Schubert singularities and Levi subgroup actions on Schubert varieties. by Reuven Hodges

Transcription

1 Schubert singularities and Levi subgroup actions on Schubert varieties by Reuven Hodges B.A. Mathematics and Computer Science Goshen College M.S. in Mathematics Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy April 11, 2017 Dissertation directed by Venkatramani Lakshmibai Professor of Mathematics

2 Dedication You see things; and you say Why? But I dream things that never were; and I say Why not? -George Bernard Shaw To my wife, Jiayun; the reason, and inspiration, for all that follows. ii

3 Acknowledgments The first person I would like to thank is my advisor Venkatramani Lakshmibai. For introducing me to the beautiful mathematics of Schubert varieties. For her patience, generosity, and humor that made the work its own reward. For her thoroughness, care, and pursuit of clarity in writing. For giving me an example of the type of mathematician that I hope to someday become. Many thanks to the professors who taught the classes at Northeastern that form the foundations of my mathematical understanding. To Valerio Toledano, Alina Marian, Robert McOwen, Gordana Todorov, Jonathan Weitsman, Egon Schulte, Chris Beasley, Alex Suciu, and many others; thank you for conveying not only the material but the excitement of mathematics. Thank you to my thesis committee Venkatramani Lakshmibai, Ivan Loseu, Gordana Todorov, and Kiyoshi Igusa for all the feedback and suggestions. Thank you to Manoj Kummini, Guido Pezzini, and Eugene Gover for writing letters of recommendation. For the guidance and the lessons. Thank you to my office-mates and friends, Gouri Seal, Chen He, and Jonier Antunes. Thank you for the invaluable mathematical discussions, but also for the commiseration on those days when it is hard being a graduate student. Thank you to my wife, Jiayun Hodges, for being there on the good days and also the bad. For giving your support, encouragement, and love. For inspiring me to be a better mathematician and person. Thank you to my parents and my brothers for always believing in me. For the continual support and kindness. And finally, thank you to mathematics. For allowing me to dream of things that never were, and ask Why not?. Northeastern University April 2017 Reuven Hodges iii

4 Abstract of Dissertation In the first part of this thesis, we study the action of the Levi part, L w, of the stabilizer Q w in GL N (C) (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,n. For the natural action of L w on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plücker embedding), we give a combinatorial description of the decomposition of C[X(w)] into irreducible L w -modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic group Q that is a subgroup of Q w. Using this combinatorial description, we give a classification of all Schubert varieties X(w) in the Grassmannian G d,n for which C[X(w)] has a decomposition into irreducible L w -modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical L w -varieties. These classes include all smooth Schubert varieties, all determinantal Schubert varieties, as well as all Schubert varieties in G 2,N and G 3,N. Also, as an important consequence, we get interesting results related to the singular locus of X(w) and multiplicities at T -fixed points in X(w). In the second part, we construct free resolutions for a large class of closed affine subvarieties of the affine space of symmetric matrices. The class includes the determinental varieties in the space of symmetric matrices, whose free resolutions were studied by Józefiak-Pragacz-Weyman. We use techniques developed by Kummini-Lakshmibai-Sastry-Seshadri, and the geometry of Schubert varieties in the Lagrangian Grassmannian, to construct these resolutions. The approach is algebraic-group theoretic and suggests a framework for extending these results to determinental varieties in other matrix spaces. iv

5 Table of Contents Dedication ii Acknowledgments iii Abstract of Dissertation iv Table of Contents v Disclaimer vi Introduction Levi subgroup actions on Schubert varieties Free resolutions of Schubert singularities Part I: Levi Subgroup Actions on Schubert Varieties Preliminaries Standard monomial theory The straightening algorithm Young diagrams and tableaux Schur and Weyl modules The Littlewood-Richardson rule Decomposition results Blocks, heads, and partitions in degree A partial order on the set of degree r heads The relation between the partial order and the action The skew semistandard tableaux associated to a degree r standard monomial 37 v

6 1.2.5 The module structure and implications Multiplicity consequences of the decomposition Sphericity consequences of the decomposition Singularities and the L-action in degree The smooth and singular locus Multiplicity at a point Part II: Free Resolutions of Schubert Singularities Preliminaries Notation and conventions in type A Notation and conventions in type C Opposite cells in Schubert varieties in the symplectic flag variety Homogeneous bundles and representations Properties of Schubert desingularization in type C Free resolutions Cohomology of homogeneous vector-bundles vi

7 Disclaimer I hereby declare that the work in this thesis is that of the candidate alone, except where indicated in the text, and as described below. Chapter 0.1 and 1.1 to 1.5 are joint work with Venkatramani Lakshmibai, and comprises the paper Levi Subgroup actions on Schubert varieties, induced decompositions of their coordinate rings, sphericity and singularity consequences, 50 pages, available at arxiv: Chapter 0.2 and 2.1 to 2.4 are joint work with Venkatramani Lakshmibai, published in the Pacific Journal of Mathematics as Free resolutions of some Schubert singularities in the Lagrangian Grassmannian, 23 pages, available at arxiv: vii

8 Introduction This thesis is comprised of two parts, both of which study Schubert varieties and their invariants. The first considers the action, by left multiplication, of certain reductive subgroups of the general linear group on Schubert varieties in the Grassmannian. This action in turn induces an action of the reductive subgroup on the homogeneous coordinate ring of the Schubert variety for the Plücker embedding. The primary goal of the first part of this thesis is to give a combinatorial description of the decomposition of the homogeneous coordinate ring into irreducible representations for this induced action. Once this is achieved, it is used to classify the Schubert varieties whose homogeneous coordinate rings have a decomposition that is multiplicity free. This leads to a proof that certain Schubert varieties are in fact spherical varieties for the action, by left multiplication, of the maximal reductive subgroup of their stabilizers. One particularly interesting consequence of this work is that it sheds light on the invariants of the torus fixed points in the Schubert variety. We give an alternate, and quite simple, description of the singular locus as well as a method for partitioning the set of torus fixed points into subsets whose points all occur with the same multiplicity in the Schubert variety. The second part gives a minimal free resolution for the coordinate rings of a certain class of closed subvarieties of the the affine space of symmetric matrices. This class includes the determinental varieties whose free resolutions were studied by Józefiak-Pragacz-Weyman. The free resolutions are constructed by realizing these closed subvarieties as the opposite cells of certain Schubert varieties in the Lagrangian Grassmannian and then using the Kempf-Lascoux-Weyman geometric technique for constructing free resolutions. This geometric technique involves two main steps. The first step is the construction of a specific commutative diagram which realizes a resolution of the singularities of the closed affine subvariety as a vector subbundle of a trivial bundle over a projective variety. Once this commutative diagram is constructed, the free resolution is given in terms of the cohomology groups of wedge powers of the quotient bundle. The second step of the technique involves the 1

9 computation of these cohomology groups. In our case the projective variety is a Grassmannian, and so methods for computing the cohomology groups of vector bundles on Grassmannians are discussed. These two parts of the thesis are introduced in much greater detail in the following two sections. 0.1 Levi subgroup actions on Schubert varieties Let GL N (C) be the group of invertible N N matrices over C. Let B be the Borel subgroup of GL N (C) consisting of upper triangular matrices, and T the maximal torus consisting of diagonal matrices. For 1 d N 1, let G d,n denote the Grassmannian variety consisting of d-dimensional subspaces of C N. The T -fixed points in G d,n are denoted by [e w ] for w I d,n := {i = (i 1,..., i d ) 1 i 1 < < i d N}. The Schubert varieties in G d,n are the Zariski closures of B-orbits through the T -fixed points, with the canonical reduced scheme structure. That is for w I d,n the Schubert variety X(w) := B[e w ]. We denote the homogeneous coordinate ring of X(w) for the Plücker embedding by C[X(w)]. Let R be a reductive group with B R a Borel subgroup. Suppose that X is an irreducible R-variety, then X is a spherical R-variety if it is normal and has a dense open B R -orbit. Our initial goal was to understand when a Schubert variety X(w) is spherical for the left multiplication action of reductive subgroups of GL N (C) that stabilize X(w). Using Proposition we relate this sphericity question to the module structure of C[X(w)] under the induced action of these reductive subgroups. Fix a w I d,n. There is a canonical choice of reductive subgroups of GL N (C) that stabilize X(w). Let Q w be the stabilizer in GL N (C) of X(w); it is clearly a parabolic subgroup of GL N (C). Let L w be the Levi part of Q w, it is a reductive group. We have a natural action of L w on C[X(w)]. The main result of this part is an explicit description of the decomposition of C[X(w)] into irreducible L w -submodules. In fact, our description holds for a much larger class of reductive subgroups, that is, for the Levi part L of any parabolic subgroup Q Q w (cf. Theorem and Corollary ). Though it would be enough to give such a decomposition for L w and deduce the result for any L L w by using branching rules, our procedure is independent of the choice of L; further, using our description for any L we are able to deduce interesting branching rule formulas (cf. Remark ). Our proof uses the standard monomial basis for C[X(w)]. As a graded ring, we have that C[X(w)] r, 2

10 r N has a vector space basis given by the set Std r of all standard monomials of degree r. We give the decomposition for C[X(w)] r as an L-module, which we describe briefly below, in terms of certain Weyl modules associated to L. Given X(w), and a Levi subgroup L as above, our first step involves capturing certain Schubert subvarieties X(), characterized by the property that L is contained in the Levi part of Q, the stabilizer in GL N (C) of X(). A combinatorial description of all such H w := {τ W P τ w} is given in Proposition We refer to these as the heads of type L and denote the subset of heads of type L in H w by Head L. The critical part of this step is showing how L gives rise to a nice partition of the Hasse diagram of H w into disjoint subdiagrams, each containing a unique head of type L. Then for a τ H w we define τ Head L to be the unique head in the disjoint subdiagram containing τ. Finally for Head L we define WStd := {τ H w τ = } and Std := {p τ Std 1 τ WStd }. Thus WStd is the collection of all elements connected to in the disjoint Hasse diagram. This gives us the decompositions Std 1 = Std Head L and C[X(w)] 1 = Std 1 = Std. Head L This decomposition of C[X(w)] 1 would hold for any partition of the Hasse diagram of H w into disjoint subdiagrams each containing a unique head of type L. However, for the partition we consider we have that the Std are in fact irreducible L-submodules; our partition is the unique partition for which this is the case. Thus the above decomposition is an L-module decomposition of the degree one part of C[X(w)]. The second step is to extend this idea to higher degrees. For τ 1,..., τ r H w we define the degree r head of (τ 1,..., τ r ) to be the sequence ( τ1,..., τr ). Then Head L,r is defined to be the set of degree r heads ( 1,..., r ) such that 1 r. As before for Head L,r we define Std := {p τ1 p τr Std r = ( τ1,..., τr )}. Remarkably these Std for Head L,r once again partition the set Std r. This is by no means readily apparent and is due to the fact that given τ 1,..., τ r H w such that τ 1 τ r we have τ1 τr (cf. Proposition ). Thus we have the decompositions Std r = Head L,r Std 3

11 and C[X(w)] r = Std r = Head L,r Std. Unfortunately, the latter decomposition is no longer a L-module decomposition. This is due to the way that the L-action interacts with the standard monomial straightening rule which results in certain Std not being L-stable. Thus in higher degrees our decomposition must be modified. To achieve this we introduce the partial order str on the set of degree r heads Head L,r. For := ( 1,.., r ) Head L,r we define and Std str = {p τ1 p τr Std r str ( τ1,..., τr )} The next step is to show that Std > str = {p τ1 p τr Std r > str ( τ1,..., τr )}. Std str and Std > str are L-stable. To show this we equivalently check that they are Lie(L) stable. Once we have done this we define U to be a L-module complement of inside of. Std > str Std str A final result is required before the description of the degree r decomposition. The U are L- submodules of Std r. As L is equal to a product of general linear groups we should be able to describe any L-module, in particular the U, in terms of tensor products of Weyl modules. As vector spaces U = Std. In view of this we first describe a vector space map from Std to a certain tensor product of (skew) Weyl modules denoted W, using the combinatorics of the standard monomials. This map as well as some character arguments are then used to conclude that the L-module U has the form W D r, where D r is a tensor product of certain determinant representations. We are now ready to state our main result(cf. Theorem ): Theorem Let Head L,r. There exists a L-module U L-module isomorphisms: a. Std str b. Std r = = U Head L,r U. Std > str. such that we have the following c. U = W D r 4

12 The U may not be irreducible L-modules, but their decomposition into irreducibles can now be computed simply by calculating the decomposition of certain tensor products of Weyl modules. This is done in Corollary where we give the explicit decomposition of C[X(w)] into irreducible L-modules. The above decomposition may then be applied to the classification of those Schubert varieties whose coordinate rings have a multiplicity free decomposition into irreducible L-modules. To make the statement of our multiplicity results more tractable we restrict our discussion to L = L w, though our techniques are applicable for any L. With the classification accomplished we then show that the class of Schubert varieties whose coordinate rings have a multiplicity free decomposition into irreducible L w -modules includes all smooth Schubert varieties, all determinental Schubert varieties(cf. Definition ), as well as all Schubert varieties in G 2,N and G 3,N. We note that if C[X(w)] has a multiplicity free decomposition into irreducible L w -modules then X(w), the cone over X(w), is a spherical L w -variety. We then use this to show that X(w) is a spherical L w -variety. We conclude that all smooth Schubert varieties, all determinental schubert varieties(and determinental varieties), as well as all Schubert varieties in G 2,N and G 3,N are spherical L w -varieties. We also get that the coordinate ring of any determinental variety has a multiplicity free decomposition into irreducible L w -modules. As a further important consequence we get some interesting results relating the singularities of X(w) and the degree 1 heads Head. We first give a description of the singular locus of X(w) in terms of maximal degree 1 heads (cf. Corollary (a)). Using this, we show that the set of T -fixed points in the smooth locus of X(w) is precisely {[e τ ], τ WStd w } (cf. Corollary (a)); here, WStd w is the set of all elements of H w connected to w in the disjoint Hasse diagram. Further, we prove that all elements in WStd (the set of all elements of H w connected to in the disjoint Hasse diagram) have associated T -fixed points occurring with the same multiplicty in X(w) (cf. Proposition ). Finally, we show that the set of T -fixed points in the smooth locus of X() contains {[e τ ], τ WStd }, with equality under certain conditions on Q w and Q (cf. Corollary ). Note that a reader interested in these singularity results need only read Section and Section 1.5. Thus this part is really at the crossroads of representation theory, combinatorics, and geometry. We hope to extend the results of this part, using similar techniques, to any Schubert variety in GL N /Q, where Q is any parabolic subgroup, as well as to Schubert varieties in the Lagrangian and Orthogonal Grassmannians. The combinatorial results that one may obtain for the spherical 5

13 Schubert varieties (by virtue of them being spherical varieties) should also be interesting. 0.2 Free resolutions of Schubert singularities The material in this portion of the introduction is from the introduction to the paper [LH15]. In [Las78], Lascoux constructed a minimal free resolution of the coordinate ring of the determinantal varieties (consisting of m n matrices (over C) of rank at most k, considered as a closed subvariety of the mn-dimensional affine space of all m n matrices), as a module over the mn-dimensional polynomial ring (the coordinate ring of the the mn-dimensional affine space). Extending this, in [KLPS15], the authors construct free resolutions for a larger class of singularities, viz., Schubert singularities, i.e., the intersection of a singular Schubert variety and the opposite big cell inside a Grassmannian. Józefiak-Pragacz-Weyman[JPW81] constructed a minimal free resolution of the coordinate ring of the determinantal varieties (in the space of symmetric matrices) as a module over the coordinate ring of the space of symmetric matrices. In this part we construct free resolutions for a certain class of closed subvarieties of the affine space of symmetric matrices, which includes the symmetric determinantal varieties. The technique adopted in [KLPS15] is algebraic group-theoretic, and we follow this approach. We now describe the results of this part. Let n be a positive integer. Let V = C 2n and let (, ) be a non-degenerate skew-symmetric bilinear form on V. Let H = SL(V ) and G = Sp(V ) = {Z SL(V ) Z leaves the form (, ) invariant}. We take the matrix of the form, with respect to the standard basis of V, to be F = 0 J J 0 where J is the anti-diagonal (1,..., 1), in this case of size n. To simplify our notation we will normally omit specifying the size of J as it will be obvious from the context. We may realize Sp(V ) as the fixed point set of the involution σ : H H given by σ(z) = F (Z T ) 1 F 1 (cf. [Ste68]). Denoting by T H and B H the maximal torus in H consisting of diagonal matrices and the Borel subgroup in H consisting of upper triangular matrices, respectively, we have that T H and B H are stable under σ and we set T G = TH, σ B G = BH. σ It is easily checked that T G is a maximal torus in G and B G is a Borel subgroup in G. 6

14 Thus we obtain W G W H where W G, W H denote the Weyl groups of G, H respectively (with respect to T G, T H respectively). Further, σ induces an involution on W H : and Thus we obtain w = (a 1,, a 2n ) W H, σ(w) = (c 1,, c 2n ), c i = 2n + 1 a 2n+1 i W G = W σ H W G = {(a 1 a 2n ) S 2n a i = 2n + 1 a 2n+1 i, 1 i 2n}. (here, S 2n is the symmetric group on 2n letters). Thus w = (a 1 a 2n ) W G is known once (a 1 a n ) is known. We shall denote an element (a 1 a 2n ) in W G by just (a 1 a n ). Further, for w W G, denoting by X G (w) (resp. X H (w)), the the associated Schubert variety in G/B G (resp. H/B H ), we have that under the canonical inclusion G/B G H/B H, X G (w) = X H (w) G/B G, scheme-theoretically. Let P = P n, the maximal parabolic subgroup of G corresponding to omitting the simple root α n, the set of simple roots of G being indexed as in [Bou68]. Let 1 k < r n be positive integers, and let w W k,r (cf. (2.2.2)). Our main result (cf. Theorem ) is a description of the minimal free resolution of the coordinate ring of Y P (w) := X P (w) O G/P, the opposite cell of X P (w), as a module over the coordinate ring of O G/P. For this, as in [KLPS15], we use the Kempf-Lascoux- Weyman geometric technique of constructing minimal free resolutions; in fact we use the same notation and description of this technique as in [KLPS15]. Suppose that we have a commutative diagram of varieties Z A V V q Y A q (0.2.1) where A is an affine space, Y a closed subvariety of A and V a projective variety. The map q is first projection, q is proper and birational, and the inclusion Z A V is a sub-bundle (over V ) of the trivial bundle A V. Let ξ be the dual of the quotient bundle on V corresponding to Z. Then the derived direct image Rq O Z is quasi-isomorphic to a minimal complex F with i+j F i = j 0 H j (V, ξ) C R( i j). 7

15 Here R is the coordinate ring of A; it is a polynomial ring and R(k) refers to twisting with respect to its natural grading. If q is such that the natural map O Y Rq O Z is a quasi-isomorphism, (for example, if q is a desingularization of Y and Y has rational singularities), then F is a minimal free resolution of C[Y ] over the polynomial ring R. In applying this technique in any given situation, there are two main steps involved: one must find a suitable Z and a suitable morphism q : Z Y such that the map O Y Rq O Z is a quasiisomorphism and such that Z is a vector-bundle over a projective variety V ; and, one must be able to compute the necessary cohomology groups. We carry this out for opposite cells Y P (w), w W k,r. As the first step, we establish the existence of a diagram as above, using the geometry of Schubert varieties. We now describe this briefly: We take A = O and Y = Y G/P P (w). Let P be the two-step parabolic subgroup of G, and let w P r k, n be the minimal representative of w P in W P (that is, the set of minimal coset representatives in W, under the Bruhat order, of W/W P, where W P is the Weyl group of P ). Let w := (k+1,.., r, n,.., r+ 1, k,.., 1) S n, the Weyl group of GL n. Let Z P ( w) := Y P (w) XP (w) X P (w)(= (O G/P P/ P ) X P (w) ). Then it turns out that Z P ( w) is smooth (cf ), and is a desingularization of Y P (w). Write p for the composite map Z P ( w) O G/P P/ P P/ P where the first map is the inclusion and the second map is the projection. We have (cf. Theorem ) that p identifies Z P ( w) as a sub-bundle of the trivial bundle O X G/P P r k(w ) over X P (w ), which arises as the restriction r k (to X P r k(w )) of a certain homogeneous vector-bundle on GL n /P r k. Denoting V := X P r k (w ), we get: Z P ( w) O V G/P V q Y P (w) q O G/P (0.2.2) In this diagram, q is a desingularization of Y P (w). Since it is known that Schubert varieties have rational singularities, we have that the map O Y Rq O Z is a quasi-isomorphism, so F is a minimal resolution. As the second step, we need to determine the cohomology of the bundles t ξ over V. In the above situation, V = X P r k(w ) GL n /P r k. As is easily seen, X P r k (w ) is a Grassmannian, namely, GL r /P r k ; further, the bundles t ξ (on GL r /P ) are homogeneous, but are not of Bott-type, r k namely, they are not completely reducible (so one can not apply Bott-algorithm for computing the cohomology). This can be resolved in two ways. In [OR06] the authors determine the cohomology of general homogeneous bundles on Hermitian symmetric spaces, and thus their results can be 8

16 used to determine H (V, t ξ). Alternatively, using a technique from [Wey03], we may compute the resolution of a related space (whose associated homogeneous vector bundle is of Bott-type) from which we retrieve the resolution of the coordinate ring of Y P (w) as a subcomplex. 9

17 Part I: Levi Subgroup Actions on Schubert Varieties 1.1 Preliminaries Standard monomial theory In this section we fix the notation that will be used throughout this part of the thesis. For a more in depth introduction to these topics both [LB15] and [LR08] may be consulted. Fix a positive integer N, and let {e 1,..., e N } be the standard basis of C N. We will do all computations over the field C. We denote by GL N the invertible N N matrices over C. Let T be the standard maximal torus comprised of diagonal matrices, and B the standard Borel subgroup comprised of upper triangular matrices. Let X(T ) := Hom alg.gp (T, C ) be the character group of T ; it is a free abelian group of rank N with a basis {ɛ i, 1 i N}, ɛ i being the character which sends a diagonal matrix diag(t 1,..., t N ) to its i-th entry t i. The elements of X(T ) will be referred to (formally) as weights. We will often simplify our notation by referring to an element of X(T ) by the sequence (a 1,..., a N ), a i Z, which corresponds to the weight a i ɛ i X(T ). A weight (a 1,..., a N ) such that a 1 a N 0 is called a dominant weight(cf. [FH91]). Recall that the set of all dominant weights gives an indexing of the set of all irreducible polynomial representations of GL N. Let V be a finite-dimensional T -module. Then we have the decomposition V = χ X(T ) where V χ is T -weight space consisting of all vectors v V such that t v = χ(t)v, for all t T. If v V χ we say that v has weight χ, and write wt(v) = χ. Let m χ = dim V χ. We define the 10 V χ

18 character of V, denoted char (V ), as the element in Z[X(T )], the group algebra of X(T ), given by char (V ) := m χ e χ. Remark If G is a product, say G = GL M GL N, by a dominant weight of G, we shall mean a sequence (a 1,..., a M, b 1,..., b N ), a i, b j Z, with a 1... a M 0 and b 1... b N 0. Given a GL M -module V and a GL N -module W, consider the GL M GL N -module V W given by the natural diagonal action; we say that V W is a polynomial representation of GL M GL N, if V, W are polynomial representations of GL M, GL N respectively. By char( V W ), we shall mean the element (char V, char W ) of Z[X(T M )] Z[X(T N )], where T M, T N denote the maximal tori, consisting of diagonal matrices in GL M, GL N respectively. More generally for G = GL N1 GL Nr these notions extend in the obvious way. Let Φ be the root system of GL N (cf. [LR08, Chapter 3]). It is the set {ɛ i ɛ j 1 i, j N}, where ɛ i ɛ j is the element of X(T ) which sends diag(t 1,..., t n ) in T to t i t 1 j in C. Our choice of the torus T and the Borel subgroup B induces a set of positive roots Φ + = {ɛ i ɛ j 1 i < j N} and simple roots S = {α i := ɛ i ɛ i+1 1 i N 1}. The Weyl group W of GL N is generated by the simple reflections s αi for α i S and is isomorphic to the symmetric group S N of permutations on N symbols under the map sending s αi to the transposition (i, i + 1). For every 1 d N 1 there is a maximal parabolic subgroup P d of GL N that corresponds to the subgroup of all matrices with a block of size N d d in the lower left corner with all entries equal to zero. P d = 0 N d d GL N Remark There is a bijection between the subsets A {1,..., N 1} and the parabolic subgroups of GL N, given by P A = d {1,...,N 1}\A The one-line notation for elements of W is a sequence (a 1,..., a N ) and corresponds to the permutation that sends i a i. For the parabolic subgroup P A, A {1,..., N 1}, W PA is the subgroup of P d. W generated by {s αi i A}. And W P A is the subset of W made up of minimal representatives, under the Bruhat order, in W of elements of W/W PA. For all 1 d N 1 we have that W P d can be represented as the set {(a 1,..., a d ) 1 a 1 < < a d N}. 11

19 The Grassmannian G d,n is the set of all d-dimensional subspaces of C N. For U G d,n fix a basis {u 1,..., u d } of U and define the map G d,n P( d C N ) U [u 1 u d ] This is the well-known Plücker embedding. Set I d,n := {i = (i 1,..., i d ) 1 i 1 < < i d N}. Then {e i := e i1 e id } i Id,N is the standard basis for d C N. Let {p j } j Id,N be the basis for ( d C N ) dual to {e i } i Id,N. This dual basis gives a set of projective coordinates for P( d C N ), called the Plücker coordinates. These coordinates have a particularly nice description for points in G d,n in terms of determinants: for U G d,n fix a basis {u 1,..., u d } of U as above and let A be the N d matrix with columns u 1,..., u d. Then the Plücker coordinate p j (U) is the d-minor with row indices j 1,..., j d of A. The Plücker embedding equips G d,n with a projective variety structure, realized as the zero set of the well known quadratic Plücker relations. The Grassmannian is a homogeneous space for the action of GL N induced on P( d C N ) by the GL N action on C N and hence d C N. Let e id := e 1 e d, then [e id ] P( d C N ) and the GL N orbit through [e id ] is G d,n, and the isotropy subgroup at [e id ] is precisely. Thus we obtain the P d identification G d,n = GL N. /P d It is no coincidence that I d,n = W P d. The T -fixed points in GL N are } for w W P d. /P d {wp d Under the identification of GL N with G /P d d,n we see that } gets identified with the point {wp d [e i1 e id ] where w is the element of W with first d entries equal to {i 1,..., i d }, which corresponds to (i 1,..., i d ) W P d. Thus the T -fixed points of G d,n are [e i1 e id ] for i I d,n = W P d. It will prove useful to emphasize the homogeneous space identification of G d,n and so subsequently we shall use W P d to index the T -fixed points as well as the Plücker coordinates. Remark Let τ W P d, say τ = (i 1,..., i d ). Denote the integers {1,..., N} \ {i 1,..., i d } by j 1,..., j N d (arranged in ascending order). We identify the Plücker co-ordinate p τ with the element e j1 e jn d in N d C N. Thus, the weight of p τ is ɛ j1 + + ɛ jn d. The weight is given by the sequence χ τ := (χ 1,..., χ N ) where 0 i τ χ i := 1 i / τ for all 1 i N. For w W P d the Schubert variety in G d,n associated to w is XP d(w) := B [e w ] =, the B BwP d /P d orbit closure through the T -fixed point [e w ], equipped with the canonical reduced scheme structure. 12

20 There is a natural partial order on W P d, referred to as the Bruhat order, induced by the partial order on the set of Schubert varieties given by inclusion. For τ := (i 1,..., i d ), w := (j 1,..., j d ) W P d we have τ w if and only if i 1 j 1,..., i d j d. Then [e τ ] X(w) if and only if X(τ) X(w) and this is if and only if τ w. We have the Bruhat decomposition X(w) = B[e τ ]. τ w Note that for the choice of w = (N d + 1,..., N) W P d we have that the Schubert variety X(w) is in fact G d,n itself. Now consider the projective embedding X(w) G d,n P( d C N ). Let C[X(w)] be the homogeneous coordinate ring of X(w) for this projective embedding. As a C-algebra it is generated by p τ, τ w. This follows from the fact that p τ ([e w ]) = δ τ,w, which implies that p τ X(w) 0 if and only if [e τ ] X(w), which occurs if and only if τ w. Thus for τ 1,..., τ r, w W P d with τ i w for all 1 i r, we have p τ1 p τr C[X(w)] r. Definition We define the monomial p τ1 p τr to be standard if τ 1 τ r. It is standard on X(w) if in addition w τ 1. Theorem (cf. [LR08, Theorem ])Monomials of degree r standard on X(w) give a C-basis for C[X(w)] r The straightening algorithm The generation portion of Theorem usually relies on exhibiting an inductive process that takes a nonstandard monomial and writes it as a sum of standard monomials. This is called straightening the nonstandard monomial, and the entire process is referred to as the straightening process. The straightening process on the Grassmannian is comprised of an inductive step usually referred to as a shuffle. Let τ := (i 1,..., i d ), φ := (j 1,..., j d ) W P d with τ φ, that is p τ p φ is not standard. This implies there exists a t, t d such that i m j m, for all 1 m t 1, and i t < j t. Let [τ, φ] denote the set of permutations σ 1, other than the identity permutation, of the multiset {i 1,..., i t, j t,..., j d } such that σ 1 (i 1 ) < < σ 1 (i t ) and σ 1 (j t ) < < σ 1 (j d ). 13

21 Define α σ 1 := (σ 1 (i 1 ),..., σ 1 (i t ), i t+1,..., i d ) and β σ 1 := (j 1,..., j t 1, σ 1 (j t ),..., σ 1 (j d )) (here, for a d-tuple (l 1,..., l d ), (l 1,..., l d ) denotes the d-tuple obtained from (l 1,..., l d ) by arranging the entries in ascending order). Then p τ p φ = ±p α σ 1 p β σ 1. σ 1 [τ,φ] Note that it is possible to keep track of the signs in the above summation but we omit this step since it is not needed for our consideration. It is not difficult to see that either α σ 1 = 0, due to a repeated entry, or α σ 1 > τ. For the same reasons either β σ 1 = 0, due to a repeated entry, or β σ 1 < φ. We will refer to this as the ordering property of the shuffle. A single shuffle is not always sufficient to straighten the monomial p τ p φ. It may be the case that for a σ 1 [τ, φ] the monomial p α σ 1 p β σ 1 is not standard. And so we must apply a shuffle to p α σ 1 p β σ 1. Suppose α σ 1 = (k 1,..., k d ) and β σ 1 = (l 1,..., l d ). Since α σ 1 β σ 1, there is a t, t d such that k m l m, for all 1 m t 1, and k t < l t. Then for a σ 2 [α σ 1, β σ 1] we define (α σ 1) σ 2 := (σ 2 (k 1 ),..., σ 2 (k t ), k t +1,..., k d ) and (β σ 1) σ 2 := (l 1,..., l t 1, σ 2 (l t ),..., σ 2 (l d )). Then p α σ 1 p β σ 1 = ±p (α σ 1 ) σ 2 p (β σ 1 ) σ 2. σ 2 [α σ 1,β σ 1 ] And this process may continue as there may be monomials p (α σ 1 ) σ 2 p (β σ 1 ) σ 2 that are not standard and we must apply another shuffle. However this process will eventually terminate after a finite number of steps, guaranteed by the fact that there are only finitely many degree 2 monomials and the ordering property of the shuffles(cf. [LR08, Chapter 4]). After substituting and combining like monomials we get that p τ p φ = A α,β p α p β with A α,β C, α β ( ) α,β where for each α, β with A α,β 0 we have α = (((α σ 1) σ 2) ) σ M, β = (((β σ 1) σ 2) ) σ M for some M > 0 and some σ M [(((α σ 1) σ 2) ) σ M 1, (((β σ 1) σ 2) ) σ M 1],..., σ 2 [α σ 1, β σ 1], σ 1 [τ, φ]. For a fixed α, β, their description in terms of M and σ 1,..., σ M may not be unique, as a particular standard monomial in the summation may be the result of multiple different chains of shuffles, which is why A α,β may equal integers other than 1, 0, 1. In addition because of the ordering property of the shuffles, for each α, β with A α,β 0 we have that α = (((α σ 1) σ 2) ) σ M > (((α σ 1) σ 2) ) σ M 1 > > α σ 1 > τ and β = (((β σ 1) σ 2) ) σ M < (((β σ 1) σ 2) ) σ M 1 < < β σ 1 < φ. We refer to ( ) as the result of the degree 2 straightening process applied on the nonstandard monomial p τ p φ. 14

22 Finally a degree r nonstandard monomial may be straightened by inductively applying the degree 2 straightening algorithm. That process is the degree r straightening process. To straighten a monomial on X(w) all that is required is to apply the straightening process for the Grassmannian and then to note that in the resulting sum of standard monomials, any that are standard but not standard on X(w) are equal to zero on X(w) (cf. Definition ) Young diagrams and tableaux This section for the most part follows the terminology of [Ful97],[FH91], and [Sta99]. Let λ = (λ 1,..., λ k ) be a collection of nonnegative integers with λ 1 λ k, then for λ := λ λ k we say that λ is a partition of λ. We call the λ i the parts of λ. It will be useful at times to make this notation more succint by rewriting λ = (λ 1,..., λ k ), replacing any maximal chain λ i,..., λ i+j 1 where λ i = = λ i+j 1 = a with a j. We identify a partition λ with its Young diagram, also denoted λ for simplicity of notation, which is a collection of left justified boxes with λ i boxes in the ith row for 1 i k. These boxes are referred to by specifying row and column, with the leftmost column denoted column 1, and the topmost row denoted row 1. Example The partition (4, 2, 2, 1) = (4, 2 2, 1) is identified with the Young diagram. The conjugate partition λ is the partition whose diagram is the transpose of the diagram of λ, or equivalently, it is defined by setting the part λ i := #{j λ j i}. The conjugate partition of (4, 2, 2, 1) from Example is (4, 3, 1, 1). A partition that has all parts equal to the same value is a rectangle, and one that has all parts equal to one of two values is a fat hook. A fat hook with all parts except the first equal to 1 is a hook. The partition (3,3,3,3) is a rectangle, the partition (4,4,4,2,2) is a fat hook, and the partition (5,1,1) is a hook. If we have a second partition µ we write µ λ if the diagram for µ is contained in the diagram for λ, or equivalently, if µ i λ i for i 1. If µ λ we may define the skew diagram λ/µ which is obtained by deleting the leftmost µ i boxes from row i of the diagram λ for each row of λ. The number of boxes in the skew diagram is equal to λ/µ := λ µ. It is important to note here the fact that λ = λ/(0), and so many definitions made for skew diagrams may be specialized to diagrams for partitions. Example The skew diagram for λ/µ = (4, 2, 2, 1)/(2, 1) is 15

23 The π-rotation of a skew diagram λ/µ, written (λ/µ) π, is obtained by rotating λ/µ through π radians. Define λ/ µ to be the skew diagram obtained by deleting all empty rows and columns from the skew diagram λ/µ. If λ (m n ) for m, n positive integers, the (m n )-complement of λ is denoted λ, with λ k = m λ n k+1. If λ (m n ) for m, n positive integers, then (m n )/λ is a skew diagram and ((m n )/λ) π is always a partition and is equal to λ. Example The π-rotation of (4, 2, 2, 1)/(2, 1) is (5, 5, 3, 2)/(4, 2, 2). Let m = 5, n = 5 then the partition (4, 2, 2, 1) (m n ) and we have the (m n )-complement of λ is λ = (5, 4, 3, 3, 1). As noted above the π-rotation of (m n )/(4, 2, 2, 1) is also (5, 4, 3, 3, 1). If λ (m n ) for m, n positive integers, there is a unique shortest lattice path of length m + n dividing the boxes of λ and the boxes of (m n )/λ starting at the bottom-left corner of the rectangle (m n ) and ending at the top-right corner of the rectangle. The m n -shortness of λ is the length of the shortest line segment in this path. Example Let λ = (4, 2, 2, 1) and m = λ 1 = 4, n = λ 1 = 4. Then the lengths of the line segments for the lattice path are (1, 1, 1, 2, 2, 1) and so the 4 4 -shortness of λ is 1. Lemma Let λ, µ be two partitions and let m, n, p, and q be positive integers such that λ (m n ), µ (p q ) and the skew diagrams (m n )/λ and (p q )/µ have no empty rows or columns. If ((m n )/λ) π = ((p q )/λ ) π then m = p, n = q, and λ = µ. Proof. Set γ = ((m n )/λ) π and ν = ((p q )/µ) π. Then γ, ν are partitions. We have γ k = m λ n k+1 for 1 k n and γ k = 0 for k > n. Similarly ν k = p µ q k+1 for 1 k q and ν k = 0 for k > q. By assumption we have that γ = ν so m λ n k+1 = p µ q k+1 for k 1. Suppose n q. Without loss of generality assume that q > n. Then γ q = 0, which implies that ν q = 0, that is that p µ 1 = 0. But this would mean that p = µ 1 which would indicate that (p q )/µ had an empty first row. This is a contradiction of our initial assumption and thus n = q. Now we know that λ n = 0 because (m n )/λ has no empty columns. We also know that µ n = µ q = 0 because (p q )/µ has no empty columns. But then γ 1 = ν 1 implies that m λ n = p µ n. Thus m = p. Now the fact that m = p, n = q trivially implies that λ = µ. 16

24 A tableaux on λ/µ is an assignment of a positive integer to each box of λ/µ. A semistandard (Young) tableaux, often abbreviated SSYT, is a tableaux where the values in each box increase weakly along each row, and increase strictly down each column. A standard (Young) tableaux is a semistandard tableaux where the values in each box increase strictly along each row. Example A tableaux, semistandard tableaux, and standard tableaux on (4, 2, 2, 1)/(2, 1) If we fix a partition λ and a bound M on the maximum value that can be assigned to a box in a tableaux T we may define the schur function s λ by s λ = T a SSYT on λ # of 1 s in T x 1 x # of M s in T M. In the same way for a skew diagram λ/µ we may define the skew schur function s λ/µ. Both the schur functions and the skew schur functions are symmetric functions, and the schur functions form a vector space basis of the ring of symmetric functions in the variables x 1,..., x M. Thus the product of two schur functions, which is itself a symmetric function, can be written as a sum of schur functions s λ s µ = ν c ν λ,µs ν and this is one of many equivalent ways of defining the Littlewood-Richardson coefficients c ν λ,µ. Note that the above sum is over all partitions ν such that ν = λ + µ. The Littlewood-Richardson coefficients are also critical in describing the expansion of the skew schur functions s λ/µ in terms of the schur functions, namely s λ/µ = ν c λ µ,νs ν and the above sum is over all partitions ν such that ν = λ µ. In the special case where for a fixed skew schur function s λ/µ all the c λ µ,ν are either 0 or 1 we say that s λ/µ is multiplicity-free. The reason for this designation will become clear in Section The study of the Littlewood-Richardson coefficients is a rich subject with applications in the decomposition of tensor products of GL N representations in characteristic zero, intersection numbers on the Grassmannian, and the eigenvalues of sums of Hermitian matrices (cf. [FH91], [HL12],[TY10], and [Ful00]). 17

25 We will need a few standard identities whose derivations can be found in [Sta99]. The first are two non-trivial symmetries of the Littlewood-Richardson coefficients with regard to the partitions; The second are two important identities for skew schur functions, c λ µ,ν = c λ ν,µ and c λ µ,ν = c λ µ,ν. ( ) s λ/µ = s λ/ µ ( ) and s λ/µ = s (λ/µ) π ( ) which state that deletion of empty rows and columns of a skew diagram as well as π-rotation do not affect the associated skew schur function. Any skew schur function whose associated skew diagram has no empty rows or columns is called basic, and so ( ) implies that any skew schur function is equivalent to a basic skew schur function. The final result we will need, due to C. Gutschwager[Gut10], and to Thomas and Yong[TY10], is a characterization of all multiplicity-free basic skew schur functions. Theorem The basic skew schur function s λ/µ is multiplicity-free if and only if λ and µ satisfy one or more of the following conditions: a. µ or λ is the zero partition b. µ or λ is a rectangle of m n -shortness 1 c. µ is a rectangle of m n -shortness 2 and λ is a fat hook (or vice versa) d. µ is a rectangle and λ is a fat hook of m n -shortness 1(or vice versa) e. µ and λ are rectangles where m = λ 1, n = λ 1, and λ is the m n -complement of λ Schur and Weyl modules The skew Schur functions s λ/µ are the characters of certain representations of GL n for some n 1, where n is the bound on the entries in the boxes(cf. [FH91]). Given a standard tableaux on the skew diagram λ/µ with the bound on the entries in the boxes equal to d we may define two subgroups of the symmetric group S d 18

26 Row λ/µ := {σ S d σ permutes the enties in each row among themselves} and Col λ/µ := {σ S d σ permutes the entries in each column among themselves}. In the group algebra C[S d ] we introduce two elements, called the Young symmetrizers and Υ λ/µ W := σ Row λ/µ ρ Col λ/µ Υ λ/µ S := σ Row λ/µ ρ Col λ/µ sign(ρ)σρ sign(ρ)ρσ. Let V = C n with standard basis {e 1,..., e n }. The symmetric group S d acts on the dth tensor product V d on the right by permuting the factors, while GL n acts on V on the left and thus diagonally on V d on the left. The fact that this left action of GL n commutes with the right action of S d is the source of Schur-Weyl duality and gives the relationship between the irreducible finite-dimensional representations of the general linear and symmetric groups. The Schur Module S λ/µ (V ) and Weyl Module W λ/µ (V ) are defined to be S λ/µ (V ) := (V d )Υ λ/µ S and W λ/µ (V ) := (V d )Υ λ/µ W. These are GL n representations spanned by all the young symmetrized tensors in V d. In characteristic zero these representations are related by the identity S λ/µ (V ) = W λ /µ (V )[Wey03, Proposition (c)] and for our purposes it will prove to be convenient to focus on the Weyl Modules W λ/µ (V ). Given a tableaux T of λ/µ numbered with {1,..., n} we can associate to T a decomposable tensor λ 1 e T = i=1 e T (,i) where e T (,i) is the tensor product, in order, of those basis vectors whose indices appear in column i. Multilinearity implies that W λ/µ (V ) is spanned by e T Υ λ/µ W as T ranges over all tableau. In fact we may do better, as the following theorem illustrates. Theorem The set {e T T is a semistandard tableaux on λ/µ} is a C-basis for W λ/µ (V ). 19

27 Proof. This is a well known result, and a sketch of the details may be found in [FH91, Exercise 6.15 and 6.19] Note that the above construction works for any Young diagram λ merely by noting that λ has the same diagram as the skew diagram λ/(0). In fact the Weyl Module W λ (V ) := W λ/(0) (V ) is an irreducible GL n representation and any Weyl Module W λ/µ (V ) can be written as a direct sum of these irreducible modules by W λ/µ (V ) = ν W ν (V ) cλ µ,ν ( ) where the c ν λµ are the Littlewood-Richarson coefficients defined in Section 1.1.3, and the direct sum is over all partitions ν such that ν = λ µ. Remark The above decomposition implies that W λ/µ (V ) will be multiplicity free when the skew Schur function s λ/µ is multiplicity free, and the skew diagrams for which this occurs are enumerated by Theorem Definition Let r be a positive integer. Define the GL M representation det r M : GL M C, det r M(g) = (det(g)) r, g GL M. Then det r M is defined to be the dual of det r M. We have the following isomorphism of GL M -modules (W λ (C M )) det r M = W (rm )/λ (C M ) = W ((rm )/λ) π (C M ). ( ) A proof of the first isomorphism may be found in [Mag98, Theorem 6(c)] although the notation used is different from ours, in particular our W λ is denoted S λ. The second isomorphism follows from ( ) The Littlewood-Richardson rule We will need to be able to calculate the value of certain Littlewood-Richardson coefficients in Section 1.3. To do this we recall the Littlewood-Richardson Rule from [Ful97, Section 5]. Given a semistandard tableaux or a semistandard skew tableaux T we define the row word, denoted w row (T ), of the tableaux T to be the entries of T read from left to right and bottom to top. A row word w row (T ) = x 1,..., x r is called a reverse lattice word if in every reversed sequence x r, x r 1,..., x s+1, x s the number i appears at least as often as i + 1 for all i and all 1 s < r. Example Consider the following two skew tableaux 20

28 The associated row words are 2,1,3,2,1,1 and 2,1,3,2,2,1. The first is a reverse lattice word and the second is not because 1,2,2 has more 2 s than 1 s. A semistandard skew tableaux T is called a semistandard Littlewood-Richardson skew tableaux if w row (T ) is a reverse lattice word. A semistandard skew tableaux has weight ν = (ν 1,..., ν m ) if the tableaux has ν 1 1 s, ν 2 2 s,..., and ν m m s. With these definitions we may state the Littlewood-Richardson Rule. Proposition ([Ful97, Proposition 5.3]) The Littlewood-Richardson coefficient c λ µ,ν is equal to the number of semistandard Littlewood-Richardson skew tableaux on the shape λ/µ with weight ν. Lemma Let n 1. a. Let λ = (3 n, 2), µ = (1), and ν = (3 n, 1). Then c λ µ,ν = 1. b. Let λ = (3 n, 1, 1), µ = (1), and ν = (3 n, 1). Then c λ µ,ν = 1. c. Let λ = (3 n, 2), µ = (2), and ν = (3 n 1, 2, 1). Then c λ µ,ν = d. Let λ = (3 n, 2), µ = (1, 1), and ν = (3 n 1, 2, 1). Then c λ µ,ν = 1. Proof. Note that for (c) and (d) when n = 1 we consider ν = (3 0, 2, 1) = (2, 1). (a): By ( ) we have that c λ µ,ν = c λ ν,µ. So according to Proposition to find c λ ν,µ we must calculate the number of semistandard Littlewood-Richardson skew tableaux on the shape λ/ν with weight µ. But λ/ν = (3 n, 2)/(3 n, 1) is a single box and there is clearly only one semistandard Littlewood-Richardson skew tableaux with weight µ = (1), that is with a single 1. Thus c λ ν,µ = 1. (b): This proceeds in exactly the same way as part (a). (c): By ( ) we have that c λ µ,ν = c λ ν,µ. We have that λ/ν = (3 n, 2)/(3 n 1, 2, 1) is two disconnected boxes, and there is only one semistandard Littlewood-Richardson skew tableaux with weight µ = (2), it is the tableaux with a 1 in each box. Thus by Proposition we have c λ ν,µ = 1. (d): Similarly to part (c) we need to calculate the number of semistandard Littlewood-Richardson skew tableaux of shape (3 n, 2)/(3 n 1, 2, 1), but in this case with weight µ = (1, 1). There are two possible fillings with this weight, however only the one with a 1 in the upper right box and a 2 in the lower left box has a row word that is a reverse lattice word. Thus c λ µ,ν = c λ ν,µ =