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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 978635, 58 pages
Diagrammatics for Soergel Categories
Department of Mathematics, Columbia University, New York, NY 10027, USA
Received 4 May 2010; Accepted 31 December 2010
Academic Editor: Alistair Savage
Copyright © 2010 Ben Elias and Mikhail Khovanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.
In this paper , Soergel gave a combinatorial description of a certain category of Harish-Chandra bimodules over a simple Lie algebra . This category was and continues to be of primary interest in the infinite-dimensional representation theory of simple Lie algebras. Soergel discovered a functor from this category to a full subcategory of bimodules over a certain ring , the objects of which are now commonly called Soergel bimodules. The category of Soergel bimodules is additive and monoidal, unlike the original category which is abelian, but it still has sufficient information to describe the original category. Soergel constructed an isomorphism between the Grothendieck ring of his category and the integral form of the Hecke algebra of the Weyl group of . Hence, Soergel's construction provides a categorification of the Hecke algebra.
Given a -dimensional -vector space and a generic , there are commuting actions of the quantum group and the Hecke algebra of the symmetric group on . These actions turn the quotient of the quantum group and of the Hecke algebra by the kernels of these action into a dual pair. A categorical realization of the triple was given by Grojnowski and Lusztig  via categories of perverse sheaves on products of flag and partial flag varieties, also see [3–6].
Many foundational ideas about categorification were put forward by Igor Frenkel in the early 90s (a small fraction of these ideas formed a part of the paper ). In particular, Frenkel conjectured  that quantum groups and not just their finite-dimensional quotients can be categorified. These conjectures remained open until recently, when categorifications of quantum and were discovered in [9, 10], with a related but different approach developed in [11, 12]. In the categorifications [9, 10] of quantum groups, 2 morphisms are given by linear combinations of planar diagrams, modulo local relations.
The parallel objective would be to categorify the Hecke algebra in the same spirit, using planar diagrams. Soergel had already provided a categorification, so it remains to ask whether his category can be rephrased diagrammatically. Diagrammatics should also provide a presentation of the category by generators and relations. A similar question was recently posed by Libedinsky , who essentially produced such a description for categorifications of Hecke algebras associated to “right-angled” Coxeter systems.
Here, we answer this question positively in the case of the Hecke algebra associated to the symmetric group. This is, of course, the Hecke algebra that appears in the Schur-Weyl duality for . For notational convenience, we use , not , as our parameter and define a diagrammatical version of the category of Soergel bimodules that categorifies the Hecke algebra of the symmetric group .
In some sense, diagrammatic categorifications are very “low-tech,” in that they can be described easily and do not rely on heavy machinery. While one can prove that Soergel bimodules categorify the Hecke algebra using only elaborate commutative algebra (see , although it is never stated explicitly), showing that indecomposable bimodules descend to the Kazhdan-Lusztig basis of the Hecke algebra utilizes Kazhdan-Lusztig theory [15, 16]. This, in turn, is related to fundamental developments in geometric representation theory like D-modules on flag manifolds [17, 18] and perverse sheaves . One hopes that a diagrammatic approach will help to visualize and work with these sophisticated constructions, in the same way that the categorifications of quantum groups [9, 10] have led to an improved understanding of perverse sheaves on quiver varieties (see ). One also hopes that this approach can shed light on categorifications of representations of the Hecke algebra coming from the context of category (see ).
We start with an intermediate category whose objects are finite sequences of numbers between 1 and . An object is represented graphically by marking points in the standard position (say, having coordinates ) on the -axis and assigning labels to marked points from left to right. Morphisms between and are given by linear combinations (with coefficients in a ground field ) of planar diagrams embedded in the strip . These diagrams are decorated planar graphs, where edges may extend to the boundary . Each edge carries a label between 1 and , so that the induced labellings of the lower and upper boundaries are and , respectively. In the interior of the strip, we allow (1)vertices of valence 1, (2)vertices of valence 3 with all 3 edges carrying the same label, (3)vertices of valence 4 seen as intersections of and -labelled lines with , (4)vertices of valence 6 with the edge labelling , , , , , , reading clockwise around the vertex, (5)boxes labelled by numbers between 1 and which float in the regions of the graph.
We impose a set of local relations on linear combinations of these diagrams, including invariance of diagrams under all isotopies. A subset of the relations says that is a Frobenius object in the category .
The space of morphisms in between and is naturally a graded vector space. Allowing grading shifts and direct sums of objects, then restricting to grading-preserving morphisms, and finally passing to the Karoubian closure of the category results in a graded -linear additive monoidal category . Our main result (Theorem 4.22 in Section 4) is an explicit equivalence between this category and the category of Soergel bimodules.
The category is monoidal, and can be viewed as a 2-category with a single object. It may be easier to tackle the diagrammatics after reading an introductory reference on diagrammatics for 2-categories. Such an introduction can be found in . This may make it easier to explore similarities with the categorifications of quantum groups in [9, 10], where regions of diagrams are labelled by integers in  and integral weights of in . Boxes floating in regions are superficially analogous to floating bubbles of [9, 10]. Unlike the diagrammatic categorifications in [9, 10], our lines do not carry dots and are not oriented.
There is another way to view our diagrammatics, which is not developed in this paper. Rouquier [22, 23] defined an action of the Coxeter braid group associated to on the category of complexes of Soergel bimodules up to homotopies, which is related to a braid group action using Harish-Chandra bimodules that had been known for some time. These complexes were later used in an alternative construction  of a triply graded link homology theory  categorifying the HOMFLY-PT polynomial [25–28]. In this approach, a product Soergel bimodule is depicted by a planar diagram given by concatening elementary planar diagrams lying in the -plane that consist of strands going up, with and -st strands merging and splitting, see [24, Figures 2 and 3]. Morphisms between product bimodules can be realized by linear combinations of foams—decorated two-dimensional CW-complexes embedded in with suitable boundary conditions. Foams have been implicit throughout papers on triply graded link homology (see  for instance, where various arrows between planar diagrams can be implemented by foams), and explicitly appear in the papers on their doubly-graded cousins, see [30, 31] and references therein.
Foams are 3-dimensional objects; they are two-dimensional CW-complexes embedded in that produce cobordisms along the -axis direction between planar objects corresponding to product Soergel bimodules. The planar diagrams of our paper are two-dimensional encodings of these foams, essentially projections of the foams onto the -plane along the -axis.
It was shown in  that the action of the braid group on the homotopy category of Soergel bimodules extends to a (projective) action of the category of braid cobordisms. Thus, the homotopy category of produces invariants of braid cobordisms, so that our planar diagrammatics carry information about four-dimensional objects. This informational density indicates the efficiency of such encodings.
Addendum 1. Since this paper first appeared, the diagrammatics developed here have led to several developments which we briefly mention here. In  it is shown that Rouquier's braid group action lifts functorially to the braid cobordism category. In [34, 35], a functor is given from the category to categories of foams used in link homology. Together, these papers show that the encodings mentioned above are more than simply heuristic. In , the Temperley-Lieb algebra is categorified as a quotient of . Additional statements relating this paper to either newer papers or to previous versions of this paper are found sparsely under a similar “Addendum” heading.
Henceforth, we will fix a positive integer . Indices , , and will range over if not otherwise specified. Finite ordered sequences of such indices (allowing repetition) will be denoted , as well as and . The length of the sequence will be denoted . For sequences of length where the single entry is , we use and interchangeably. Occasionally will also be used as an index, and whenever this occurs we make the tacit assumption that , so that all indices used remain between 1 and . The same goes for , , and the like. We denote the length 0 sequence by the empty set symbol .
We work over a field , usually assuming that Char , and sometimes specializing it to .
Given a noetherian ring , the category is the full subcategory of -bimodules consisting of objects which are finitely generated as left -modules. If is graded, the category is the analogous subcategory of graded -bimodules and grading-preserving homomorphisms.
2.1. Hecke Algebra
Let be a Coxeter system of a finite Weyl group , with length function , and let be the identity. The Hecke algebra is an algebra over (we follow Soergel's use  of the variable ; related variables are denoted in the literature by and ), which is free as a module with basis . Multiplication in this basis is given by when , and for . is the identity element in and will often be written as 1.
In the case we are interested in presently, , and consists of the transpositions for . The element will be denoted . The Hecke algebra has a presentation over , being generated by subject to the relations Clearly then, is also generated as an algebra by , , and the relations above transform into We often write the monomial as where . Notice that .
Let be the involution of determined by . It extends to an involution of given by In particular, .
Kazhdan and Lusztig  defined a pair of bases and for , which immediately proved to be of fundamental importance for representation theory and combinatorics. The two bases are related via a suitable involution of , and the elements of the second Kazhdan-Lusztig basis are determined by the two properties where has negative -degree strictly less than for and . There is no simple formula expressing in terms of , but observe that and . For a good introduction to the Kazhdan-Lusztig basis, see .
Let be the -linear map given by and if . Thus, simply picks up the coefficient of in . The easily checked property for any implies that , for all , so that is a trace map and turns into a symmetric Frobenius -algebra.
Denote by a -antilinear anti-involution defined uniquely by . The anti-involution and -antilinearity conditions say that and , for and .
Consider the pairing of -modules given by
It satisfies the following properties: (1)the pairing is semi-linear, that is, while , for , (2) is self-adjoint, that is, and , (3)if with then . Such a monomial is called an increasing monomial, and an increasing sequence. When , the sequence is empty and .
Remark 2.1. It is not difficult to observe that is the unique form satisfying these three properties. This is because the Hecke algebra has a spanning set over consisting of monomials , and every monomial may be reduced, by cycling the last to the beginning and by applying the Hecke algebra relations, to an increasing monomial. This is a simple combinatorial argument that we leave to the reader.
2.2. Soergel Bimodules
In , Soergel introduced a category of bimodules which categorified the Hecke algebra, and later generalized his construction to any Coxeter group . Within the category , for a certain graded -algebra ( an infinite field of characteristic ), he identified indecomposable modules for , such that the only indecomposable summands of tensor products of 's are for . Thus, the subcategory of generated additively by the has a tensor product, and its Grothendieck ring is isomorphic to , under the isomorphism sending to . Moreover, every shows up as a summand of some tensor product of various for . While the general may be difficult to describe, has an easy description.
Henceforth we specialize to the case where and . We also make one additional change from Soergel's conventions.
Remark 2.2. Soergel defines to be the coordinate ring of the -dimensional reflection representation of , while we find it easier to consider , the coordinate ring of the -dimensional standard representation . This is akin to treating instead of , and a similar convention is adopted in . The bimodules are defined in  to be the coordinate rings of unions of “twisted diagonals” in , and can be defined analogously for . Now and the entire story of translates to by base extension. Conversely, is a quotient of by the first elementary symmetric polynomial , which is a central element of our category, so that the entire story of translates easily to under the quotient. We will interest ourselves with the story below because the ring is slightly more intuitive, and mention briefly the changes required to deal with in Section 4.6. Since we only use and below, we will denote them as and instead to avoid an apostrophe catastrophe.
With these conventions, we now make the story explicit.
Notation 2.3. Let be the ring of polynomials in variables, with the natural action of . The ring is graded, with . If is the subgroup of generated by transpositions , then we denote the ring of invariants under as or . Thus are the invariants under the transposition .
Since is an -algebra, is a bifunctor sending two -bimodules to an -bimodule. Henceforth, with no subscript denotes tensoring over , while denotes tensoring over the subring . Most commonly we will just use for various indices .
Definition 2.4. The Soergel bimodule is , where denotes a grading shift.
Notation 2.5. We denote by the tensor product .
Note that and We reiterate this important point: a typical element of a tensor product of generators can be expressed (up to linear combination) by a choice of polynomials, one in each slot separated by the tensors. Multiplication by an element of in any particular slot is an -bimodule endomorphism.
For each there is a map of graded vector spaces , called the Demazure operator, sending . This map is -linear. Since is -invariant, it is not hard to see that is also -invariant. Since , this implies that is a free graded -module of rank two, with homogeneous generators 1 and . In other words, there is an isomorphism of graded -modules, sending , with inverse .
From the isomorphism of graded -modules just illustrated, one can deduce other isomorphisms. For instance, as graded left (or right) -modules. Repeating this, we see that is a free left -module of rank , and properly belongs in . Finally, we can deduce an isomorphism , which unlike the previous isomorphisms is actually an isomorphism of -bimodules.
Remark 2.6. Let us make this slightly more explicit. To give the isomorphism of left -modules , note that
Rewriting a term as a sum of terms like above will happen often, and we refer to it as forcing the polynomial across the tensor. If is -invariant then it may be slid across leaving nothing behind, while an arbitrary when forced leaves terms with either 1 or behind (alternatively, we may choose to leave 1 and behind, if it is more convenient). We consistently use the term “slide” instead of “force” when the polynomial is invariant so it can be moved across without any ado.
Inside , taking an element and forcing to the left (or right) is now an -bilinear operation, since multiplication on the left or right will only affect or , not . This gives the isomorphism of -bimodules, via with inverse . These maps are -bimodule morphisms, since the only polynomials which can slide from to in both the source and the target of the map are those polynomials in .
Remark 2.7. We also remark on spanning sets for as -bimodules. For instance, we've seen that has a spanning set . The bimodule for has a spanning set , since any polynomial in the middle can be forced to the left leaving at most behind (or , which we choose when ), and that can be slid to the right; thus generates it as a -bimodule. The bimodule has a spanning set . This is because all polynomials anywhere between the two tensors may be slid out, leaving somewhere in-between. As an exercise, the reader may generalize this argument to an arbitrary and find a spanning set as a -bimodule, consisting of terms, where is the number of pairs such that and for between and . Between such a pair, one either places a linear “unslideable” term like , or just 1. Note that is equal to minus the number of distinct indices in .
2.3. The Soergel Categorification
Several subcategories of and will play a role in what follows. Let be the full subcategory of whose objects consist of for all sequences of indices ; these are called Bott-Samelson bimodules. Since is a commutative ring, the Hom spaces in are in fact enriched in . Let be the subcategory of whose objects are finite direct sums of various graded shifts of objects in and the morphisms are all grading-preserving bimodule homomorphisms. Finally, let be the Karoubi envelope of , a category equivalent to the full subcategory of which contains all summands of objects of
In general, the Karoubi envelope is the category which formally includes all “summands,” where a summand of an object is identified by an idempotent corresponding to projection to that summand. Since is abelian, it is idempotent-closed, and the Karoubi envelope of can be realized as a subcategory in . We refer the reader to  for basic information about Karoubi envelopes.
This final category (the category of Soergel bimodules) is a -linear additive monoidal category with the Krull-Schmidt property. Soergel showed that, when is an infinite field of characteristic other than 2, the indecomposable bimodules in this category are enumerated by elements of the Weyl group and grading shifts (Theorem 6.16 in ). They are denoted by for and . An indecomposable is determined by the condition that it appears as a direct summand of , where and is a reduced presentation of , and does not appear as direct summand of any , for sequences of length less than .
The Hecke algebra is canonically isomorphic to , the Grothendieck group of . Multiplication by corresponds to the grading shift: . Multiplication in the Hecke algebra corresponds to the tensor product of bimodules: The isomorphism takes to and to .
Remark 2.8. Nothing prevents one from defining a category where the field is replaced with in the definitions of the previous section. Thus one could define the category for any ring. However, one does not have control over the size of the Grothendieck group in this instance. When defined over a field of characteristic , we may use Theorem 6.16 of  to classify indecomposables and get results about the Grothendieck group. When , Soergel has shown that the Kazhdan-Lusztig basis satisfies . This is unknown in general.
Relation (2.2) lifts to isomorphisms of graded bimodules in
It is important to note that these isomorphisms take place in , not , since the latter does not have grading shifts or direct sums of objects. However, the same information can be encapsulated in inclusion and projection morphisms of various degrees, which do live in . This will be explored in Section 4.5.
The first isomorphism has already been made explicit in Remark 2.6. We have chosen a specific isomorphism; other choices were possible. The second and third isomorphisms come from the following isomorphisms in :
These isomorphisms will be made explicit in Section 4.5.
The ring is canonically isomorphic to the Grothendieck group of the category of finitely generated graded free -modules. Under this isomorphism goes to 1 and to . In particular, given any graded free -module , its image in the Grothendieck group will be its graded rank, calculated by choosing a homogeneous -basis of and letting . The bar involution on lifts to the contravariant equivalence that takes to , the latter naturally viewed as an -module.
In the category , given any objects , the space is a graded free finitely generated left -module. By extension, the same is true of the module . Shifting the grading of will shift the grading of this Hom space in the same direction, while shifting will shift the Hom space in the opposite direction. Therefore, the bifunctor categorifies a semilinear form which sends
The bimodule self-biadjoint, that is, that and via some adjunction maps. This will become explicit in Section 3.1. In fact, every bimodule in has a biadjoint bimodule such that tensoring with on the left (resp., right) is biadjoint to tensoring with on the left (resp., right). Due to a cyclicity property (see the next section) any homomorphism of bimodules dualizes to a canonical homomorphism , so that can be made into an antiequivalence of , lifting the anti-involution . Notice that takes to itself and to where is given by reading from right to left.
Unsurprisingly, the semilinear product on above (induced by the Hom bifunctor) agrees with the one defined in Section 2.1. To check this, following Remark 2.1 and using the self-biadjointness of , we only need to show the following claim.
Claim 2.9. When is an increasing sequence, is a free left -module of rank 1, generated by a morphism of degree , the length of .
Proof. We only sketch this result. An -bimodule map from to is clearly determined by an element of on which right and left multiplication by polynomials in are identical. Any element of is of the form , and clearly for , and . Hence can be the image of 1 under a bimodule map from if and only if .
Thus can be an image if and only if and , if and only if . Such elements form a left -module generated by the case , , or in other words by . The element has degree 1 in , so we deduce that . Let us call the corresponding map , .
One may use the same argument for the general case. Suppose is increasing and has length , then is a free left -module of degree , with generators ranging over terms where either or . As an exercise, the reader may find the criteria for a general element to be ad-invariant under , and verify that the only possible bimodule maps are -multiples of the following iterated version of : The first map is , the second map is , and so forth. This generator is a map of degree , so that and .
Remark 2.10. We have swept the calculation under the rug, so the dependence of this claim on the fact that is increasing is unclear. In general, when has a repeated index there will be additional maps from to . Roughly speaking, certain symmetry conditions are placed upon polynomials in order for them to slide across certain tensors. The duplication of an index will yield a redundant symmetry condition that places fewer constraints on an ad-invariant element than would be expected from the length of the sequence. We suggest the reader try to find all the maps from to in the length 2 case, first when has no repeated index and then when has a repeated index. This should illustrate the main idea.
Because it is a crucial statement which we use again and again, we restate the overall result and give a reference.
Proposition 2.11. Given two sequences and , is a free graded left (or right) -module of rank , where is the standard trace on defined in Section 2.1.
Proof. This is deduced from the above discussion. For Soergel's proof, see , although it is somewhat obscured. Theorem 5.15 in that paper and especially its proof together state this result, once one unravels exactly what means. Propositions 5.7 and 5.9 state that , since sends to and sends to 1.
The facts below will not be used in this paper.
For a Soergel bimodule the space of bimodule homomorphisms is just the 0th Hochschild cohomology of . Thus, unraveling the definitions, Calculations in Hochschild cohomology can be used to provide a proof of the claim above. One could also define a trace map which is the decategorification of the functor HH0 of taking the 0th Hochschild homology.
Hecke algebra has a trace more sophisticated than or , called the Ocneanu trace , which describes the HOMFLY-PT polynomial. The categorification of the Ocneanu trace utilizes all Hochschild homology groups rather than just HH0, see [24, 43, 47].
The Rouquier complexes mentioned in the introduction are described here. Invertible elements that satisfy the braid relations become  invertible complexes in the homotopy category of the Soergel category (with sitting in cohomological degree −1). This aligns with the fact that in the Hecke quotient of the braid group. Their inverses become inverse complexes with in cohomological degree 1, agreeing with . The homomorphism from the braid group into the Hecke algebra is categorified by a projective functor from the category of braid cobordisms between -stranded braids to the category of endofunctors of the homotopy category of the Soergel category .
Remark 2.12. Note that this convention for Rouquier complexes is opposite that found in , which is to say that we have flipped with . Presumably this arises because we are using as a parameter for the grading shift, and not . The choice of is more natural for the calculation of graded ranks of Hom spaces.
2.4. Diagrammatic Calculus for Bimodule Maps
We follow the standard rules for the diagrammatic calculus of bimodules, or more generally for the diagrammatic calculus of a monoidal category. An excellent and thorough explanation of these rules can be found in , so we will provide a quick summary. A planar diagram will represent a morphism of -bimodules, with the following conventions. A horizontal slice or line segment in this diagram will represent an object (an -bimodule). A rectangle inside the plane will represent a morphism from its bottom horizontal line segment to its top horizontal line segment.
The -bimodule is denoted by a point (on a horizontal line segment) labelled . The tensor product of bimodules is depicted by a sequence of labelled points on a horizontal line segment, so that tensor products are formed “horizontally”. A vertical line labelled denotes the identity endomorphism of , and similarly labelled lines placed side by side denote the identity endomorphism of the tensor product. More general bimodule maps are represented by some symbols connecting the appropriate lines, and are composed “vertically”, and tensored “horizontally”. All diagrams are read from bottom to top, so that the following diagram represents a bimodule map from to :
A horizontal line segment which does not contain any marked points represents as a bimodule over itself, the monoidal identity. The empty rectangle represents the identity endomorphism of . Planar diagrams without top and bottom endpoints (without boundary) represent more general endomorphisms of .
The structure of bimodule categories (or more generally strict 2-categories) guarantees that a planar diagram will unambiguously denote a morphism of bimodules.
We will be using so many such pictures that it will become cumbersome to continuously label each line by an index. Generally, the calculations we do will work independently for each , and can be expressed with diagrams that use lines labelled , and the like. In these circumstances, when there is no ambiguity, we will fix an index and draw a line labelled with one style, a line labelled with a different style, and so forth, maintaining the same conventions throughout the paper. We use different styles of lines because most printers are black and white, but we recommend that you do your calculations at home in colored pen or pencils instead; we even refer to the labels as “colors” throughout this paper.
We use the styles above when referring to indices , , , and , where will be used unambiguously for any index which is “far away” from any other indices in the picture (in other words, when drawing a picture only involving -colored strands, we require , while for a picture involving both and we require or ).
Proposition 2.13. Suppose one chooses the subset of the morphisms in , including the identity morphism of each object, as well as the following morphisms: (1)the generating morphism from to , (2)some isomorphisms that yield the Hecke algebra relations, as well as the respective projections to and inclusions from each summand in (2.10) and (2.12), (3)the unit and counit of adjunction that make into a self-biadjoint bimodule.
Consider the subcategory generated monoidally over the left action of by these morphisms, that is, it includes left -linear combinations, compositions, and tensors of all its morphisms, then is a full subcategory, and thus it is actually .
Proof. For any objects in , there is an inclusion of graded left -modules (since it is clearly an inclusion of -modules, and all generating morphisms are homogeneous). One can define for any graded left -module by choosing generators of , where is the ideal of positively graded elements, and it is a simple argument that a submodule of a free graded -module with the same graded rank is in fact the entire module. So we need only show that Hom spaces in have the same graded rank.
We can define a semilinear form on the free -algebra generated by by the formula . The existence of isomorphisms and projection maps will give us the direct sum decompositions (2.10)–(2.12) in , with the resulting implications for Hom spaces. Therefore the Hecke algebra relation (2.2) are in the kernel of this semilinear form, so it descends to a form on the Hecke algebra. Each will be self-adjoint. When is increasing, contains the generator of the free rank one -module , since that generator is the tensor of the generating morphisms from to various (see the proof of Claim 2.9). Hence it is in fact the entire module, so , and .
By unicity, this inner product agrees with our earlier inner product on the Hecke algebra. In particular, the graded ranks agree, and the inclusion is full.
Below we will construct a category of diagrams via generators and local relations, where the Hom spaces will be graded -bimodules. We will construct a functor from to , showing that our diagrams give graphical presentation of morphisms in . The morphisms in the image of will include all the morphisms enumerated in Proposition 2.13, hence the functor will be full. Calculating the Hom spaces in between certain objects (corresponding to , for increasing), we may use a similar argument to the above proposition to show that they are free -modules of the same graded rank as the Hom spaces in , then the functor will be faithful, and an equivalence of categories. This describes in terms of generators and relations.
Let be the category whose objects are finite direct sums of formal grading shifts of objects in , but whose morphisms only include degree 0 maps. Finally, let be the Karoubi envelope of . The functor lifts to functors and , as in the picture below, with all three horizontal arrows being equivalences of categories.(2.24)
We will define the category originally without reference to isotopy, in order to make the definition of the functor entirely straightforward, using the standard rules for diagrammatics for bimodules. The category would be entirely unchanged if one used different pictures to represent each morphism. However, when the “correct” pictures are chosen for the generators, then every morphism can actually be viewed as a planar graph, and moreover two embedded graphs linked by isotopy represent the same morphism. One could very well define originally using graphs, but this would obscure the definition of .
The most difficult part of the proof will be showing the faithfulness of , which involves a calculation of certain Hom spaces in the diagrammatic category. This calculation will be made possible by the planar graphs interpretation of , wherein some relatively simple graph theory can be applied to simplify pictures.
3. Definition of
This section contains a piecemeal definition of and . For pedagogical reasons, we prefer to provide commentary as we go, instead of defining the category all at once (in fact, some relations do not make sense without the commentary). We also provide some redundant relations in the first pass, because they help make the category more intuitive. However, we repeat the definition all in one place in Section 3.4, without redundant relations, where we also explicitly define the functor .
3.1. The Category : Zero Colors and One Color
This section and the next several will hold the definition of the category , which will be a -linear additive monoidal category, with -graded Hom spaces. Shortly it will become clear that Hom spaces are actually graded -bimodules. It is generated monoidally by objects , whose tensor products will be denoted .
Morphisms will be given by (linear combinations of) diagrams inside the strip , constructed out of lines colored by an index , and certain other planar diagrams, modulo local relations. The intersection of the diagram with , called the lower boundary, will be a sequence of colored endpoints, the source of the map, and the upper boundary will be the target. A vertical line colored represents the identity map from to . The monoidal structure consists of placing diagrams side by side, and composition consists of placing diagrams one above the other, in the standard fashion for diagrammatic categories.
We present the generators and relations in an order based on the number of colors they use. The one-color generators and relations will be sufficient to describe the category for , the two-color ones for , and the three-color ones for the general case. The set of all relations is invariant under all color changes that preserve adjacency, so we only display each generator for a single color , using the conventions described in Section 2.4. However, the generator exists for each index .
All the relations we will give are homogeneous with respect to the grading on generators stated.
The first class of generators, which use no colors, are the following endomorphisms of the monoidal identity :
There is one such generator for each . It is a map of degree 2, which we call multiplication by . After we apply the functor , this will actually correspond to the endomorphism of given by multiplication by . Together, these generators are called boxes. A morphism from to consisting of a sum of disjoint unions of boxes will be called a polynomial. Since the composition of multiplication by and multiplication by is multiplication by , such a sum of products of boxes will obviously correspond under the functor to multiplication by an element . As a shorthand we draw such a morphism as a box with the corresponding element inside. As a map from to , and thus a closed diagram, a polynomial may be placed in any region of another diagram. Placing boxes in the rightmost and leftmost regions of a diagram will define the -bimodule structure on Hom spaces in .
The generating morphisms which use only one color are
For the beginner, the maps are, respectively, a map from to , a map from to , a map from to , a map from to .
Remember, there is one such set of generators for each color . We give these maps names, but the names are temporary. Once we explore the meaning of isotopy invariance, we will stop distinguishing between Merge and Split, and call them both trivalent vertices. Similarly we will stop distinguishing between StartDot and EndDot, and call them both dots.
We also use a shorthand for the following compositions:
We now list a series of relations using only one color, the one-color relations, dividing them into several types of relations for ease of reference. The first set we refer to as the Frobenius relations, since they imply that is a Frobenius object in (see [48, 49] for more on Frobenius algebras). Once we define the functor , this will imply that is a Frobenius object in . Remember that the cups and caps appearing below can actually be rewritten in terms of the generators.(3.1)(3.2)(3.3)(3.4)(3.5)(3.6)(3.7)(3.8)(3.9)
For quick reference, we refer to these relations by their Frobenius algebra names. The first two are the associativity of Merge and the coassociativity of Split. The next two are the unit and counit relations. Relation (3.5) is the biadjunction relation, and the final four are cyclicity relations.
Remark 3.1. For readers not well versed in cyclicity properties and their implications towards isotopy invariance, let us quickly discuss the topic, using the easily visualized notion of a twist. Given a morphism, one can twist it by taking a line which goes to the upper boundary and adding a cap, letting the line go to the other boundary instead. An example is given below,
One can also twist a downward line back up, or twist lines on the left as well. Two morphisms are twists of each other if they are related by a series of these simple twists, using cups and caps on the right and left side. For instance, relations (3.6) and (3.7) state that the Merge is a simple twist of the Split, twisting on the left or right. If one applies the same twist to every term in a relation, one gets a twist of that relation. For instance, relation (3.4) is actually a twist of the definition of the cup.
Because of biadjointness (3.5), twisting a line down and then back up will do nothing to the morphism. Once biadjointness is shown, all twists of a relation are equivalent, because twisting in the reverse direction we get the original relation back. When a morphism has a total of inputs and outputs, twisting a single strand will often be referred to as rotation by degrees.
The above relations imply that twisting any of the above generators by 360 degrees will do nothing. A morphism is said to be cyclic with respect to a fixed set of adjunctions (i.e., cups and caps) if 360 rotation does not change the morphism. Cyclicity is useful because of the following proposition.
Proposition 3.2. Fix adjunctions of each object, which are drawn as caps and cups. If every generating morphism in a diagram is cyclic with respect to those adjunctions, then so is the entire diagram, and the morphism represented by that diagram is invariant under isotopy of the diagram.
Merge and Split are 60 rotations of each other, and each is invariant under 120 degree rotation, so we may represent them isotopy-unambiguously with pictures that satisfy the same properties. A similar statement holds for StartDot and EndDot. We will refer to these morphisms as dots and trivalent vertices from now on, because these terms encapsulate the picture up to rotation.
Remark 3.3. Henceforth, we can take more liberties in our drawings. We can draw a horizontal line colored , and even though this can not be constructed using our generators, it is isotopy equivalent to a cup or cap which can be so constructed. We can allow a diagram to have a boundary not just on the top or bottom, but also on the side. While this does not represent a morphism in our category, the line running to the side boundary can be twisted either up or down to represent a genuine morphism. A relation drawn using diagrams with side boundaries does unambiguously give a relation in .
Associativity and coassociativity are twists of each other. This relation is written in a rotation-invariant form below, and will be crucial in the sequel. We refer to this relation, which permits one to “slide” one trivalent vertex over another, as one-color associativity.(3.10)
We refer to either picture above as an “H”. The horizontal line in the right picture is exactly such a liberty as in Remark 3.3.
Note that relations (3.1), (3.5), (3.6), and (3.8) are sufficient to imply the other Frobenius relations, because of the remarks about twisting made above. Here is the proof of half of (3.7) using (3.6), as an illustrative example.
The next set of relations are known as polynomial slides, which have obvious analogies in the definitions of the modules .(3.11)(3.12)(3.13)
The appearing in the box in the last relation can be any index not equal to or . Together, these relations imply precisely that any polynomial which is invariant under can be slid across a line colored , since is generated by , , and for . Therefore, for an arbitrary polynomial , we have the following immediate consequence (see Remark 2.6).
Proposition 3.4. One may force a polynomial to the other side of a line, leaving at most behind, as follows: (3.14)(3.15)
Proof. This is proven in the same way as (2.7).
Now these are the final one-color relations. First, the dot relations (3.16)(3.17)
It is important to realize that such a relation does not apply if there is anything inside the eye of the needle, as can be seen in the following examples.
Example 3.5. Combining these relations, we have a number of simple but important consequences, which we leave as easy exercises to get the reader used to the diagrammatic calculus
(3.19)Use the first dot relation, then the needle relation and the unit relation
(3.20)Use the needle relation and associativity
(3.21)As above, with the unit relation
As the examples demonstrate, and the following proposition proves, we may remove cycles of this nice form from a one-color graph.
Proposition 3.6. The following relations hold:
More generally, for any polynomial , one has (3.24)
Proof. This is a simple consequence of (3.4), along with the needle, associativity, and unit relations.
There is another relation which is equivalent (given the others) to the first equality in (3.16)(3.25)
This relation quickly leads to the decomposition , see Section 4.5.
For a single color and two variables , the category above, modulo the relation , is equivalent to the category considered by Libedinsky  in the case of a single label . Morphisms given by dot, Merge, Split, and Cap correspond to morphisms , , , , and in [13, section 2.4]. Planar graphical notation, of paramount importance to us, is implicit in . From here on, we diverge from Libedinsky's work, by generalizing to the case of the Weyl group , while Libedinsky  investigates the right-angled case.
3.2. The Category : Adjacent Colors
We now add some generators which mix adjacent colors, which we call 6-valent vertices. Remember that the thick lines represent , and the thin lines represent ,
For the beginner, these maps are, respectively, a map from to , a map from to .
Below are the relations which deal with our new generators. In addition to the relations below, we also impose the same relations with the colors switched. The two color variants in general do not imply each other. However, it is better to think of the two colors as being arbitrary adjacent colors, rather than one being and the other ; then one views these relations as generic for adjacent colors(3.26)(3.27)(3.28)(3.29)(3.30)
It will be shown in Section 4.5 that the first relation is related to the isomorphism .
The second relation shows that the 6-valent vertex is cyclic, that drawing it as a 6-valent vertex is unambiguous, and that isotopy classes of diagrams built out of our local generators will still unambiguously designate a morphism. See Remark 3.1 for details. Because of this, we have used the liberties of Remark 3.3 when writing the last two relations. Note that (3.27) does in fact imply the color-switched version of that same relation, using (3.5).
The relation (3.29) contains a number of equalities, and it is clear that the last equality is merely a rotation of the color switch of the first equality. In fact, there are numerous redundancies amongst (3.29) and (3.30). It is a worthwhile exercise for the reader at this point to check the following statement.
Example 3.7. Assume the relation (3.28) and those before it, then any pair of equalities from (3.29) will imply both color variants of (3.30) as well as the remainder of the equalities from (3.29). Hint: adding a dot to the relation (3.29) allows one to recover (3.30), while the latter may be applied twice within the former.
An important feature to notice is that the 6-valent vertex can be visualized as two trivalent vertices, one of each color, that overlap. If one takes a graph constructed out of dots, trivalent vertices, and 6-valent vertices (our generators so far), then the subgraph formed by all edges of a specific color will have only univalent and trivalent vertices. We use the term two-color (overlap) associativity for to refer to the transformation performed by either (3.30) or the first equality of (3.29), because when viewed as an operation on the “thick”-colored graph, these operations mimic one-color associativity (3.10). Note that, under the same transformations, the “thin”-colored graph (labelled ) is transformed in a different way. However, the color-switched relations will give two-color associativity for instead.
3.3. The Category : Distant Colors
Fix , an index which is not adjacent to . In pictures involving both and , we also assume is not adjacent to . Remember that is represented by a dashed line. This new generator is called a 4-valent vertex, or a crossing.
Note that this definition also covers the same picture with the colors reversed. The colors and can be switched freely since the only requirement was that they were distant from each other.
Now, for relations involving the new generator.(3.31)(3.32)(3.33)(3.34)(3.35)(3.36)
Relation (3.35) holds when you switch and , but one color variant will follow quickly from the other by twisting and applying the first relation. In the final relation, the new color represents an index which is distant from both and . We will refer to (3.32) and (3.36) as the R2 and R3 moves, respectively, because of the obvious analogy to knot theory. The R2 relation is essentially the isomorphism , see Section 4.5.
The same statements about cyclicity and drawing diagrams with sideways boundaries apply from before (see Remarks 3.1 and 3.3). Once again, the 4-valent vertices are drawn so that morphisms are isotopy invariant.
The relations (3.32)–(3.36) imply that a -colored strand can just be pulled underneath any morphism only using colors distant from , since it can be pulled under any generating morphism, whether it be a line, a dot, a trivalent vertex, or a 6-valent vertex. In fact, thanks to (3.4), the R2 move follows from (3.33) and (3.34).
We have now listed all the generators of our subcategory: trivalent, 4-valent, and 6-valent vertices, and dots. There is one final relation, coming from the fact that and may not interact, but they do jointly interact with . The final relation will be called three-color (overlap) associativity for :(3.37)
In the above diagram, dotted lines carry label , thick lines and thin solid lines . Rotating this relation by 90 degrees, we get the same relation except with and switched, so that only color variant is needed to imply both. The “thick”-colored graph undergoes the associativity transformation. The same is true (symmetrically) with the “dotted”-colored graph.
This concludes the definition of the category .
3.4. The Complete Definition and the Functor
In order to put everything in one place with no redundancy, let us define the category again.
Definition 3.8. The category has objects given by sequences of indices in , with a monoidal structure given by concatenation. Fix two sequences and . Consider the set of all diagrams in , constructed out of vertical lines colored by indices, and out of the generating pictures below, such that the intersection of the diagram with is the sequence of points , and the intersection with is . This set is graded, where the generators have the degree indicated, then the space is defined to be the -linear span of this set of diagrams, modulo the homogeneous local relations below,
Some relations are drawn using the liberties of Remark 3.3. We also use the definition of the cap and cup,
For each color,
Here, is the color of the line, and is a color ,
For any two adjacent colors,
For any two distant colors,
For two adjacent colors and a third, distant to both,
For three mutually distant colors,
For three colors with the same adjacency as ,
Definition 3.9. Let be the functor from to specified as follows. On objects, . We define the functor on generating morphisms and extend it monoidally to all morphisms.
In doing so, we always use the isomorphism (2.6) to identify with the -bimodule spanned by a choice of polynomials. If one thinks of diagrammatically as vertical lines, then a spanning element of is a choice of polynomial for each empty region delineated by the lines (and polynomials with the appropriate symmetry may slide across the lines). We write the map explicitly for a general element when it is easy enough to do so, or we write it for a spanning set as an -bimodule (see Remark 2.7).
For a line colored ,
For lines colored and distant,
For a thin line colored and a thick line colored ,
For any ,
Claim 3.10. The above maps are -bimodule maps.
Proof. This is obviously true for EndDot, since the resulting map is no more than multiplication. StartDot is sent precisely to the generator of discussed in Section 2.3. Split and Merge have already been seen as inclusion and projection maps in the isomorphism , see Remark 2.6. For the 4-valent vertex, the only polynomials which slide all the way across or are in , so that the map is a bimodule map (that spans it was observed in Remark 2.7).
Only the 6-valent vertices remain to be checked. Consider the first of the two variants. The generating set as an -bimodule was chosen because can be slid freely between the second and third slots. We have defined the -bimodule map on generators before showing that the map is an -bimodule map at all, which is akin to putting the cart before the horse. Let us explicitly define the map on a spanning set by the following algorithm: given , first we force to the right and slide the “remainder" to the left, that is ; then we force the terms in the second slot to the left, yielding . Finally, each term can be evaluated using the given definition of on generators. This gives an explicit formula for the image of , which we only need check is invariant under: sliding an element of from to , or from to ; sliding an element of from to . Sliding elements of does not pose a problem, since we defined the map by forcing to and to , which fully respects such slides. Checking invariance under slides from to is nontrivial. However, the bulk of the work is encapsulated in the following discussion, which is useful for calculations in general.
By adding and subtracting , the image of under the first 6-valent vertex (see above) can be written more symmetrically as . The first term is a polynomial symmetric in all the relevant variables and thus can be slid anywhere. In the other two terms, can not be slid freely under a line labelled , so it is stuck in its respective position. In contrast, and are not equal, since can not be slid over a line labelled , but the images of both these elements are easier to remember, and are shown below,
The way to remember these formulae is that the variable which cannot be slid is sent to the variable which cannot be slid, from the middle on one side to the exterior on the other. It is easy to see that these calculations were done according to the algorithm above, forcing to the outside first and then evaluating on the leftover .
Now, we do the consistency check for the simplest cases. We wish to show that and are sent to the same element by the algorithm. However, this is rather easy, for in both cases, the term slides immediately to the exterior, and the term is evaluated as above, so both are sent to . Similarly, both and are sent to . The general case is not significantly more difficult than this; we leave the details to the reader.
Proposition 3.11. The functor is well-defined. That is, the relations of hold between morphisms of -bimodules in .
Checking that the relations hold is a series of simple but tedious calculations that is postponed until Section 5.1. We assume this result henceforth. In addition, we note once and for all that (as one can easily check) all relations in the definition of are homogeneous, and preserves the degree of the generators.
Addendum 2. It may strike the reader as unusual that the definition of the functor seems lopsided, while the definition of is invariant under right-left reflection, or under reversing the order of the colors , . For instance, applied to StartDot yields the element , which is actually invariant under right-left reflection but not immediately so. Had this element been rewritten as , perhaps the calculations would be more natural despite having more fractions. A worse offender is the forcing rule , which should be rewritten . In general, the elements are more natural than or , coming from the reflection representation rather than the standard representation of (see Remark 2.2 and Section 4.6).
Previous versions of this paper, however, used the style above, and so we feel compelled to stick with it to maintain consistency. Also, checking that is a functor may be easier with the current notation.
We will spend the next few sections classifying the homomorphisms in . For many of the results, proofs will be postponed until Section 5.2.
We will we using the fact, extensively discussed in the previous sections, that a morphism can be viewed unambiguously as an isotopy class of graphs with polynomials in the regions (or rather, a linear combination of these). Henceforth, the term graph only refers to colored finite graphs with boundary (embedded in the planar strip) which can be constructed out of univalent, trivalent, 4-valent, and 6-valent vertices as above. Remember that these graphs do have edges which run to the boundary, which we call boundary lines, and may have edges which meet neither the boundary nor any vertex, and thus must necessarily form a circle. We say a graph has a boundary if it has at least one boundary line. A graph divides the planar strip into regions, and there are two distinguished regions: the lefthand and righthand regions, which contain and , respectively.
We call a boundary dot any connected component of a graph which consists entirely of an edge starting at the boundary and ending in a dot. We call a double dot any connected component of a graph which consists entirely of an edge with a dot on both ends. Cutting an edge in a diagram and replacing it with two dots we call breaking the edge (see, for instance, relation (3.16)).
Given a set of graphs and a morphism in , we say that underlies if can be written as a linear combination of morphisms, each of which is given by a graph with polynomials in regions. We say that a graph reduces to if underlies every morphism that underlies. Clearly reduction is transitive, in that if reduces to , and every graph in reduces to , then reduces to . Our goal will be to find a nice set of graphs to which all other graphs reduce. We will do this by finding reduction moves, which are local moves on graphs, sending a graph to a set of graphs to which it reduces.
Let be a subset of . The -graph of a graph will be the subgraph consisting of all edges colored for . Some 6-valent vertices in the original graph may become trivalent vertices in the -graph. Similarly, some 4-valent vertices in the original graph may become 2-valent vertices in the -graph, which we ignore, connecting the incoming edges into a single edge. The -graph is itself a graph by our above definition. Most often we will just consider the -graph for a single color (i.e., ). Typically, our reduction moves will be designed to simplify the -graph for a particular , allowing us to simplify the graph one color at a time.
Remark 4.2. The rest of this paper will have numerous calculations, but they will mostly be calculations with the underlying graphs, not keeping track of polynomials, so they do not reflect how morphisms actually behave in . For lots of examples of computations in the graphical calculus, see .
4.2. One Color Reductions
In this section, we assume all graphs consist of a single color .
Definition 4.3. Consider the following “moves", or transformations. They take a subdiagram looking like Start, and replace that subdiagram with Finish. We call these the basic moves.
Remember, these are moves on graphs, not graphs with polynomials. Note that the needle move, by adding a dot on the bottom, yields a reduction from the circle to a double dot. The only moves which change the connectivity of a graph are double dot removal, which deletes a connected component, and the connecting move, which has the potential to link two components into one.
Claim 4.4. All of these moves are reduction moves in .
Proof. The associativity move follows from (3.10). That is, even if there are polynomials in the regions of the graph, the relation (3.10) can still be applied. These polynomials, being in external regions, do not interfere with the application of relations. Similarly, dot contraction/extension follow from (3.4), dot removal follows from (3.17), and the connecting move follows from (3.25).
The needle move remains. Suppose we have an arbitrary polynomial in the eye of the needle. We may use (3.24), generalizing (3.19), to replace the diagram with a dot accompanied by .
The following example of reduction should be familiar.
Definition 4.7. A simple tree with boundary lines is a connected one-color graph with boundary, whose form depends on : (1)if , then is a trivalent tree with vertices connecting all the boundary lines. Note that any two such trees are equivalent under the associativity move; (2)if , then is a single boundary dot; (3)if , then is the empty graph.
Definition 4.8. A cycle in a one-color graph is either a circle or a path from a vertex to itself which does not repeat any edges. Any cycle splits the plane into two parts, the inside and outside of the cycle. A cycle is minimal if the inside of the cycle consists of a single region.
By counting vertices, it is clear that any connected purely trivalent graph with no cycles is a simple tree. Any graph with a cycle has a minimal cycle.
We will now give a precise inductive algorithm to reduce a graph to a disjoint union of simple trees, by reducing minimal cycles.
Proposition 4.9. Consider a minimal cycle in a one-color graph . Using the associativity, needle, dot removal, and dot contraction/extension moves, we may reduce a neighborhood of the cycle (including the inside region) to a simple tree (see (4.1)).
Proposition 4.10. Using the associativity, needle, dot removal and dot contraction/extension moves, one can reduce any one-color graph to a disjoint union of simple trees. During this process, each component with no boundary lines will be replaced with the empty graph, and after this is done, no further connected components are created or destroyed or merged.
We prove both propositions together, in several steps.
Proof of Proposition 4.10 for a Graph with No Cycles. Suppose there are no cycles in . If there is a dot, the edge coming from that dot must connect to either the boundary, another dot, or a trivalent vertex. Our simplification algorithm is as follows. (1)Remove any dot connected to a trivalent vertex, using dot contraction. Repeat Step 1 until no such dots remain. This does not alter the connected components.(2)Replace any double dot with the empty set, using double dot removal. Because Step 1 did not alter the connected components, this could only be applied to double dots which arose from components which had no boundary lines.
Boundary dots are in their own connected component. Any other connected component is purely trivalent and has no cycles, so it must be a simple tree.
Proof of Proposition 4.9, Assuming Proposition 4.10 for Graphs with No Cycles. Let denote a minimal cycle of . Consider a neighborhood of in , and let be the subgraph consisting of the interior of the cycle, as in (4.1). Let us call the boundary lines of spokes, since they run into from the inside, like spokes hitting the wheel of a bicycle. Now may not have any cycles, or else would not be a minimal cycle. Moreover, the spokes of must be in distinct connected components of , or else they would create additional regions and would not be a minimal cycle. Our simplification algorithm is as follows: (1)Apply Proposition 4.10 to , replacing with a disjoint union of boundary dots, one for each spoke.(2)Use dot contraction to remove all the spokes. We are now in the situation of Example 4.6.(3)Apply associativity as in Example 4.5, reducing the length of by one. Repeat until is a needle or a circle.(4)If is a needle, apply the needle move to replace it with a dot. If associativity moves were performed in Step 3, use dot contraction to contract this dot into one of the trivalent vertices, as in Example 4.5.(5)If is a circle, apply dot extension to replace it with a needle attached to a dot, then apply needle reduction to obtain a double dot, and double dot removal to obtain the empty graph.
It is a simple observation that the result of this procedure is a simple tree, and that the only alteration of connected components which occurred was the removal of components which had no boundary lines to begin with.
Proof of Proposition 4.10 in the General Case. Suppose we have an arbitrary graph . Our simplification algorithm is as follows: (1)If has a cycle, apply Proposition 4.9 to replace its neighborhood with a simple tree. Repeat this process until has no cycles.(2)Apply the procedure for the case of no cycles above.
Note that Step 1 will terminate, which can be shown by induction on the number of internal regions (regions which do not meet the boundary of the planar strip). Each application of Proposition 4.9 reduces the number of internal regions by 1.
Corollary 4.11. Using the connecting move in addition, we can reduce the graph to a single simple tree.
Proof. It is an easy observation that when one uses the connecting move on a simple tree with boundary lines and a simple tree with boundary lines, one gets a simple tree with boundary lines, after possibly removing extraneous dots if either or equals 1.
Remark 4.12. There are two useful sets of one-color graphs with boundary lines, to which all others reduce. The first set just contains the simple tree with boundary lines, and the latter is the collection of all disjoint unions of simple trees whose number of boundary lines add up to . The former is useful because there is a single graph, so we have fewer cases to deal with. The latter is useful because it does not require the connecting move.
More importantly, these sets behave differently when we introduce polynomials into the equation. Let us assume that all boundary lines are on the top boundary, so that we are looking at a morphism in where is ( times). The following statements will not be used in this paper, and can be more easily proven after the calculation of Hom spaces.
Claim 4.13. Consider diagrams which are a simple tree, with an arbitrary monomial in the lefthand region, and either 1 or in each other region. This is a basis for over .Claim 4.14. Consider diagrams which are disjoint unions of simple trees, with an arbitrary monomial in the lefthand region, and no other polynomials. These are a spanning set for over .
The second claim is easy to see, given Proposition 4.10. Given a disjoint union of simple trees, with arbitrary polynomials, we may use relation (3.16) and the polynomial slides to force all the polynomials to the left, at the cost of potentially breaking some lines. This breakage is not a problem, since one can reduce again to a simple tree without adding more polynomials, using (3.10) and (3.4). We do not get a basis this way: consider the three different ways to break a line in a trivalent vertex diagram; there is a linear dependence relation between these diagrams and the trivalent vertex with a polynomial in the lefthand region.
Relations (3.25) and (3.11) essentially allow us to get from the second spanning set to the first, showing that the first is at least a spanning set. That it is a basis is immediate from counting the graded dimension of the Hom space, one we prove that the dimension of Hom spaces in conforms with a certain semilinear form.
The connecting move is less important than the others in the proofs, and was introduced primarily to make these remarks. It can generally be ignored below.
4.3. Broken One-Color Reductions
The reductions of the previous section do apply, as stated, to any one color graph. However, we would like to apply these moves to the -colored graph of a multicolor graph, where the moves above do not extend trivially to reduction moves in . In this section, we quickly generalize the results of the previous section to a weaker set of moves.
Definition 4.15. Consider the following reductions for one-color graphs, which take the graph on the left and replace it with the set of graphs on the right:
We call the moves above (together with the basic moves that have no weak analog) the broken or weak basic moves.
These moves behave like the basic moves of the same name, except that they may also replace the original diagram with a broken version of itself, that is, a version with some edges broken. To distinguish the original basic moves, we may call them the strict basic moves.
Proposition 4.16. Consider a minimal cycle in a one-color graph . Using the weak associativity, needle, dot removal, weak dot contraction, and dot extension moves, we may reduce a neighborhood of the cycle (including the inside region) to a disjoint union of simple trees.
Proposition 4.17. Using the weak associativity, needle, dot removal, weak dot contraction, and dot extension moves, one can reduce any one-color graph to a disjoint union of simple trees.
Of course, two distinct simple trees are no longer equivalent under the weak associativity move, but this is really irrelevant for us.
Proof. Breaking a line will never create a cycle or increase the number of trivalent vertices. Because of this, the proofs of the previous section go through almost verbatim (ignoring any statements about connected component, and occasionally replacing “a simple tree" with “a disjoint union of simple trees"). The only significant alterations that need to be made come in the proof of Proposition 4.9. In Step 4 or Step 5 one may need to remove an additional double dot. In Step 2 or Step 3, weak dot contraction and weak associativity have multiple outcomes, but each outcome that does not agree with strict dot contraction or strict associativity will have broken the cycle already, allowing us to complete the proof using the no-cycle algorithm of Proposition 4.10.
Alternatively, one could also prove these statements by induction on the number of trivalent vertices. Each weak move is equivalent to a strong move modulo diagrams with fewer trivalent vertices. The only part of the proof that ever created additional trivalent vertices was the single use of dot extension in Step 5 of the proof of Proposition 4.9. It is easy to see how Step 5 does not actually cause a problem, however, since after dot extension is applied, the needle move and double dot removal will do the trick in the same way regardless.
Addendum 3. The overall proof using weak one-color moves is slightly different than the treatment in previous versions of this paper, but it is cleaner and more straightforward.
4.4. -Colored Moves
Now we list the moves which allow us to simplify multicolor graphs.
Definition 4.18. Consider a graph whose -graph looks like one of the pictures in the start column of Definition 4.3. Let be the set of all graphs whose -graph looks like the corresponding picture in the finish column. Let be the set of all graphs whose -graph looks like any of the corresponding pictures in the finish column of definition weakbasicmoves. The strict -colored move replaces with the set . The weak -colored move replaces with the set .
For instance, strict -colored associativity will replace any graph whose -graph is with the set of graphs whose -graph is . This set is enormous, for other colors can interfere, and the -graph for some other can be arbitrarily complicated. The -colored vertices, seemingly trivalent, could come from 6-valent vertices in . In general, an -colored move will behave nicely on the -graph, but may significantly complicate the full graph.
Proposition 4.19. The weak -colored basic moves are reduction moves in , so long as they are applied to graphs which do not contain either the color or .
A color will be called extremal for a graph if it appears in but either or does not appear. Clearly, any nonempty graph will have an extremal color, such as the minimal color present.
The proof of this proposition is found in Section 5.2, as well as more precise details on what can be done. The power of the proposition can be seen immediately:
Corollary 4.20. Any graph without boundary lines can be reduced to the empty graph.
Proof. We induct on the set of colors present in the graph . If no colors are present, then is the empty graph and we are done. Else, choose an extremal color . By Proposition 4.19 we may apply the weak -colored basic moves and use Proposition 4.17 to replace every connected component of the -graph with a disjoint union of simple trees with no boundary lines. Since a simple tree with no boundary lines is the empty set, reduces to a set of graphs which do not include the color . By induction, now reduces to the empty graph.
One can apply a similar procedure to a graph with boundary lines. Choose an extremal color , and reduce the -graph to a disjoint union of simple trees, then, within each region delimited by the -graph, the colors and are now extremal, and we can reduce those. One can repeat this procedure, however, it will not produce a very simple graph in all cases. If the color has at least 3 boundary lines, the -graph may have trivalent vertices, and the graph itself may have 6-valent vertices in their place. These 6-valent vertices will produce more or colored boundary lines inside the regions delimited by the -graph. Nonetheless, we have the following simple case.
Corollary 4.21. Any graph whose boundary has at most one line of each color can be reduced to a disjoint union of boundary dots.
Proof. We know we can reduce the -graph, for an extremal color, to a disjoint union of simple trees. A simple tree with at most one boundary line is either the empty set or a boundary dot. Therefore the -graph is now either the empty set or a boundary dot, depending on whether or not appears in the boundary. The dot need not be a boundary dot in the entire graph , but it can encounter only 4-valent vertices en route to the boundary. Since a dot can be slid under a 4-valent vertex by relation (3.33), we may turn the dot into a boundary dot (its own connected component). The remaining connected components form a subgraph (also viewable in the planar strip) without the color . Induction now concludes the proof.
4.5. Is Fully Faithful
In this section, modulo the proofs of previous sections which were delayed until Section 5, we prove our main theorem.
Theorem 4.22. The functor from to is an equivalence of -linear monoidal categories with Hom spaces enriched in .
We know is a functor by Proposition 3.11, and inspection of the objects in both categories shows immediately that it is essentially surjective. To show is full, we use Proposition 2.13, which motivates the next few statements.
Corollary 4.23. For any index , the object in is self-biadjoint. This means that for any sequences and , there are natural isomorphisms and .
Proof. The first isomorphism and its inverse are shown below. That these maps compose to be the identity is exactly the relation (3.5)
The second isomorphism and its inverse are the left-right mirror of the maps above.
Note that when in the corollary above, these isomorphisms combine to yield an isomorphism , drawn as below
At this point, one could construct a semilinear product on the free algebra generated by , , via , and would be self-adjoint. If we show that the Hecke algebra relations are in the kernel of this semilinear product then it will descend to the Hecke algebra. We have several methods by which we could do this.(1)Look in , where we have direct sums and grading shifts, and prove the isomorphisms (2.10)–(2.12). (2)Look in the Karoubi envelope , find idempotents corresponding to the auxiliary modules in (2.13) and friends, and show those isomorphisms. (3)Work entirely within and show the isomorphisms (2.12) only after applying the Hom functor. For instance, showing that will be sufficient.
All these tactics are primarily the same. We illustrate the third method, although we do explore the auxiliary modules of the second method.
The relation (3.25) precisely descends to . We decompose the identity of into the sum of two idempotents, and obtain orthogonal projections from to of degrees 1 and −1, respectively,(4.2)
Stated very explicitly, we have two projection maps , and two inclusion maps , , as indicated in the diagram above, which is just relation (3.25) divided in half. Include the minus sign on the right picture into the map , then one can quickly check , , , , and . So these must be projections to and inclusions of summands, and they are maps of the correct degree.
Now, we look at and which are distant. We have the isomorphism . It is clear that we may construct for any isomorphisms or vice versa merely by precomposing with the appropriate 4-valent vertex, and (3.32) shows that these maps are inverses.
Because we can, let us discuss how we might extend our diagrammatic calculus to include additional modules from , like . Let us (temporarily) allow a new color of line in our pictures, call it , and extend the functor so that it sends to . We add new generators to our diagrammatics, and specify the image under the extended functor, as below
The definition of these bimodule maps, and the proof that they are in fact bimodule maps, is exactly akin to the discussion in Section 3.4. It is clear that composing these morphisms to get an endomorphism of will yield the identity map, and composing them to get an endomorphism of will also yield the identity map (these would be relations in the extended diagrammatic calculus). Thus we get isomorphisms in .
We now combine these techniques to deal with adjacent colors. We have isomorphisms and . If we allow the new color of line, again called , to represent the bimodule , then we may define the following maps:
Again, checking that we have well-defined bimodule maps is akin to Section 3.4. Composing the two maps that go through and to get an endomorphism of will yield the identity map of , and the same is true, respectively, of .
Then the equation (3.26), which is a decomposition of the identity of , actually follows from this relation in (4.3)
There is a more explicit statement to be derived from this relation, completely analogous to the two-line case, with projection and inclusion maps (include the minus sign on the right picture in ). Similarly we get a decomposition of via the same relation with the colors switched. The auxiliary module used here is in fact the indecomposable Soergel bimodule where .
To state the result without extending the calculus to the Karoubi envelope, one may merely observe that for any there is a map and vice versa given by precomposing with the appropriate 6-valent vertex; there is another map given by precomposing with , and a map backwards given by precomposing with , then (3.26) exactly yields that .
Thus we have shown the following claim:
We have now satisfied the requirements of Proposition 2.13, which implies the fullness of .
Claim 4.25. The functor is full.
Finally, our graphical reductions give us a classification of certain Hom spaces.
Corollary 4.26. The space , where is a length increasing sequence, is a free left (or right) -module of rank 1, generated by a homogeneous morphism of degree .
Proof. Let be the morphism consisting entirely of boundary dots. This corresponds, under the functor, to the generator of degree discussed in Claim 2.9. By Corollary 4.21, viewed as a graph underlies every morphism in . This graph has a single region, so any morphism that it underlies will be generated by under the action of on the right or left, which puts a polynomial into that single region. This gives a surjective map from to the Hom space. After applying , we know that the morphisms must surject onto , which is a free -module (see Claim 2.9). Therefore, the Hom space in must also be free.
Corollary 4.27. The semilinear form on induced by Hom spaces in agrees with the form defined in Section 2.1, so it agrees with the form induced by . Therefore is faithful.
Proof. This is now immediate, from Remark 2.1.
In conclusion, is an equivalence of categories, and Theorem 4.22 is proven.
We also get for free the following corollary, which is difficult to prove purely diagrammatically.
Corollary 4.28. Hom spaces in are free as left or right -modules.
Remark 4.29. It is worth reiterating what is proven diagrammatically, and what is proven using the functor to Soergel's category. Diagrammatically, we can prove Proposition 4.19 and Corollary 4.21, which implies that surjects onto for increasing. However, without the full functor and the nondiagrammatic knowledge of Hom spaces in Soergel's category, we do not know how to prove that the Hom space is free. We do not have a fully diagrammatic proof that the Hom spaces in are what they are, we only have a diagrammatic proof of an upper bound on the size of the Hom spaces.
4.6. The Quotient
In the remainder of this chapter, we provide some sketched statements about generalizations and variations of . We do not use these results elsewhere in the paper, and while the proofs are only sketched, they are fairly obvious and can be fleshed out without too much work.
As described in Remark 2.2, one usually constructs Soergel bimodules with respect to the -dimensional fundamental or geometric representation, instead of the -dimensional standard representation. This amounts to working over the ring , which is a quotient of by the first elementary symmetric function . Since is symmetric, it is in the center of the category (it slides freely under all lines, tensor-commuting with all morphisms), and one may easily take the quotient category, setting , without changing any of the diagrammatics. There is a functor from this quotient category to the appropriate category of Soergel bimodules in , which is an equivalence of categories, so this diagrammatical quotient category also categorifies the Hecke algebra.
One advantage to passing to the quotient is that, after inverting a suitable integer, we may remove the boxes from our list of generators. According to relation (3.17), the double dot colored is equal to . Thus linear combinations of double dots of colors will give us the -span of inside the space of linear polynomials (these are the simple roots). This span will not include in , which is why we require at least one box as an additional generator. However, it is easy to check that, if is invertible in , then is in the -span after passing to the quotient . As an example, when , the double dot is equal to , and passing to , we get . Thus, if is invertible in , one can eliminate boxes from the quotient calculus altogether, replacing them with linear combinations of double dots.
Replacing boxes with double dots is much more natural, since it emphasizes that the boxes themselves should have a color, and that the set of polynomials depends in a natural way upon the coxeter group . Viewing them as boxes may help make the proofs in this paper more intuitive, however.
4.7. Color Elimination
We have already shown that , without needing to investigate in depth any morphisms except those from to an increasing sequence. Because this pins down the size of all Hom spaces, we can deduce some additional facts about general morphisms. The following result is not used elsewhere in this paper, but may come in handy when constructing the analogous category for arbitrary Coxeter systems.
Proposition 4.30. Let be a graph where the color does not appear in the boundary, then can be reduced to graphs not containing the color at all.
We can already show this when is extremal, since by Proposition 4.19 we may apply the weak -colored basic moves and reduce the color to the empty graph. Unfortunately, as in Remark 4.29, we do not have a diagrammatic result of this proposition in general, but use the known size of Hom spaces to prove it.
For , (i.e., for a Coxeter subgraph of ) we let be the category defined analogously but where edges can only be labelled by colors in . If colors which do not appear on the boundary are not needed in the graph after reduction, then the natural inclusion of into is fully faithful, which one would expect. The full faithfulness of this inclusion is equivalent to the above proposition.
However, as discussed in the previous section, the boxes which are allowed to appear should also depend on . For the rest of this section, we assume we are working in the quotient. Let be the category defined analogously, where edges are labelled in , and where the only polynomials appearing are in the subring generated by double dots colored in . This is really the category we should look at. Given a morphism in , we can use polynomial forcing rules to guarantee that the only polynomials not in appear in the lefthand region. We do not prove it here, is simply ; that is, the objects are unchanged, and morphism spaces undergo base change.
Proposition 4.31. Let be a Coxeter subgraph of , and let , , designate the corresponding constructions for this Coxeter graph, then there is a functor from to which is an equivalence of categories.
Proof. We will only sketch this result. The set will be a disjoint union of various subgraphs , each isomorphic to for some . For each we know that , and we have all the results aforementioned. Moreover, for any and objects and , we have natural isomorphisms constructed with 4-valent crossings. Thanks to the distant sliding rules, these natural isomorphisms commute in the proper way with all morphisms in or . The category can be constructed via some universal “symmetric monoidal product" construction, by taking the product over the categories . The same holds true for , which will yield the result.
Given the sketchy result above, we can calculate the rank of Hom spaces in by using the standard trace map on . This trace map commutes with the standard trace on under the inclusion . Thus we see that, for objects and in , the space is a free module of some rank , and the space is a free module of the same rank . Therefore is a free module of rank as well, and the inclusion of in is fully faithful.
We quickly sketch an alternative proof of Proposition 4.30, which assumes that we know that Hom spaces in conform to the standard trace. We can think of any graph with boundary as a morphism in from to . We have already “inductively calculated” the space of such morphisms, along the lines of Remark 2.1 and Proposition 2.13. Namely, if has no repeated colors, then every morphism is a polynomial with boundary dots. If has repeated colors, we use idempotent decompositions to consider the Hom space as the sum of the Hom spaces of its summands, or apply biadjunction to cycle the sequence , until we have expressed the Hom space in terms of for various nonrepeating. Since neither biadjunctions nor idempotent decompositions add any new colors to the graph, we have just constructed a generating set of morphisms in a way which does not involve any colors which were not in to begin with.
5.1. Is a Functor
We need now to check that each of our relations holds true for Soergel bimodules, after application of . The relations are listed in full in Section 3.4. These checks are entirely tedious and straightforward, and require little imagination. With a little imagination, however, there are some tricks which make checking most relations trivial.
Again, we always use relation (2.6) to identify elements of with tensors of polynomials.
For brevity, we will call any element which is a tensor product a 1-tensor. EndDot, Split, and the 4-valent and 6-valent vertices all send 1-tensors to 1-tensors. This simple fact is already enough to prove relations (3.1), (3.32), (3.33), (3.34), and (3.36), since the bottom bimodule in each of these pictures is generated by the 1-tensor; we call this the 1-tensor trick. Caps and Merges kill a 1-tensor, while Cups and StartDots send it to a sum of two-linear terms.
There are several choices to make when checking the various relations. Once the twisting relations are shown, one is free to prove any twist of the other relations. Also, one may choose which set of generators of the source bimodule to check equality on. Finally, whenever a Cup or a StartDot appears, there are two equivalent ways to write the result, since in , and one might be easier than the other to evaluate.
The first trick is to choose the set of generators which makes the calculation the easiest. Let us remind ourselves of the arguments of Remark 2.7. An arbitrary module of length will have generators as an -bimodule, where is the number of different colors that appear. Let us use the term -pair to denote two instances of the index in , separated only by colors . Let denote the set of all -pairs for all ; the size of is . As we force polynomials to the right or left, a variable (or alternatively ) might get stuck between the two -colored lines of an -pair, and this independently of each other -pair or -pair for . The following claim is easy to show from the forcing rules.
Claim 5.1. Each will be generated as an -bimodule by any set of linearly independent tensors, for which we have a bijection between and the power set of , satisfying the following property: If a tensor corresponds to a subset , then each -pair in corresponds to a distinct linear factor of which is either or somewhere inside the -pair.
It is a mouthful, but an example clarifies it. The following is an example of a set of generators for , where so we need 8 generators:
Each picture represents a tensor, where by convention a blank area is filled with a 1-tensor. Reading across, the first picture corresponds to the empty subset of . The second picture corresponds to the -pair, the third to the -pair, and the fourth to the