Combinatorics of -orbits and Bruhat–Chevalley order on involutions
Samara State University
Samara, 443011, Ak. Pavlova, 1, Russia,
1. Introduction and main results
1.1. Let be the symmetric group on letters. Bruhat–Chevalley order on is fundamental in a multitude of contexts. For example, it describes the incidences among the closures of double cosets in the Bruhat decomposition of the general linear group . An interesting subposet of Bruhat–Chevalley order is induced by the involutions, i.e., the elements of order 2 of (we denote this subposet by ). Activity around was initiated by R. Richardson and T. Springer [RichardsonSpringer], who proved that the inverse Bruhat–Chevalley order on encodes the incidences among the closed orbits under the action of the Borel subgroup on the symmetric variety .
The poset of involutions was also studied by F. Incitti [Incitti1], [Incitti2] from a purely combinatorial point of view. In particular, he proved that this poset is graded, calculated the rank function and described the covering relations. In [BagnoCherniavsky], E. Bagno and Y. Chernavsky present another geometrical interpretation of the poset , considering the action of standard Borel subgroup (i.e., the group of upper-triangular invertible matrices) of on symmetric matrices by congruence. Note that all geometric interpretations deal with the closures of orbits for various actions of the Borel subgroup. The purpose of the paper is to incorporate coadjoint orbits into the picture.
Let be the space of strictly upper-triangular matrices and its dual space. Since acts on by conjugation, one can consider the dual action of on . To each involution one can assign the -orbit (see Subsection Combinatorics of -orbits and Bruhat–Chevalley order on involutions for precise definitions). Our main result is as follows.
Theorem 1.1. Let . The orbit is contained in the Zariski closure of if and only if with respect to Bruhat–Chevalley order.
Note that in [Melnikov1], [Melnikov2], [Melnikov3] A. Melnikov described the incidences among the closures of-orbits on the variety of upper-triangular -nilpotent matrices in combinatorial terms of so-called link patterns and rook placements. (In [BoosReineke], M. Boos and M. Reineke generalize the results of Melnikov to all 2-nilpotent matrices; see also B. Rothbach’s paper [Rothbach].) In some sense, our results are “dual” to Melnikov’s results.
The paper is organized as follows. In the rest of this Section, we define orbit associated to involution from the perspective of representation theory, combinatorics and geometry. Namely, in Subsection Combinatorics of -orbits and Bruhat–Chevalley order on involutions, we give precise definitions and explain the role of orbits in A.A. Kirillov’s orbit method in representation theory of the unipotent radical of . In Subsection Combinatorics of -orbits and Bruhat–Chevalley order on involutions, we briefly recall Melnikov’s results and define the partial order on in combinatorial terms in the spirit of [Melnikov2]. Then, we formulate Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions claiming that encodes the incidences among the closures of , . In Subsection Combinatorics of -orbits and Bruhat–Chevalley order on involutions, we formulate Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions claiming that the restriction of Bruhat–Chevalley order to coincides with . Next, in Subsection Combinatorics of -orbits and Bruhat–Chevalley order on involutions, we present a conjectural approach based on the geometry of tangent cones to Schubert varieties.
In Section Combinatorics of -orbits and Bruhat–Chevalley order on involutions, we prove Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions (see Propositions 2 and LABEL:prop:only_if). In Subsection LABEL:sst:Bruhat, using Incitti’s results, we prove Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions. This concludes the proof of our main result. Section LABEL:sect:proofs_Lemmas contains the proofs of technical (but important) Lemmas used in the proof of Proposition LABEL:prop:only_if. Finally, in Section LABEL:sect:remarks, we present a formula for the dimension of (see Proposition LABEL:prop:dim_Omega). We also formulate a conjecture about the closure of and check it in some particular cases (see Subsection LABEL:sst:closure_conj). A short announcement of our results was made in [Ignatev].
Acknowledgements. A part of this work was carried out during my stay at Moscow State University. I would like to thank Professor E.B. Vinberg for his hospitality. Financial support from RFBR (grant no. 11–01–90703-mob_st) is gratefully acknowledged.
I am very grateful to A.N. Panov for useful discussions. I would like to express my gratitude to E.Yu. Smirnov. He was the first to explain me the role of rook placements in combinatorics of the orbit closures. I also would like to thank A. Melnikov for suggesting the idea of Conjecture LABEL:conj_closure. I thank two anonymous referees for useful comments on a previous version of the paper.
1.2. Let be the general linear group, its standard Borel subgroup (etc.). Let be the unitriangular group (i.e., the group of upper-triangular matrices with ’s on the diagonal). Group acts on by conjugation, so the dual action of on is induced. For and , is defined by
Let denote the orbit of under this action. Let denote the orbit of under the action of . (Clearly, .)
In 1962, Kirillov showed [Kirillov1] that there is a bijection between the set of -orbits on and the set of equivalence classes of unitary irreducible representations of in Hilbert spaces. (The proof was adapted for unipotent groups over finite fields by D. Kazhdan in [Kazhdan1].) Further, it turned out that all the principle questions about representations can be answered in terms of orbits (see [Kirillov2] for the details). However, a complete description of is unknown and seems to be a very difficult problem.
An element satisfying is called an ivolution. Let be the set of involutions of . To one can assign the orbits of the groups and by the following rule. Write as a product of disjoint cycles: , where for and for . Denote
and put . Clearly, is a basis of . Here for , where is the usual matrix unit. Hence one can consider the dual basis of . Now, to each map one can assign the -orbit by putting , where
(If , then and .) We say that is associated with and . Set for all , and . (In other words, .) Lemma 1 shows that , where the union is taken over all maps .
It turned out that almost all -orbits on studied so far are associated with involutions.
Example 1.2. i) Being an orbit of a connected unipotent group on an affine variety, any -orbit is a Zariski-closed irreducible subvariety of . Let be an orbit of maximal dimension (such an orbit is called regular). Then either or for some (in the last case must be even). Here , , and . Conversely, all ’s are regular [Kirillov1, §9, Example 2].
ii) An orbit is called subregular if it has the second maximal dimension. Pick and put to be the involution such that
Then is subregular for all . Subregular orbits were described by Panov in [Panov].
iii) Let . The orbit of is called elementary. Evidently, it is associated with the involution . Elementary orbits are described in [Mukherjee1].
Thus, orbits associated with involutions play an important role in representation theory. (See [Andre1], [Andre2], [AndreNeto1], [Ignatev1] and [Ignatev2] for further examples and generalizations to other unipotent algebraic groups.) They were completely described by Panov in [Panov]. In particular, for a given orbit , he presented the set of equations defining this orbit as a closed subvariety of . On the contrary, -orbits are not closed, so the natural question arises: given two orbits and , , when ? (Here denotes the Zariski closure of a subset .) By Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions, this occurs if and only if , where denotes Bruhat–Chevalley order.
1.3. Let be the variety of upper-triangular matrices of square zero:
Group acts on by conjugation. For a given , let denote the orbit of under this action. To one can assign the orbit by the following rule. Write as a product of disjoint cycles: , where for and for . Denote by the matrix of the form , and put . By [Melnikov1, Theorem 2.2], one has
To each one can also assign the matrix by putting
where acts on a matrix by replacing all entries of the first rows and the last columns by zeroes. Let us define a partial order on . Given , we put if , i.e., for all .
Example 1.3. Let , , . Then
so and .
Remark 1.4. Note that this partial order has an interpretation in terms of so-called rook placements. Namely, can be treated as a rook placement on the triangle board with boxes labeled by pairs , : by definition, there is a rook in the th box if and only if . Then is just the number of rooks located non-strictly to the South-West of the th box.
As above, let be the closure of a subset with respect to Zariski topology. By [Melnikov2, Theorem 3.5], one has the following nice combinatorial description of the orbit closures in :
In [Melnikov3], an interpretation of this result in terms of link patterns is given.
Now, let be the space of strictly lower-triangular matrices (with zeroes on the diagonal). We can identify it with by putting
Thus, in the sequel we identify with . Note that under this identification, for all , and , where denotes the strictly lower-triangular part of , that is
Let . Then is identified with , so is identified with , where denotes the transposed matrix to . In fact, our goal is to describe in combinatorial terms. To do this, let us define another partial order on . Given , , put if , i.e., for all . Here is the matrix defined by the rule
As above, acts on a matrix by replacing all entries of the first rows and the last columns by zeroes.
Example 1.5. Let , , . Then
so and .
Remark 1.6. Of course, this partial order has an interpretation in terms of rook placements. Namely, can be treated as a rook placement on the triangle board with boxes labeled by pairs , : by definition, there is a rook in the th box if and only if . Then , , is just the number of rooks located non-strictly to the South-West of the th box.
The following theorem plays a key role in the proof of the main result of the paper (cf. (1)).
Theorem 1.7. Let be involutions in and the corresponding -orbits in . Then
The proof will be presented in the next Section (see Proposition 2 for the proof of “only if” direction and Proposition LABEL:prop:only_if for the proof of “if” direction).
Remark 1.8. Note that there is no natural analogue of the variety in the space . Actually, one can put
Clearly, this subset of is stable under the action of , but it is neither open nor closed, if . (For , .) Indeed, it contains the orbit , where , (as in Example Combinatorics of -orbits and Bruhat–Chevalley order on involutionsi)). It follows from [Kirillov1, §9, Example 2] and Lemma 1 that belongs to if and only if for all . Here
Hence is an open subset of , so and is not closed.
On the other hand, suppose is open. Consider . Then must be an open subset of . However, Lemma 1 together with [Panov, Theorem 1.4] imply that
which is obviously not an open subset of , a contradiction. Note, however, that is an irreducible constructive subset of (as a union of orbits containing a dense subset of ). Note also that, unlike of the adjoint case considered by Melnikov, the closure of a given , , is not a subset of (see Subsection LABEL:sst:closure_conj for a conjectural description of ).
1.4. Recall that the Bruhat–Chevalley order on is defined in terms of the inclusion relationships of double cosets in . Namely, , where denotes the permutation matrix corresponding to . Let . By definition, if . Let be a reduced expression of as a product of simple reflections , , and . It’s well-known that
Further, let be an arbitrary – matrix with at most one in every row and every column. Denote by the matrix such that
(see the previous Subsection for the definition of ).
Remark 1.9. Notice that is just the number of rooks located non-strictly to the South-West of the th box. As above, for a given matrix , let denote the strictly lower-triangular part of . Then .
Let , . Then (see, e.g., [Proctor])
Suppose , . It follows immediately from Remark Combinatorics of -orbits and Bruhat–Chevalley order on involutions that implies . In fact, the second ingredient of the proof of Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions is the fact that these conditions are equivalent, i.e., the order on induced by Bruhat–Chevalley order coincides with .
Theorem 1.10. Let be involutions in . Then
The proof based on the computing the covering relations for and on Incitti’s results is presented in Subsection LABEL:sst:Bruhat. Note that this Theorem together with Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions imply our main result.
1.5. Before starting with the proof of Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions, we will briefly describe another (conjectural) approach to orbits associated with involutions in terms of tangent cones to Schubert varieties. Since, the flag variety can be decomposed into the union , where is called the Schubert cell. By definition, the Schubert variety is the closure of in with respect to Zariski topology. Note that is contained in for all . One has if and only if . Let be the tangent space and the tangent cone to at the point (see [BileyLakshmibai] for detailed constructions); by definition, and if is a regular point of , then . Of course, if , then .
Let be the tangent space to at . It can be naturally identified with in the following way. Since , is isomorphic to the factor , where is the Lie algebra of and is the Lie algebra of . In turn, is naturally isomorphic to . Next, acts on by left multiplications. Since is invariant under this action, the action on is induced. One can easily check that this action coincides with the action of on defined above [Kirillov3, Section 3, Theorem 1]. Further, the tangent cone is -invariant, so it splits into a union of -orbits.
It is well-known that is a subvariety of of dimension [BileyLakshmibai, Chapter 2, Section 2.6]. Let . is irreducible as the closure of an orbit. By Proposition LABEL:prop:dim_Omega, , so is an irreducible component of of maximal dimension. For , for all . (See [PanovEliseev] for an explicit description of tangent cones.) Unfortunately, we can not prove the irreducibility of for all for an arbitrary . On the other hand, we do not know counterexamples to the equality . This allows us to formulate
Conjecture 1.11. Let be an involution. Then the closure of the -orbit coincides with the tangent cone to the Schubert variety at the point .
Note that this conjecture implies that if , then .
2. Proof of the Main Theorem
2.1. The goal of this Subsection is to prove the “only if” direction of Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions. Fix an involution . Recall notation from Subsection Combinatorics of -orbits and Bruhat–Chevalley order on involutions. Let be the subgroup of diagonal matrices. Clearly, . (In other words, for a given , there exist unique , such that .)
Lemma 2.1. One has111Cf. [Melnikov1, Subsection 3.3]. .
Proof. Let be a map. If , then , so . On the other hand, let be an element of . Then there exist , such that , so , where . Thus, .
Lemma 2.2. Let . Then for all .
Proof. Fix a map . Lemma 1 shows that it’s enough to check that if , , then for all , because . Pick an element . It’s well-known that there exist such that
where (the product is taken in any fixed order). Hence we can assume for some , . Then
Hence if and , then . If (and so ), then the th row of is obtained from the th row of by adding the th row of multiplied by . Similarly, if (and so ), then the th column of is obtained from the th column of by subtracting the th column of multiplied by . In both cases, , as required.
Proposition 2.3. Let , be involutions in . If , then222Cf. [Melnikov2, Lemma 3.6]. .
Proof. Suppose , This means that there exists such that . Denote
Clearly, is closed with respect to Zariski topology. Lemma Combinatorics of -orbits and Bruhat–Chevalley order on involutions shows that , so . But , hence , a contradiction.
2.2. Now, let us start with the proof of much more difficult “if” direction of Theorem Combinatorics of -orbits and Bruhat–Chevalley order on involutions.First, we need some more notation (cf. [Melnikov2, Subsections 3.7–3.14]). There exists a natural partial order on . Namely, given , , we set if and ; we also set , if and . Let and , i.e., and . Denote
Suppose exists. Further, suppose that there are no such that , . Then denote by the involution such that
Example 2.4. It’s very convenient to draw the corresponding ’s as rook placements. For example, if , , then , so