Abstract

A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).

1. Introduction

The Orthogonal Grassmannian is defined in Section 2. A Richardson variety in the Orthogonal Grassmannian (The Richardson varieties in the ordinary Grassmannian are also studied by Stanley in [1], where these varieties are called skew Schubert varieties. Discussion of these varieties in the ordinary Grassmannian also appears in [2].) is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian with an opposite Schubert variety therein. In particular, the Schubert and opposite Schubert varieties are special cases of the Richardson varieties. In this paper, we provide an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any -fixed point of . It should be noted that the local properties of the Schubert varieties at -fixed points determine their local properties at all other points, because of the action; but this does not extend to the Richardson varieties, since Richardson varieties only have a -action.

In Raghavan and Upadhyay [3], an explicit description of the initial ideal (with respect to certain conveniently chosen term orders) for the ideal of the tangent cone at any -fixed point of a Schubert variety in the Orthogonal Grassmannian has been obtained. In this paper, we generalize the result of [3] to the case of the Richardson varieties in the Orthogonal Grassmannian.

Sturmfels [4] and Herzog and Trung [5] proved results on a class of determinantal varieties which are equivalent to the results of [6–8] for the case of the Schubert varieties (in the ordinary Grassmannian) at the -fixed point . The key to their proofs was to use a version of the Robinson-Schensted-Knuth correspondence (which we will call the “ordinary” RSK) in order to establish a degree-preserving bijection between a set of monomials defined by an initial ideal and a “standard monomial basis.” The difficulty in extending this method of proof to the case of the Schubert varieties (in the ordinary Grassmannian) at an arbitrary -fixed point lies in generalizing this bijection, which is done in the three papers [6–8]; the work done in [8] is slightly more general, since it applies to the Richardson varieties, and not just to the Schubert varieties. These three bijections, when restricted to the Schubert varieties in the ordinary Grassmannian, are in fact the same bijection (This supports the conviction of the authors in [6] that this bijection is natural and that it is in some sense the only natural bijection satisfying the required geometric conditions.), although this is not immediately apparent. In the work of Kreiman in [8], this “generalized bijection” has been viewed for the first time as an extended version of the “ordinary” RSK correspondence, which he calls the Bounded-RSK correspondence; this viewpoint was not present in [6, 7]. Although the formulations of the bijections in [6, 7] are similar to each other, the formulation of the bijection in [8] is in terms of different combinatorial indexing sets.

The work done in [3] by Raghavan and Upadhyay does not involve any version of the RSK correspondence, unlike the work done by Herzog and Trung in [5]. The work done in [3] relies on a degree-preserving bijection between a set of monomials defined by an initial ideal and a “standard monomial basis,” and this bijection is proved by Raghavan and Upadhyay in [9]. It is mentioned in [9] that it will be nice if the bijection proved therein can be viewed as a kind of  “Bounded-RSK” correspondence, as done by Kreiman in [8] for the case of the Richardson varieties in the ordinary Grassmannian. This paper fulfills the expectation of [9] that one should be able to view the bijection there as a generalized-bounded-RSK correspondence, which we call here the Orthogonal-bounded-RSK correspondence (OBRSK). Kreiman has mentioned in his paper [8] that he believes in the possibility of adapting the methods of his paper ([8]) to the Richardson varieties in the Symplectic and the Orthogonal Grassmannian as well; this paper supports Kreiman’s belief for the Orthogonal Grassmannian case.

2. The Orthogonal Grassmannian and Richardson Varieties in It

Fix an algebraically closed field of characteristic not equal to . Fix a natural number , a vector space of dimension over , and a nondegenerate symmetric bilinear form on . For an integer such that , set . Fix a basis of such that equals if and is otherwise. Denote by the group of linear automorphisms of that preserve the bilinear form and also the volume form. A linear subspace of is said to be isotropic, if the bilinear form vanishes identically on it. Denote by the closed subvariety of the Grassmannian of -dimensional subspaces consisting of the points corresponding to maximal isotropic subspaces. The action of on induces an action on . There are two orbits for this action. These orbits are isomorphic: acting by a linear automorphism that preserves the form but not the volume form gives an isomorphism. We denote by the orbit of the span of and call it the (even) Orthogonal Grassmannian. One can define the Orthogonal Grassmannian in the case when the dimension of is not necessarily even. But it is enough to consider the case when the dimension of is even, the reason being the following (see also Section 2.1 of [9] for the same): suppose that the dimension of is odd; say dimension of . Let and be a vector space of dimension with a nondegenerate symmetric form. Let be a basis of such that equals if and is otherwise. Put and . Take to be an element of the field such that . We can take to be the subspace of spanned by the vectors , and, and a basis of to be these vectors in that order.

There is a natural map from to : intersecting with an isotropic subspace of of dimension gives an isotropic subspace of of dimension , and we denote this map by . The map is onto, for every isotropic subspace of (and hence of ) is contained in an isotropic subspace of of dimension . In fact, more is true: the map is two-to-one. The map being two-to-one, it is also elementary to see that the two points in any fiber lie one in each component of . We therefore get a natural isomorphism between and . Therefore, now onwards we call the (even) Orthogonal Grassmannian (as defined above for a -dimensional vector space ) the Orthogonal Grassmannian.

Let be the Plücker embedding (where denotes the Grassmannian of all -dimensional subspaces of ). Thus is a closed subvariety of the projective variety , and hence inherits the structure of a projective variety.

We take (resp., ) to be the subgroup of consisting of those elements that are upper triangular (resp., lower triangular) with respect to the basis and the subgroup of consisting of those elements that are diagonal with respect to . It can be easily checked that is a maximal torus of ; and are the Borel subgroups of which contain . The group acts transitively on , and the -fixed points of under this action are easily seen to be of the form for in , where is the set of subsets of of cardinality satisfying the following two conditions: (i)for each , , the subset contains exactly one of , , and (ii)the number of elements in the subset that exceed is even.

We write for the set of all -element subsets of . There is a natural partial order on and so also on : if and only if ,…, . For , define the complement of as , and denote it by .

The -orbits (as well as -orbits) of are naturally indexed by its -fixed points: each -orbit (as well as -orbit) contains one and only one such point. Let be arbitrary, and let denote the corresponding -fixed point of . The Zariski closure of the - (resp., -) orbit through , with canonical reduced scheme structure, is called a Schubert variety (resp., opposite Schubert variety) and denoted by (resp., ). For , the scheme-theoretic intersection is called a Richardson variety. Each -orbit (as well as -orbit) being irreducible and open in its closure, it follows that -orbit closures (resp., -orbit closures) are indexed by the -orbits (resp., -orbits). Thus the set becomes an indexing set for the Schubert varieties in , and the set consisting of all pairs of elements of becomes an indexing set for the Richardson varieties in . It can be shown that is nonempty if and only if ; that for , if and only if .

3. Statement of the Problem and the Strategy of the Proof

3.1. Initial Statement of the Problem

The problem that is tackled in this paper is this: given a -fixed point on a Richardson variety in , compute the initial ideal, with respect to some convenient term order, of the ideal of functions vanishing on the tangent cone to the Richardson variety at the given -fixed point. The term order is specified in Section 3.4, and the answer is given in Theorem 7.

For the rest of this paper, , and are arbitrarily fixed elements of such that . So, the problem tackled in this paper can be restated as follows. Given the Richardson variety in and the -fixed point in it, find the initial ideal of the ideal of functions vanishing on the tangent cone at to , with respect to some conveniently chosen term order. The tangent cone being a subvariety of the tangent space at to , we first choose a convenient set of coordinates for the tangent space. But for that we need to fix some notation.

3.2. Basic Notation

For this subsection, let us fix an arbitrary element of . We will be dealing extensively with ordered pairs , , such that is not and is an entry of . Let denote the set of all such ordered pairs, and set , , , , , and . We will refer to as the diagonal.

Figure 1 shows a drawing of . We think of and in as row index and column index, respectively. The columns are indexed from left to right by the entries of in ascending order and the rows from top to bottom by the entries of in ascending order. The points of are those on the diagonal, the points of are those that are (strictly) above the diagonal, and the points of are those that are to the South-West of the polyline captioned “boundary of ”—we draw the boundary so that points on the boundary belong to . The reader can readily verify that and for the particular picture drawn. The points of indicated by solid circles form an extended -chain (see Figure 1); the definition of an extended -chain is given later in Section 3.5.

Figure 1 

Figure 2 

We will be considering monomials in some of these sets. A monomial, as usual, is a subset with each member being allowed a multiplicity (taking values in the nonnegative integers). The degree of a monomial has also the usual sense: consider the underlying set of the monomial and look at the multiplicity with which each element of this underlying set appears in the monomial; the degree of the monomial is the sum of these multiplicities. The intersection of any two monomials in a set also has the natural meaning; it is the monomial whose underlying set is the intersection of the underlying sets of the given two monomials and the multiplicity of each element being the minimum of the multiplicities with which the element occurs in the two given monomials.

Given any two monomials and consisting of elements of , let set and set denote the underlying sets of and , respectively. We say that (as monomials) if set , and the multiplicity with which every element occurs in the monomial is less than or equal to the multiplicity with which the same element occurs in the monomial . Given two monomials and consisting of elements of such that (as monomials), we can define a monomial called the “monomial minus” of from (denoted by ) as follows. Take any element of . If the multiplicity with which occurs in is and the multiplicity with which occurs in is , then the multiplicity with which occurs in the monomial is . And any element in occurs in the monomial with the same multiplicity with which it occurs in . This finishes the description of .

Remark 1. Note that in this subsection, and was any element of , was not necessarily in . In particular, all the above basic notations will hold true, if we take as well.

3.3. The Ideal of the Tangent Cone to at

Let be the element of which was fixed at the beginning of this section. Consider the matrix of size whose columns are numbered by the entries of and the rows by ; the rows corresponding to the entries of form the identity matrix, and the remaining rows form a skew symmetric matrix whose upper half entries are variables of the form , where .

Let be the Plücker embedding (where denotes the Grassmannian of all -dimensional subspaces of ). For in , let denote the corresponding Plücker coordinate. Consider the affine patch of given by . The affine patch of the Orthogonal Grassmannian is an affine space whose coordinate ring can be taken to be the polynomial ring in variables of the form with .

For , consider the submatrix of the above mentioned matrix given by the rows numbered and columns numbered . Such a submatrix is of even size and is skew-symmetric about antidiagonal, therefore its determinant is a square. The square root, which is determined up to sign, is called the Pfaffian. Let denote this Pfaffian. Set . From [10] we can deduce a set of generators for the ideal of functions on vanishing on (see Section 4.2 of [9] for the special case of the Schubert varieties). The following equation gives the generators: We are interested in the tangent cone to at or, what is the same, the tangent cone to at the origin. Observe that is a homogeneous polynomial. Because of this, itself is a cone and so is equal to its tangent cone at the origin. The ideal of the tangent cone to at is therefore the ideal in (1).

3.4. The Term Order

We now specify the term order on monomials in the coordinate functions (of the tangent space to at the torus fixed point ; it is easy to see that coordinate ring of the tangent space to at is the polynomial ring in variables of the form , where ) with respect to which the initial ideal of the ideal of the tangent cone is to be taken.

Let be a total order on satisfying all of the following conditions.(i), if , , and the row indices of and are equal. (ii), if , , the row indices of and are equal, and the column index of exceeds that of . (iii) if , , and the row index of is less than that of . (iv), if , , and the column indices of and are equal. (v), if , , the column indices of and are equal, and the row index of exceeds that of . (vi), if , , and the column index of is less than that of . Note here that the first conditions above are the same as the conditions put on the total order as mentioned in Section 1.6 of [3]. Recall that in the paper [3], initial ideals of ideals of tangent cones at torus fixed points to the Schubert varieties in Orthogonal Grassmannians were computed, and the paper [3] did not deal with the Richardson varieties. The last conditions above arise in this paper as an addition to the conditions put on the total order (as mentioned in Section 1.6 of [3]), because here we are dealing with the Richardson varieties in .

Let be the term order on monomials in given by the homogeneous lexicographic order with respect to .

Remark 2. The total order on satisfying the 6 properties mentioned above can be realized as a concrete total order on if we put the following extra condition on it.Let , and denote the row index of , the row index of , the column index of , and the column index of , respectively. If , , , and , then when and when .

3.5. Extended -Chains and Associated Elements of

For this subsection, let be an arbitrarily fixed element of (not necessarily an element of , unless otherwise stated). For elements of , we write if and (Note that these are strict inequalities.). A sequence of elements of is called an extended -chain. Note that an extended -chain can also be empty. Letting to be an extended -chain, we define and . We call (resp., ) the positive (resp., negative) parts of the extended -chain . We call an extended -chain positive (resp., negative) if (resp., ). The extended -chain is called nonempty if it has at least one element in it. Note that if we specialize to the case when , then whatever is called a -chain in Section of [9] is a positive extended -chain over here. To every extended -chain , we will now associate subsets and of (each of even cardinality), but for that we first need to fix some notation and recall certain terminology from [3, 9].

Definition 3 (Pr and Pro). Given any subset of , let denote the monomial whose underlying set is given by is a vertical or horizontal projection of some element of (both vertical and horizontal, as defined in Section 2.3 of [9]), and the cardinality of any element of this underlying set inside the monomial equals the number of elements in whose one of the projections (vertical or horizontal) is . For in , define . The involution is just the reflection with respect to the diagonal . For a subset of , the symbol has the obvious meaning. One calls if . Given any symmetric subset of , let one denote by the set , by the set , and by the monomial formed by taking the union of the subset with the monomial . Let one make the definition of more precise. The multiplicity with which any element occurs in the monomial is equal to the sum of the multiplicities with which the element occurs in the subset and the monomial . So for any symmetric subset of , is a monomial consisting of elements from the diagonal. Similarly for any subset of , is also a monomial consisting of elements from the diagonal.

Let . Let be a positive extended -chain in . Two consecutive elements and of are said to be connected if the following conditions are both satisfied: (i)their legs are “intertwined”, that is, ,(ii)the point belongs to . Consider the coarsest equivalence relation on the elements of generated by the above relation. The equivalence classes of with respect to this equivalence relation are called the connected components of the -chain .

For any in , the elements and of the diagonal are called, respectively, the vertical and horizontal projections of . Given the positive extended -chain as above, let be the monomial of associated to defined as follows.

First suppose that is a connected -chain in . Observe that if there is at all an integer () such that the horizontal projection does not belong to , then . Define

For a -chain that is not necessarily connected, let be the partition of into its connected components, and set

Note that even if we take to be in (and not merely in ) and define for any positive extended -chain exactly in the same way as we did above, there is no logical inconsistency. Hence we extend the definition of to any positive extended -chain , where . Clearly is a symmetric subset of . Hence the monomial is well defined for any positive extended -chain , where .

Definition 4 (the flip map ). For any and any element , let be the element of given by . So is an invertible map from to (note here that if , then needs not always to belong to ); let one denote the inverse of by . The map naturally induces an invertible map from the set of all monomials in to the set of all monomials in . One continues to call the induced map and its inverse .

Definition 5 (the subsets and of ). Given any non-empty extended -chain , one will now associate subsets and of (each of even cardinality) with it. Let Similarly, let It is an easy exercise to check that and thus defined are actually subsets (not monomials) of and that each of them has even cardinality.

Definition 6 (elements of   associated  with   and  ). For this definition, we let to be an arbitrary element in . Note that given any subset of of even cardinality, one can naturally associate an element of to it by removing those entries from which appear as column indices in the elements of the set and then adding to it the row indices of all the elements of . It is easy to check that the resulting element actually belongs to . One denotes the resulting element by . If is empty, then is taken to be itself.
Let and . These are the two elements of that we can naturally associate with the subsets and of .

3.6. The Main Theorem and a Strategy of the Proof

Recall that the ideal of the tangent cone to at is the ideal given by (1). Let be as in Section 3.4. For any element ; let denote the initial term of with respect to the term order . We define to be the ideal inside the polynomial ring . For any monomial in , let us denote by the product of all the elements , where runs over the elements in with multiplicities.

Let denote the set such that either (i) or (ii) below holds}.

(i) is nonempty, and . (ii) is nonempty, and .

The main result of this paper is the following.

Theorem 7. .

Example 8. Let , , and . Clearly then . It can be verified that the set consists of elements of the following types.(i)Any extended -chain in which contains the element . (ii)Any extended -chain in which contains the element and a nonempty positive part. (iii)All positive extended -chain in    except the singleton--chain .
One can therefore determine the corresponding initial ideal using Theorem 7.

Remark 9. It follows from the statement of Theorem 7 that the set of all monomials in which contain at least one extended -chain such that form a vector space basis of the initial ideal over the field ; see, for example, Example 8. In the special case when the Richardson variety is a Schubert variety, it is easy to see that the previous statement of this remark says exactly what has been said in the main theorem (Theorem ) of [3].

Remark 10. Since , it follows that the dimension as a vector space of the graded piece of of degree equals the cardinality of the set of all monomials in of degree which do not contain any extended -chain such that .
Since and have the same Hilbert function, the same is true with (in the previous statement) replaced by . Hence if denotes the coordinate ring of the tangent cone to at , then the dimension as a vector space of the homogeneous piece of of degree equals the cardinality of the set of all monomials in of degree which are such that for every extended -chain in the monomial, and .

From Remark 10, the following corollary is immediate (the proof is similar to that of Corollary of [9]).

Corollary 11. The multiplicity of at equals the number of monomials in of maximal cardinality that are square-free and are such that, given any extended -chain in any of these monomials, one has: and .

We now briefly sketch the proof of Theorem 7 (omitting details, which can be found in Section 8). In order to introduce the main combinatorial objects of interest and outline a strategy of the proof, we will first need to prove that the set , and this proof will follow from whatever is said in Remark 12.

Remark 12. Let be a nonempty extended -chain in such that . If is nonempty and , then it can be proved that , the proof being exactly the same as that in Section 4 of [3]. Then since is an ideal and , it follows that .
If is nonempty and , look at , where is the flip map as defined in Definition 4 from the set of all monomials in to the set of all monomials in . Then is a positive extended -chain in . We need to prove that , for which it is enough to show that . To prove that , we will proceed in a way equivalent to the proof done in Section 4 of [3]. But there is a subtle difference between what is proved in Section 4 of [3] and what we are going to prove here; namely, Whatever was proved in Section 4 of [3] can be rephrased in the language of this paper as “Every positive extended -chain satisfying the property that belongs to the initial ideal of the ideal of the tangent cone,” but here we are going to prove that “Every negative extended -chain satisfying the property that belongs to the initial ideal of the ideal of the tangent cone.”
Because of this subtle difference, we need to construct certain gadgets for negative extended -chains, which will play a role similar to the role of the objects like the new forms, and corresponding to positive extended -chains (For positive extended -chains, such objects are already defined in [3]). This construction is given in the following paragraph.
Consider the positive extended -chain . We can construct new forms, and for in the same way as they were constructed in [3], note here the fact that may or may not belong to does not really affect the construction of the new forms, and for . Then we apply the map to these objects constructed for , the resulting objects are the analogues of the new forms, and for the negative extended -chain . We apply a similar treatment to any other monomial related to that we happen to encounter if we replace the “-chain ” in Section 4.2 of [3] with “the positive extended -chain .”
In Section 2.4 of [3], an element of corresponding to any -chain (the notion of a -chain being as in Section 1.7 of [3]) has been defined. The analogous element of for the negative extended -chain (we call it here) can be obtained from by following the natural process: the column indices of elements of occur as members of ; these are replaced by the row indices to obtain the desired element of for . It is easy to check that belongs to and that . Since we already have that , it follows that . These facts about will be needed to produce an analogue of the main proof of [3] in our present case. To be more precise, these facts about give the analogues of Propositions and of [3], and these two propositions had been used quite crucially inside the main proof of [3].
With all these analogues constructed for negative extended -chains, we can proceed in an “equivalent” manner (Here, by “equivalent” we mean keeping track of the subtle difference as mentioned above and working accordingly.) as in the paper [3] and end up proving the desired fact, namely, .