated.

The Temperley-Lieb algebra is obtained from the Hecke algebra by setting the elements to zero, for . These elements lift in the Soergel categorification to objects . The obvious way one might hope to categorify would be to take the quotient of the category by each object .

To mod out an additive monoidal category by an object , one must kill the monoidal ideal of in , that is, the morphism space in the quotient category is exactly modulo the submodule of morphisms factoring through for any . If the category is drawn diagrammatically, one needs to only kill any diagram which has as a subdiagram.

We have not truly drawn diagrammatically, only . The object we wish to kill is not an object in ; the closest thing we have is the corresponding idempotent, the doubled 6-valent vertex. However, this is not truly a problem, due to the following proposition, whose proof we leave to the reader.

Proposition 32. Let be an additive category, and let be an object in , and let be an idempotent in . Let be the quotient of by the morphism . Let and be the respective Karoubi envelopes. Finally, let be the quotient of by the identity of the object . Then, there is a natural equivalence of categories from to .
The analogous statement holds when one considers graded Karoubi envelopes.

Remark 33. Note that has more objects than , but they are still equivalent. For instance, and are distinct (isomorphic) objects in , but are the same object in .

So to categorify , one might wish to take the quotient of by the doubled 6-valent vertex, and then take the Karoubi envelope. This is easy to do diagrammatically, which is one advantage to the diagrammatic approach over the -bimodule approach. The quotient of will no longer be a category which embeds nicely as a full subcategory of bimodules. One might worry that Krull-Schmidt fails, or that to understand its Karoubi envelope one must classify all idempotents therein. Thankfully, our calculation of HOM spaces will imply easily that its graded additive closure is Krull-Schmidt and is already idempotent closed, so it is equivalent to its own Karoubi envelope (see Section 3.3).

3. The Quotient Category

3.1. A Motivating Calculation

As discussed in the previous section, our desire is to take the quotient of by the doubled 6-valent vertex, and then take the graded Karoubi envelope.

An important consequence of relations and is that from which it follows, using , that so the (monoidal) ideal generated in by a doubled 6-valent vertex is the same as the ideal generated by the 6-valent vertex.

Claim 1. The following relations are all equivalent (the ideals they generate are equal)

Proof. : add a dot, and use relation .
: add a dot to the top, and use .
: apply to the middle of the diagram.
: stretch dots from the blue strands towards the red strand using , and then apply to the middle.
: use relation .
: use .

Modulo 6-valent vertices, the relations and become and above. All other relations involving 6-valent vertices, namely, , , and , are sent to zero modulo 6-valent vertices. Relation implies both and without reference to any graphs using 6-valent vertices. So, if we wish to rephrase our quotient in terms of graphs that never have 6-valent vertices, the sole necessary relation imposed by the fact that 6-valent vertices were sent to zero is the relation .

Suppose, we only allow ourselves univalent, trivalent, and 4-valent vertices, but no 6-valent vertices, in a graph . Then, the -graph of , which consists of all edges colored and all vertices they touch, will be disjoint from the - and -graphs of . The distant sliding property implies that the -graph and the -graph of do not interact effectively, when and are distant. This will motivate the definition in the next section.

3.2. Diagrammatic Definition of

Definition 34. We let be the monoidal category, with hom spaces enriched over graded vector spaces, defined as follows. Objects will be sequences of colored points on the line , which we will call or . Consider the set whose elements are described as follows: (1) for each , consider a planar graph in the strip, which is drawn with edges colored (see Definition 21); (2) the only vertices in are univalent vertices (dots) and trivalent vertices; (3) the graphs and are disjoint. All graphs are pairwise disjoint on the boundary; (4) we consider isotopy classes of this data, so that one may apply isotopy to each individually so long as it stays appropriately disjoint.

This set has a grading, where the degree of a graph is for each dot and for each trivalent vertex, and the degree of an element of this set is the sum of the degrees for each graph . Just as in Definition 23, each element of the set has a top and bottom boundary which is an object in , and will be thought of as a map from the bottom boundary to the top. We let be the graded vector space with basis given by elements of the set above with bottom boundary and top boundary , modulo the relations through , , and the new relation . As a reminder, the new relation is given here again.

As before, composition of morphisms is given by vertical concatenation, the monoidal structure is given by horizontal concatenation, and relations are to be interpreted monoidally. This concludes the definition.

Phrasing the definition in this fashion eliminates the need to add distant sliding rules, for these are now built into the notion of isotopy. Note that as we have stated it here, and may have edges which are embedded in a tangent fashion, or even entirely overlapped. However, such embeddings are isotopic to graph embeddings with only transverse edge intersections, which arise as 4-valent vertices in our earlier viewpoint.

Proposition 35. The category is isomorphic to modulo the 6-valent vertex.

Proof. Due to the observations of Section 3.1, this is obvious.

Hom spaces in are in fact enriched over graded -bimodules, by placing double dots as before. However, they will no longer be free as left or right -modules, as we will see.

Remark 36. Note that tree reduction (see Remark 24) can now be applied to any tree of a single color in , regardless of what other colors are present, since the only colors which can intersect the tree are distant colors which do not actually interfere.
We denote by the graded Karoubi envelope of , and by the graded additive closure of . However, we will show that is already idempotent closed, so that and are the same.

It is obvious that in , and from the relation and the simple calculation (using dot forcing rules) that

For the same reasons as in Section 2.4, we still have for distant , and in . Therefore, is a potential categorification of , and induces an adjoint pairing and a trace map on . At this point, we have not shown that the category is nonzero, so this pairing could be .

3.3. Using the Adjoint Pairing

Proposition 37. Let be an enriched category which is a potential categorification of , whose objects are for sequences . Let be its additive graded closure, and let be its graded Karoubi envelope. Suppose that the induced trace map on is equal to . Then, the set of for and 321-avoiding forms an exhaustive irredundant list of indecomposables in . In addition, is Krull-Schmidt and idempotent closed (so and are equivalent), and categorifies .

This proposition is an excellent illustration of the utility of the induced adjoint pairing. We prove it in a series of lemmas, which all assume the hypotheses above.

Lemma 38. The object in has no nontrivial (homogeneous) idempotents when is 321-avoiding. Moreover, if both and are 321-avoiding, then in if and only if and in .

Proof. Two 321-avoiding monomials in are equal only if they are related by the relation . Since this lifts to an isomorphism in , we have .
If an object has a 1-dimensional space of degree 0 endomorphisms, then it must be spanned by the identity map, and there can be no nontrivial idempotents. If an object has endomorphisms only in nonnegative degrees, then it can not be isomorphic to any nonzero degree shift of itself. If two objects and are such that both and are concentrated in strictly positive degrees, then no grading shift of is isomorphic to , since there can not be a degree zero map in both directions.
Therefore, we need only show that (for 321-avoiding monomials) has endomorphisms concentrated in nonnegative degree, with a 1-dimensional degree 0 part, and that when , is in strictly positive degrees. This question is entirely determined by the pairing on , since it only asks about the graded dimension of Hom spaces.
When is empty, we already know that , which has degree 0 coefficient , and is concentrated in nonnegative degrees.
We know how to calculate in when and are monomials, and either or is not (see Section 2.2). We draw as a crossingless matching, draw upside-down and place it below , and close off the diagram: if there are circles in the diagram, then . In particular, if , then the Hom space will be concentrated in nonnegative degrees, with 1-dimensional degree 0 part. If , then the Hom space will be concentrated in strictly positive degrees.
We leave it as an exercise to show that, if is a crossingless matching (i.e., a 321-avoiding monomial) then the closed diagram for has exactly circles. The following example makes the statement fairly clear, where is upside-down:

In this example has all 3 kinds of arcs which appear in a crossingless matching: bottom to top, bottom to bottom, and top to top. Each of these corresponds to a single circle in the diagram closure.

Similarly, there are fewer than circles in the diagram for whenever the crossingless matchings are nonequal. Consider the diagram above but with the region removed. One can see that no circles are yet completed, and each boundary point of ’s region is matched to the other by an arc. The number of circles is maximized when you pair these boundary points to each other, and this clearly gives the matching . For any other matching , two arcs will become joined into one, and fewer than circles will be created.

Lemma 39. is idempotent closed, and its indecomposables can all be expressed as grading shifts of for 321-avoiding . It has the Krull-Schmidt property.

Proof. Since the Temperley-Lieb relations allow one to reduce a general word to a 321-avoiding word, one can show that every is isomorphic to a direct sum of shifts of for 321-avoiding , using isomorphisms and direct sum decompositions instead of the analogous Temperley-Lieb relations. Clearly these shifted are all indecomposable, since they have no nontrivial idempotents; these are then all the indecomposables. Since every indecomposable in has a graded local endomorphism ring (with maximal ideal given by positively graded morphisms), is idempotent closed and Krull-Schmidt (see [33], Section 2.2).

The Krull-Schmidt property implies that isomorphism classes of indecomposables form a basis for the Grothendieck group.

Proof of Proposition 37. There is a -linear map of rings , which is evidently bijective because it sends the 321-avoiding basis to the 321-avoiding basis. Since , we are done.

This proposition shows that Lemma 1 implies Theorem 2.

Remark 40. In analogy to the paper [15], the bulk of the proof of Theorem 2 lies in proving that hom spaces induce a particular adjoint pairing. Beyond that, we have mostly stated the obvious. Let us note that what is obvious for and is not obvious at all when dealing with and . In particular, if we are given a category which is a potential categorification of as in Proposition 37, we can not conclude that categorifies . We summarize the differences here.
It is clear (for both Hecke and Temperley-Lieb) that the map is well defined and surjective. The two main subtleties are the difference between and , and the injectivity of the map.
In general, one likes to examine the additive Grothendieck group only of idempotent closed categories with the Krull-Schmidt property, because this guarantees that indecomposables form a basis for the Grothendieck group. Thus it is convenient that is already idempotent closed. Thankfully, we have a result of Soergel [11] that proves that , as was discussed in Remark 12.
To show injectivity of the map in the case, we can identify a basis for which is sent to a complete set of indecomposables, and then we can evaluate the trace map to show that these indecomposables are pairwise nonisomorphic. For , we do not currently know what the indecomposables (i.e., idempotents) are, nor do we know their preimage in . If we knew a class of indecomposables which decategorified to the Kazhdan-Lusztig basis, then we could use a similar argument to the above to show that they and their shifts form an exhaustive irredundant list of indecomposables in , and therefore that the map is injective. Soergel discusses this in the last chapter of [11]. This is actually a deep question, shown by Soergel ([34], see also [8, 11]) to be equivalent to proving a version of the Kazhdan-Lusztig conjectures. In any case, the result depends on the base field , and no simple proof has been found. In particular, to prove that the graded Karoubi closure of the diagrammatic category categorifies the Hecke algebra (for certain ), we must pass to the world of bimodules where Soergel’s powerful geometric techniques will work. In particular, there is currently no proof of injectivity if one defines the category diagrammatically over .
It should be emphasized that the story of is a particularly easy one (as is its Kazhdan-Lusztig theory). No high-powered technical machinery is needed, and the proofs of idempotent closure and injectivity are self-contained and diagrammatic. In fact, the arguments in this paper do work entirely over , as can be checked. Dividing by two must be allowed in order to split the identity of into idempotents, as in ; however, it is likely that the arguments would work over as well. Working over is discussed more extensively in [22].

Remark 41. A category analog of the fact that 321-avoiding monomials lift to indecomposable Soergel bimodules, which remain indecomposable upon passage to the Temperley-Lieb quotient, can be found in Lemma of [14].

3.4. Reductions

When we say that a graph or a morphism “reduces" to a set of other graphs, we mean that the morphism is in the -span of those graphs. We refer to a one-color graph, each of whose (connected) components is either a simple tree with respect to its boundary or a double dot, as a simple forest with double dots. If there are no double dots, it is a simple forest without double dots. Tree reduction implies that any graph without cycles reduces to a simple forest with double dots. Note also that circles in a graph are equal to needles with a dot attached, and can be treated just like any other cycle.

If there were only one color, we could iterate the following rule (which is an implication of the dot forcing rules and ) to break cycles:

We do something similar for the general case.

Proposition 42. In , any morphism reduces to one where, for each , the -graph is a simple forest with double dots. Moreover, we may assume that all double dots are in the lefthand region.

Proof. We use induction on the total number of cycles (of any color) in the graph. Suppose there is a blue colored cycle: choose one so that it delineates a single region (i.e., there are no other cycles inside). There may be blue “spokes” going from this cycle into the interior, but no two spokes can meet, lest they create another region. By tree reduction on the spokes, we can assume that any blue appearing inside the cycle is in a different blue component than the cycle. Other colors may cross over the cycle, into the interior. If we view the interior of the cycle as a graph of its own, it has fewer total cycles so we may use induction. Since the boundary of the interior contains no blue color or colors adjacent to blue, they may be assumed to appear in the interior only in the form of double dots next to the cycle. Using dot-forcing rules, we reduce to two graphs: one with the cycle broken, and one with all these double dots on the exterior of the cycle. The former reduces by induction. For the latter, only distant colors enter the cycle, so they can be slid out of the way to leave an empty blue cycle, which is by the rule above.
We need to only do the base case, where the graph has no cycles. The dot-forcing rules imply that double dots may be moved to any region of the (multicolored) graph, at the cost of breaking a few lines. Breaking lines will never increase the number of cycles. Therefore, if we have a graph without cycles, tree reduction implies that we actually have a simple forest with double dots, and dot forcing allows us to move these double dots to the left. The breaking of lines may require more tree reduction, yielding more double dots, but this process is finite.

Remark 43. This proposition and its proof will apply to graphs in any connected simply connected region in the plane.

Corollary 44. For any nonrepeating , is generated (as a left or right -module) by a single element of degree , pictured below

Proof. A simple forest with double dots and at most one boundary edge is no more than a boundary dot with double dots. Thus, any morphism reduces to a boundary dot for each color, accompanied by double dots.

To show Lemma 1, we need to only investigate for increasing , since we have already shown that the values of are determined by their values for increasing . This space will be an -bimodule where the left and right action are the same (since the lefthand and righthand regions are the same in any picture with no bottom boundary), so we view it as an -module, and we have just shown that it is cyclic. Let be the ideal which is the kernel of the map sending ; we call it the TL ideal of . Proving the Lemma 1 is to find and show that the graded dimension of is .

Remark 45. Since the space is a free -module, all polynomials in must have arisen from reducing to some morphism which contained the relation (3.5) to a “nice form," that is, plus double dots. In other words, letting be the morphism pictured below, we want to plug into a bigger graph, reduce it to a nice form, and see what we get.

Remember that is actually just a 6-valent vertex with two dots attached (one red and one blue). This bigger graph, into which is plugged, will actually be a graph on the punctured plane or punctured disk with specified boundary conditions on both the outer and inner boundaries. The difficult graphical proofs of this paper just consist in analyzing such graphs. This is done by splitting the punctured plane into simply connected regions and using the above proposition.

3.5. Generators of the TL Ideal

The sequence is assumed to be nonrepeating.

Proposition 46. The TL ideal of contains over all .
The TL ideal of contains , where if , otherwise.

We will prove that these actually generate the ideal in Proposition 58 but postpone the proof as it is long and unenlightening.

Proof. Adding 4 dots to , or 6 dots to a 6-valent vertex, we get

This is . Even though we are not allowing 6-valent vertices in our diagrams, we will sometimes express as to avoid having to consider sums of graphs (it’s easier for me to draw!).

To obtain the other , note the following equalities under the action of on :

From this it follows by explicit calculation that

Now, when we surround a polynomial with a -colored circle and use (3.11), we are left with a -colored double dot times , so we get .

Combining this with the calculations we just made, we see that a circle around will yield up to sign, and so forth. We now have numerous ways to express : for any , take with 4 dots to get , and then surround it with concentric circles whose colors, from inside to out, are and then

Clearly the colors of the increasing sequence and those of the decreasing sequence are distant, so a sequence like is also okay, or any permutation which preserves the order of the increasing and the decreasing sequence individually.

For very similar reasons, is in the TL ideal of . Adding two or three dots to , we get several more equations.

Again, for a variety of these pictures, we use shorthand like

These give you in the case where at least one of . Again by , putting a polynomial in the eye of a -colored needle will yield next to a -colored boundary dot.

This gives us several ways to draw .

If neither nor are in , then and is pictured as above, but with additional boundary dots being put below to account for . Since these extra dots are generally irrelevant, we often do not bother to draw them.

If and , we have two ways of drawing . One can take , connect one input to the outer boundary, add dots, and surround it with circles colored

Alternatively, take some , add dots to , and surround it with circles forming an increasing sequence and a decreasing sequence , except that the final -colored circle is a needle

The case of and is obvious.

If both , then we have several choices again. If , then we must use

but in general, we may either repeat with a -needle instead of a -circle

or repeat with a -needle instead of a -circle

In any case, it is clear that the polynomials above are in the TL ideal, and the claim is proven.

Let us quickly consider the redundancy in this generating set of the ideal. When let and .

Corollary 47. Suppose that is nonempty, and fix an index . Then, is generated by for , . None of these generators are redundant.
None of the generators of are redundant.

Proof. We leave the checks of irredundancy to the reader, but a proof will also arise as a byproduct in the next section (see Remark 51).
Suppose that but . If , then so that is redundant. If , then . A similar statement holds for . In the same vein, if but , then for a given , only one of or is needed, and if , then any two of the three pairwise relations will imply the third.

3.6. Graded Dimensions

In this section, fix a nonrepeating sequence . We assume in this section that the generators of are precisely the polynomials described in Proposition 46.

Notation 48. An element of can be written as a polynomial in , so let be a general monomial. Choose any , possibly empty. Given a monomial , let be the subset containing and all indices such that . For a fixed subset , let be the subset of all monomials with . This inherently depends on the choice of .

Under the map , the image of will be graphs where the colors appearing are precisely . Every color in appears as a boundary dot, and every corresponds to a double dot of that color. The case only occurs when and .

To find a basis for , we will use the Bergman Diamond Lemma [35] for commutative rings.

Definition 49. Let be a free commutative polynomial ring, where monomials are given a partial order with the DCC, compatible with multiplication in that . Let be an ideal generated by relations of the form , where is a monomial and is a linear combination of monomials which are each less than in the partial order. A reduction is an application of a relation to replace with , but not the other way around (a reduction always lowers the partial order on each term in a polynomial). One says a polynomial reduces to if can be obtained from by a series of reductions applied to monomials in . A monomial is called irreducible if it does not have as a factor for any relation . An inclusion ambiguity is a monomial , where for some , and for some . An overlap ambiguity is a monomial , where for some and for some . Each ambiguity has two natural reductions, and one says the ambiguity is resolvable if the two reductions are then jointly reducible to the same element.

Lemma 50 (Bergman Diamond Lemma for Commutative Rings, [35]). With these definitions in place, if every inclusion and overlap ambiguity is resolvable, then the images of the irreducible monomials form a basis for .

This process may become more transparent from the example below; in addition, Bergman’s paper has a number of nice examples for the trickier, noncommutative version. We treat two separate cases, when and when .

Claim 2. Let . We place the lexicographic order on monomials in , so that . The relation for will be rewritten as , which replaces with a sum of monomials all lower in the order. For each , the irreducible monomials in are precisely , where is the minimal index in and (note: the exponent of is ). When , is irreducible. Irreducibles form a basis for .

Proof. A monomial is irreducible if never appears as a factor for any . Because of this, the classification of irreducible monomials in each is obvious. There are no inclusion ambiguities between relations, since they are all homogeneous and degree 3. There are two kinds of overlap ambiguities, both labelled by a choice of .
For the first ambiguity, one can reduce by either reducing or . Applying the former reduction, which has a term given by that can be further reduced, yielding . Applying the latter reduction, , which has a term given by that can be further reduced, yielding . Since these agree, the ambiguity is resolvable.
For the second ambiguity, one can reduce by either reducing or . A very similar calculation shows that this ambiguity is resolvable as well. Therefore, the Bergman diamond lemma implies that irreducibles form a basis for the quotient.

Remark 51. This also proves that none of the is redundant. Removing from the ideal, we may apply the same Bergman diamond lemma argument to say that irreducibles form a basis for the quotient. However, with no , the monomial is irreducible, and the quotient is larger than before. A similar statement can be made about the below.

When , the graded rank of the irreducibles in is . When is empty, the only element of is . So the graded rank of is , but since every may either appear or not appear, independently of every other. Hence . Putting it all together, the graded rank is . Hence we have proven the following claim.

Claim 3. The graded dimension of is exactly .

Claim 4. Let , and fix . We choose a different order on indices, where , and then place the lexicographic order on monomials. The relation for will be rewritten in decreasing order format as either for , or for , where the sum is over between and . Then, the irreducible monomials in are precisely for . Irreducibles form a basis for .

Proof. An irreducible polynomial will be a polynomial which does not have as a factor, for , and does not have as a factor for . The classification of irreducibles in is now obvious. There are no ambiguities whatsoever, so we are done by the Bergman diamond lemma.

The graded rank of irreducibles in is , for the length of (remember that ). Thus, the graded rank of is , and the graded rank of is . Hence, one considers the following.

Claim 5. The graded dimension of is exactly .

This is clearly sufficient to prove Lemma 1, modulo Proposition 58.

3.7. Weyl Lines and Disoriented Tubes

We now give two alternate interpretations of the TL ideals . We continue to assume that is nonrepeating and generates .

Definition 52. Let be the reflection representation of , such that is the coordinate ring of . Note that the linear equations which cut out reflection-fixed hyperplanes are precisely for . A Weyl line is a line in through the origin which is defined by the intersection of reflection-fixed hyperplanes; it is given by a choice of transversely intersecting reflection-fixed hyperplanes. Given a nonrepeating sequence , one says that a Weyl line is transverse to if it is transverse to (i.e., not contained in) the hyperplanes for each .

Proposition 53. The TL ideal of is the ideal associated with the union of all Weyl lines transverse to (with its reduced scheme structure).

Example 54. Let . One can check that cuts out 7 lines in , namely, (1) , (2) , (3) , (4) , (5) , (6) , (7) , These 7 lines are precisely the 7 lines cut out by the intersection of pairs of reflection-fixed hyperplanes. There are 6 reflection-fixed hyperplanes, given by equations , , , , , and , or alternatively, by for . Intersecting pairs of hyperplanes will give a line, and occasionally this line is forced to lie in a third hyperplane, as in the list above. One can check that this list covers all pairs of hyperplanes which give distinct lines as their intersection.

Proof. This is not difficult to show, but since we have not seen it elsewhere, we provide a complete proof. First, we show by induction on that the ideal cuts out the Weyl lines with the reduced scheme structure. The case is trivial (and is also obvious).
For any , consider the hyperplane as an -dimensional space , with an action of . Giving a Coxeter structure with simple reflections for (note that in the quotient), it is quite easy to see that is the reflection representation of . Moreover, the Weyl hyperplanes are cut out by (where so it may be left out of the sum) for , and the equivalent polynomials also have the same formulae, and are indexed by . Therefore, for , the images of are just , and the same for and . Moreover, if either or equals , then on , and is redundant on , being equal to some . By induction, cut out the Weyl lines with the reduced scheme structure on , and therefore the vanishing set of agrees with the Weyl lines on .
If all , then it is easy to see that the cut out a single line with the reduced scheme structure, namely, . This is a Weyl line, the intersection of all . We wish to show that this is the only Weyl line transverse to all . We can show this by induction as well (again, the base case is easy). Suppose we are given transverse hyperplanes . If any two involve the index , that is, and , then we may replace the pair with and since they have the same intersection (and is not already in the set, or the intersection would not be transverse). So, we may assume that at most one of the chosen hyperplanes involves the index , but then we have transverse hyperplanes which only involve indices , which must then be mutually transverse to . Letting be the hyperplane viewed as a reflection representation as above, we have transverse hyperplanes which cut out a Weyl line transverse to for all . By induction, that Weyl line is (which holds true modulo ), but repeating the same argument for the index instead, we leave out the th term and get modulo . Together, all these equalities imply that everywhere.
One might be worried, because of the restrictions used in the induction step, that does not give the reduced structure on the Weyl lines at the origin. However, is a homogeneous ideal which cuts out a reduced 0-dimensional subscheme of , so that its vanishing on is the cone of a reduced scheme, and hence is reduced. This concludes the proof that cuts out the Weyl lines with the reduced scheme structure.
For , and the vanishing of is contained in that of . Choose . If then is equal to , where depending on whether , but either way we get that . Then for , and so forth. Therefore only intersects the vanishing of at the origin (as sets). It is clear that, on the open set where for all , the polynomials and have the same vanishing (as schemes), since they differ by a unit. The same cone argument shows that gives the reduced structure at the origin.

Remark 55. In particular, is contained in every ideal, and the category is manifestly -linear.

Remark 56. Let be the union of all Weyl lines in . The previous results should lead one to guess that the Temperley-Lieb algebra should be connected to the geometry of the action on via , in much the same way that the Hecke algebra is connected to the reflection representation via (see [11]). However, at the moment, we have no way to formulate the category in terms of coherent sheaves on (i.e., -bimodules) or the derived category thereof. Describing using sheaves on seems like an interesting question.
As an example of the difficulties, let be the bimodule , where the tensor is over ; this should be the equivalent of the Soergel bimodule . Then, there is a degree-1 map sending to (the boundary dot on the top), but there is no degree-1 map (the boundary dot on the bottom); such a map should send to . There is only a degree-3 map, sending to (the boundary dot with a double dot). A similar problem occurs again: the trivalent vertex seems to be defined only in one direction.

Now, we describe briefly the topological intuition associated with the category , and another way to view . These remarks will not be used in the remainder of the paper, nor will we give a proof. The reader should be acquainted with the section on -foams in Vaz’s paper [16].

Remark 57. Let be the functor from to the category of disoriented cobordisms , as defined in Vaz’s paper. If is the double dot colored , then one can easily see that sends to a tube connecting the th sheet to the th sheet, with a disorientation on it. If the double dot appears in a larger morphism , such that in , the th sheet and the th sheet are already connected by a saddle or tube, then adding another tube between them does nothing more than adding a disoriented handle to the existing surface. Note that the map previously defined will connect the th sheet to the th sheet for any .
Suppose that the th, th, and th sheets are all connected in a cobordism. Then adds a handle on the left side of the th sheet, adds a handle on the right side, and these two disoriented surfaces are equal up to a minus sign in . This fact is essentially the statement that:

In other words, the algebra maps to , sending to the disoriented tube between the th and th sheet. The ideal is clearly in the kernel of this action when applied to the cobordism . In fact, it is precisely the kernel, using the argument of Proposition in [36]: for any distinct monomials in a basis for , their image in will have independent evaluations with respect to some closure of the cobordism. We do not do the calculation here. The usual arguments involving adjoint pairings imply that the faithfulness of the functor can be checked on . Therefore, the functor is faithful.

3.8. Proof of Generation

Proposition 58. The TL ideal is generated by over all .
The TL ideal is generated by all , where if , otherwise.

We wish to determine the ideal generated by inside , for nonrepeating . As discussed in Remark 45 (where is defined), our goal is to take any graph on the punctured plane, with as its outer boundary and as its inner boundary, plug into the puncture, and reduce it to something in the ideal generated by the pictures of Section 3.5.

Our coloring conventions for this chapter will be that blue always represents the index , red represents , and other colors tend to be arbitrary (often, the number of other colors appearing is also arbitrary). However, it will often happen that colors will appear in increasing or decreasing sequences, and these will be annotated as such. Note that blue or red may appear in the outer boundary as well, but at most once each.

Let us study , and not bother to plug in . The only properties of which we need are the following: This follows from , or just from isotopy. The same holds with colors being switched. This is because the diagram reduces to a -colored needle, with inside, but is fixed by , so it slides out of the needle, and the empty needle is equal to 0. A similar equality holds with colors switched. This follows from the above and the dot forcing rules.

The final property we use is that any graph only using colors or can slide freely across the puncture.

Note however that, say, an arbitrary edge cannot automatically slide across the puncture, because a edge might be in the way, and this could be in turn obstructed by a edge, which cannot slide across the blue at all.

The one-color reduction results apply to any simply connected planar region, so we may assume (without even using the relation ) that in a simply connected region of our choice, the -graph for each is a simple forest with double dots. Any connected component of an -graph that does not encircle the puncture will be contained in a simply connected region, and hence can be simplified; this will be the crux of the proof. The proof is simple, but has many cases.

Remark 59. We will still need to use relation as we simplify graphs.

We will treat cases based on the “connectivity” of that is, how many of the blue and red boundary lines in the inner and outer boundary are connected with each other. We will rarely perform an operation which makes the graph more connected. At each stage, we will reduce the graph to something known to be in the ideal or break edges to decrease the connectivity. We call an edge coming from the puncture an interior line and one coming from the outer boundary an exterior line.

Note also that any double dots that we can move to the exterior of the diagram become irrelevant, since the picture with those double dots is in the ideal generated by the picture without double dots. Also, any exterior boundary dots are irrelevant, since they are merely part of the map and do not interfere with the rest of the diagram at all.

Step 1. Suppose that the two interior red lines are in the same component of . Then, there is some innermost red path from one to the other, such that the interior of this path (the region towards the puncture) is simply connected. Applying reductions, we may assume that the -graph in this region consists of a blue boundary dot with double dots, and the -graph and -graph each consists only of double dots. We may assume all double dots occur right next to one of the red lines coming from the puncture. The current picture is exactly like that in , except that there may be double dots inside and other colors may be present (also, there could be more red spokes emanating from the red arc, but these can be ignored or eliminated using and tree reduction). However, the double dots may be forced out of the red enclosure at the cost of potentially breaking the red edge, and breaking it will cause the two red interior lines to be no longer in the same connected component. If there are no double dots, then all the remaining colors (which are or ) may be slid across the red line and out of the picture. Hence, we are left with the exact picture of , which is zero.
Thus, we may assume that the two red lines coming from the puncture are not in the same component. The same holds for the blue lines.

Step 2. Suppose that the component of one of the interior blue lines wraps the puncture, creating an internal region (which contains the puncture). Again, reducing in that internal region, the other interior blue line cannot connect to the boundary so it must reduce to a boundary dot (with double dots), the reds may not connect to each other so each reduces to a boundary dot, and as before we are left in the picture of except possibly with double dots and other colors. If there are no double dots, all other colors may be slid out, and the picture is zero by . Again, we can put the double dots near the exterior, and forcing them out will break the blue arc. It is still possible that some other cycle still allows that component to wrap the puncture; however, this process needs to only be iterated a finite number of times, and finitely many arcs broken, until that component no longer wraps the puncture.
So, we may assume that the component of any interior line, red or blue, does not wrap the puncture. That component is contained in a simply connected region, so it reduces to a simple tree. Hence, we may assume that the components of interior lines either end immediately in boundary dots, or connect directly to an external line of the same color (at most one as such exists of each color).

Step 3. Suppose that there is a blue edge connecting an internal line directly to an external one. Consider the region :

Then is simply connected. Other colors in may leave to cross through the blue line; however, the colors may not. Therefore, reducing within , we may end the internal blue line in a boundary dot and eliminate all other instances of the color blue (since they become irrelevant double dots on the exterior), reduce red to a simple forest where the two interior lines are not connected (again, ignoring irrelevant double dots), and reduce to either the empty diagram or an external boundary dot (depending on whether ). Once this has been accomplished, the absence of the color implies that we may slide freely across the puncture! The color can be dealt with in the entire disk, which is simply connected, so it reduces to the empty diagram or an external boundary dot (depending on whether ), with extraneous double dots. Then we may deal with color , and so forth.

Thus, the existence of the blue edge implies that all colors can be ignored: they appear in irrelevant double dots, in irrelevant boundary dots, or not at all. Similarly, the existence of a red edge allows us to ignore all colors .

Step 4. Let us consider only the components of graphs which do not meet the internal boundary.

Lemma 60. Consider a component of a graph on a punctured disk which does not meet the internal boundary and which meets the external boundary at most once. Then it can be reduced to one of the following, with double dots on the exterior: the empty graph, a boundary dot, a circle around the puncture, a needle coming from the external boundary, with its eye around the puncture.

Proof. Suppose that the component splits the punctured plane into regions. If the component is contained in a simply connected part of the punctured plane, we are done. This is always true for . So we may suppose that and, we have two distinguished regions: the external region, and the region containing the puncture. Any other region is one of two kinds, as illustrated in the following equality (due to :

On the right side we have a region which is contained in a simply connected part and thus can be eliminated by reduction (see Proposition 42). On the left side the region is not contained in a simply connected part nor does it contain the puncture. However, any such region can be altered, using as in the heuristic example above, into a cycle of the first kind. Therefore, we may assume there are exactly 2 regions.

In the event that there are two regions, we have a cycle which surrounds around the puncture and may have numerous branches into both regions, internal and external. However, each branch must be a tree lest another region be created. These trees reduce in the usual fashion, and therefore the internal branches disappear, and the external branches either disappear or connect directly to the single exterior boundary. Thus, we have either a needle or a circle. Double dots, as usual, can be forced out of the way possibly at the cost of breaking the cycle, and reducing to the case .

Let us now examine the remaining cases. We will ignore all parts of a graph which are double dots on the exterior, or are external boundary dots.

Case. Both a blue edge and a red edge connect an internal line to an external line. Then, as in Step 3, all other colors can be ignored, and the entire graph is

This, as explained in Section 3.5, is .

Case 2. A blue edge connects an internal line to an external line, and both red internal lines end in boundary dots. As discussed in Step 3, we may ignore all colors , and both colors and do not appear in a relevant fashion outside of what is already described. We may ignore the presence of any double dots. However, there may be numerous circles and needles colored which surround the puncture and cross through the blue line, in an arbitrary order.

Claim 6. The sequence of circles and needles can be assumed to form an increasing sequence of colors, from until the final color, and assuming that only the final color may be a needle.

Proof. If the innermost circle/needle is not colored , then it may slide through the puncture, and will evaluate to zero by . So, suppose the innermost is . If it is a needle, not a circle, then there can be no more -colored circles, and no -colored circles. Color can be pulled through the middle so resolved on the entire disk and hence can be ignored, and so too with and higher. This is the “needle" analogy to the conclusion of Step 3: the existence of an -colored needle around the puncture and the lack of or on the interior of the needle will allow us to ignore all colors .

So, suppose it is a -colored circle. If the next circle/needle is colored then it slides through the circle and the puncture and evaluates to zero. If the next circle/needle is also colored , then we may use the following calculation to ignore it. The calculation begins by using

Thus, we may assume that the next circle/needle is colored . Again, if it is a needle, then we can ignore all other colors, and our picture is complete.

Similarly, the next circle/needle can not be colored lest it slide through, and it can not be colored lest we use . If it is colored , then we may use the following calculation to ignore it. The calculation begins by using (3.5), and assumes green and purple are adjacent

Thus, we can assume the next circle/needle is colored . If it is a needle, then all colors and higher can be ignored. Additional circles of color could run through the needle, but these could be slid inwards and reduced as before. So, if it is a needle, our picture is complete.

Finally, the next circle/needle can not be colored lest it slide, lest we use , lest we use , or lest we slide it inside and reduce it as above. Hence it is colored , and if it is a needle, we are done. This argument can now be repeated ad infinitum.

Thus, our final picture yields as in or .

Note that the case of a red edge works the same way, with a decreasing sequence instead of an increasing sequence.

Case 3. All the internal lines end in boundary dots. We may assume that the remainder of the graph consists of circles/needles around this diagram, but we have no restrictions at the moment on which colors may appear.

Claim 7. We may assume that the colors in circles/needles form an increasing sequence from up, and a decreasing sequence from down (these sequences do not interact, so without loss of generality. we may assume the increasing sequence comes first, then the decreasing one). Only the highest and lowest color may be a needle.

Proof. The method of proof will be the same as the arguments of the previous case.

Consider the innermost circle/needle. If it is colored or , then we may use to reduce the situation to a previous case. If it is colored or then it slides through the puncture. So we may assume it is or . If it is a -colored (resp., -colored) needle, then the usual arguments imply that all colors (resp., ) can be ignored. This same argument with needles will always work, so we will not discuss the circle/needle question again, and speak as though everything is a circle.

Assume that the first colors appearing are an increasing sequence from to and then a decreasing sequence from to . Note that either sequence may be empty. If the next color appearing is then it slides through the whole diagram and the puncture, and evaluates to zero. If the decreasing sequence is nonempty and the next color is then we use ; if it is and then we slide it as far in as it will go and use . If the decreasing sequence is nonempty and the next color is then one can push it almost to the center, and use the following variant of :

In this picture, green is and is the only thing in the way of the blue circle. The first equality uses (3.5), and the second equality uses , and eliminates the terms which vanish due to .

Continuously, if the decreasing sequence is empty and the next color is , then we may use as above. Any colors which are do not depend on the increasing sequence, and instead use the exact analogs for the increasing sequence.

Hence, in any case in which the next color appearing is not , or , or the beginning of a new increasing/decreasing series, we may simplify the diagram to ignore the new circle. Induction will now finish the proof.

Therefore, the resulting diagram is equal to , matching up either with or .

Since every possible graph can be reduced to a form which is demonstrably in the ideal generated by , we have proven that these elements do in fact generate the TL ideal .

4. Irreducible Representations

In this section, we may vary the number of strands appearing in the Temperley-Lieb algebra. When appears, it designates the Temperley-Lieb algebra on strands, but designates the algebra on strands.

4.1. Cell Modules

The Temperley-Lieb algebra has the structure of a cellular algebra, a concept first defined by Graham and Lehrer [37]. One feature of cellular algebras is that they are equipped with certain modules known as cell modules. Cell modules provide a complete set of nonisomorphic irreducible modules in many cases (such as in type ). Cell modules come equipped with a basis and a bilinear form, making them obvious candidates for categorification. We will not go into detail on cellular algebras here, or even use their general properties; instead we will describe the cell modules explicitly and pictorially for the case of , where things are unusually simple. Nothing in this section or the next is particularly original, and we state some standard results without proof.

Notation 61. Consider a crossingless matching in the planar strip between points on the bottom boundary and points on the top. We call this briefly an diagram. In the terminology of [30], there are two kinds of arcs in a diagram: horizontal arcs which connect two points on the top (let us call it a top arc), or two points on the bottom (bottom arc) and vertical arcs which connect a point on the top to one on the bottom. Elsewhere in the literature, vertical arcs are called through-strands. An diagram with exactly through-strands (and therefore no top arcs) has an isotopy representative with only “caps" (local maxima) and no “cups" (local minima) so it is called an cap diagram. A diagram with through-strands is called a cup diagram.

The set of all diagrams can be partitioned by the number of through-strands. Any diagram with through-strands can be expressed as the concatenation of an cap diagram with a cup diagram in a unique way. For an illustration of this concept, see Figure 5.

In an diagram the number of top arcs equals the number of bottom arcs, and if is the number of through-strands, then . We will typically use and to represent the number of through-strands and top arcs in an diagram henceforth.

Notation 62. Let be the set of all diagrams. Let be the endomorphism of sending each diagram to its vertical flip. We will write the operation on diagrams of reduced vertical concatenation by : places above and remove any circles. Let be the set of crossingless matchings with exactly through-strands. Let be the set of all cap diagrams, so that is the set of all cup diagrams.

Definition 63. Let be the free module spanned by , the cap diagrams. We place a right -module structure on by concatenation, where circles become factors of as usual, and any resulting diagram with fewer than through-strands is sent to . This is the cell module for cell , and it is irreducible.

Example 64. The only diagram in corresponds to the identity map in . The cell module has rank 1 over , and its generator is killed by all . We will take this as the definition of the sign representation of .

Example 65. The next cell module has rank over , having generators , (see Figure 6), such that

Given a cap diagram and a cup diagram , there are two things we can do: take the composition to obtain an element called of or take the composition to get an element of (there may be additional circles created, and the final diagram may have fewer than through-strands). Both compositions have the same closure on the punctured plane. Note that . The seemingly extraneous use of the notation is standard for cellular algebras.

Proposition 66. There is, up to rescaling, a unique pairing for which is self-adjoint, that is, . Given cup diagrams and in , one evaluates by considering the closure of , or equivalently the closure of . If the diagram has nesting number , one returns a scalar times raised to the number of circles; if it has nesting number , one returns zero. This is precisely the evaluation for some well-defined trace on supported on nesting number (which are unique up to rescaling).

4.2. Some Induced Sign Representations

Cell modules are naturally subquotients of the cellular algebra itself, viewed as a free module (see [37]). For our purposes, we will describe the cell modules as subquotients of in a different way, which will be more convenient to diagrammatically categorify. Taking the inclusion for some sub-Dynkin diagram , we can induce the sign representation of up to . This is the quotient of by the right ideal generated by . In a future paper, we will describe, for both the Hecke and Temperley-Lieb algebras, a diagrammatic way to categorify the induction of both the “sign" and “trivial" representations of sub-Dynkin diagrams, but for this paper we restrict to a specific case. For the sub-Dynkin diagram which contains every index except , let be the corresponding ideal (generated by for ), and consider the induced sign representation . Let and let . It turns out that we can embed inside , as shown explicitly below, and we will categorify both modules accordingly. For this reason, we use to denote . Note that every possible can be achieved as some with the exception of .

For the rest of this section, fix an index . We define a module over abstractly, and then prove that this module is isomorphic to .

Definition 4.5. For (and letting as always), let be the following cup diagram with top arcs, where the innermost top arc always connects to :

Let consist of all matchings of the form for . Let be the disjoint union of all for , and let be the free -module with basis . There is a distinguished element of this basis, the unique member of . Let act on on the right by viewing elements of as though they were in , using the standard multiplication rules, and then killing any terms whose diagrams are not in .

The elements of exhaust those elements of where the only simple top arcs (those connecting to for some ) connect to . Any crossingless matching with a simple top arc connecting to has an expression in as a monomial which begins with . The converse is also true. Thus, are the elements of for which every expression of the matching begins with . This motivates the definition.

While something does need to be checked to ensure that this defines a module action, it is entirely straightforward. In the Temperley-Lieb algebra, things are generally easy to prove because products of monomials always reduce to another monomial (with a scalar), not a linear combination of multiple monomials. Therefore, checking the associativity condition for being a module, say, involves showing that both sides of an equation are the same diagram in , or that both sides are 0. This module is cyclic, generated by , and is clearly in the annihilator of , so that surjects onto . One could prove the following by bounding dimensions.

Claim 8. The modules and are isomorphic.

There is a (cellular) filtration on , given by the span of , diagrams with at most through-strands (call it ). Clearly, each subquotient in this filtration has a basis given by , or in other words by the elements for . It is an easy exercise that this subquotient is isomorphic to the cell module , under the map sending to . There is one subquotient for each .

Claim 9. The module is a submodule of .

Proof. Letting and , the final term in the filtration is precisely .

Having explicitly defined the embedding , we pause to investigate adjoint pairings on .

Proposition 68. Consider the module of semi-linear pairings on where for all . Consider the functionals on this space, which send a pairing to for various , . Then these linear functionals are independent and yield an isomorphism between the space of pairings and a free module of rank .

Note that, using adjunction, one can check that .

Proof. Given diagrams , the self-adjointness of implies that the value of is an invariant of the diagram . In particular, , where refers to the image of this diagram in the quotient . Therefore, if either or is not in then the value of is zero. However, where this set runs over all with . Thus the value of the pairing on all elements is clearly determined by the values of for all such .
Consider the following map : fix , and for basis elements send to if , and send to zero otherwise. Clearly this is a well-defined semi-linear map (being defined on a -basis) and is self-adjoint. Thus we have enough pairings to prove independence.

Remark 69. Once again, all pairings are defined topologically. The closure of has nesting number exactly , which distinguishes the traces.

4.3. Categorifying Cell Modules

Categorifying the sign representation is easy. If we take the quotient of by all nonempty diagrams, we get a category where the only nonzero morphism space is the one-dimensional space . This clearly categorifies , and we will say no more.

Consider the quotient of the category by all diagrams where any color not equal to appears on the left. Call this quotient . As usual, we let be its additive grading closure, and let be its graded Karoubi envelope. We will show that , so that we really may think of entirely diagrammatically without worrying about idempotents. We claim that categorifies . Not only this, but the action of on by placing diagrams on the right will categorify the action of on .

Any monomial which goes to zero in is equal to a (scalar multiple of a) monomial where some index appears on the left. Therefore, the corresponding object will be isomorphic to , whose identity morphism is sent to zero in since it has a -colored line on the left. There is an obvious map from to the Grothendieck group of , and the action of , descended to the Grothendieck group, will commute with the action of on .

Therefore, spaces in will induce a semi-linear pairing on , which satisfies the property because is self-adjoint. As before, once we determine which pairing this is, our proof will be almost complete.

Lemma 70. The pairing induced by will satisfy , where .

Remark 71. Taking a diagram and closing it off on the punctured plane, if is the number of circles and is the nesting number, then the pairing comes from the trace on which sends this configuration to .
For a closure of an arbitrary diagram, is possible. However, for any diagram in (with extra circles thrown in), we have , since removing the circles yields precisely for some . This guarantees that evaluating the formula on an element of yields a power series with nonnegative coefficients.

The proof of the lemma may be found shortly below. Temporarily assuming the lemma, the remainder of our results are easy.

Theorem 72. is idempotent closed and Krull-Schmidt, so that . Its Grothendieck group is isomorphic to .

Proof. It is enough to check that for any corresponding to matchings in , is concentrated in nonnegative degrees with a 1-dimensional degree 0 part, and is concentrated in strictly positive degrees (see the proof of Proposition 37). This is a calculation using the semi-linear pairing.
Letting be the number of circles in a configuration on the punctured disk, and letting be the nesting number, then the evaluation will be in strictly positive degrees if , and will be in nonnegative degrees with a 1-dimensional degree 0 part if exactly, but this was precisely the calculation in the proof of Lemma 38: for arbitrary crossingless matchings and , the closure of has fewer than circles if , and exactly if they are equal.

Corollary 73. Let be the full subcategory of with objects consisting of (sums and grading shifts of) such that is an element of . This has an action of on the right. On the Grothendieck group, this setup categorifies the cell module .

Proof. That this subcategory is closed under the action of is obvious, as is the existence of a map from to the Grothendieck group. We already know the induced pairing, because the subcategory is full. Therefore, the same arguments imply that the Grothendieck group behaves as planned.

Proof of Lemma 70 To calculate the pairing, we may calculate for the following choices of : , , , , , and so forth. These are pictured below.

These sequences are split into subsequences we call “tiers", where the th sequence adds the th tier. The following property of these sequences is easily verified: each sequence is in , and remains in if one removes any subset of the final tier, but ceases to be in if one removes a single element from any other tier instead.

Fix nonempty in this sequence, and let be the subsequence with the final tier removed. It is a quick exercise to show that the lemma is equivalent to , where is the number of elements in the final tier.

Now consider an element of the endomorphism ring. Using previous results, we may assume it is a simple forest, with all double dots on the far left. Any double dot colored will be sent to zero, so we have only an action of the ring on the left. This accounts for the appearing in all the formulae.

Suppose there is a boundary dot in the morphism on any line not in the final tier. Then the morphism factors through the sequence where is with that index removed. As discussed above, is not in and therefore is isomorphic to the zero object. See the first picture below for an intuitive reason why such a morphism vanishes. Hence, the only boundary dots which can appear occur on the final tier. It is easy to check that the existence of a trivalent vertex joining three boundary lines will force the existence of a dot not on the final tier; see the second picture below. Both pictures are for the sequence , and blue will represent in all pictures in this section.

Therefore a nonzero endomorphism must be accompanied on the right by either identity maps or broken lines (pairs of boundary dots), because all other simple forests yield a zero map. Identity maps have degree zero, while broken lines have degree 2. If these pictures form a basis (along with the action of blue double dots on the left), then the graded dimension will be exactly as desired. An example with , two broken lines, and one unbroken line is shown below.

This spanning set is linearly independent in over , so any further dependencies must come from having a nonblue color on the left. Consider an arbitrary endomorphism, and reduce it using the relations to a simple forest with all double dots on the left. The actual double dots appearing are ambiguous, since there are polynomial relations in , but it is easy (knowing the generators of the TL ideal) to note that these relations are trivial modulo nonblue colors. Hence the spanning set will be linearly independent if any diagram in which started with a nonblue color on the left will still have a nonblue color (perhaps in a double-dot) on the left after reducing to a simple forest. This will be the case for any diagram with a boundary dot on .

Let red indicate any other index, and suppose that red appears on the far left. Regardless of what index red is, unless there is a dot on , the identity lines of block this leftmost red component from reaching any red on the boundary of the graph. Take a neighborhood of a red line segment which includes no other colors and goes to . Excising this neighborhood, we get a simply connected region where the only relevant red boundary lines are the two which connect to the ends of the segment. Red then will reduce to a simple forest with double dots on the left, which in this case yields either a red double dot or a red circle (potentially with more double dots).

However, no colors adjacent to red can interfere on the interior of a red circle, so the circle evaluates to zero. Therefore, the diagram evaluates to zero or has at least one red double dot on the left. We may ignore the red double dot and reduce the remainder of the diagram, and so regardless of what else is done, the final result will have a red double dot on the left.

Acknowledgments

The author was supported by NSF Grants DMS-524460 and DMS-524124. The author would like to thank Mikhail Khovanov, Catharina Stroppel, and the referees for many thoughtful comments.