Matrix Schubert varieties for the affine Grassmannian Jason Cory Brunson Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Mark Shimozono, Chair Ezra Brown Nicholas Loehr Leonardo Mihalcea January 31, 2014 Blacksburg, Virginia Keywords: Schubert polynomials, affine Grassmannian, matrix Schubert varieties Copyright 2013, Jason Cory Brunson Matrix Schubert varieties for the affine Grassmannian Jason Cory Brunson (ABSTRACT) Schubert calculus has become an indispensable tool for enumerative geometry. It concerns the multiplication of Schubert classes in the cohomology of flag varieties, and is typically conducted using algebraic combinatorics by way of a polynomial ring presentation of the cohomology ring. The polynomials that represent the Schubert classes are called Schubert polynomials. An ongoing project in Schubert calculus has been to provide geometric foundations for the combinatorics. An example is the recovery by Knutson and Miller of the Schubert polynomials for finite flag varieties as the equivariant cohomology classes of matrix Schubert varieties. The present thesis is the start of a project to recover Schubert polynomials for the Borel–Moore homology of the (special linear) affine Grassmannian by an analogous process. ThisrequiresfinitizinganaffineSchubertvarietytoproduceamatrixaffineSchubert variety. This involves a choice of “window”, so one must then identify a class representative that is independent of this choice. Examples lead us to conjecture that this representative is a k-Schur function. Concluding the discussion is a preliminary investigation into the combinatorics of Gröbner degenerations of matrix affine Schubert varieties, which should lead to a combinatorial proof of the conjecture. This work was partially funded by NSF grant DMS-0652641 and DMS-0652648. I am in- debted to the members of the Focused Research Group on “Affine Schubert Calculus: Combi- natorial, geometric, physical, andcomputationalaspects” forextremelyhelpfulconversations and emotional support. Contents Introduction 1 1 Finite flags 3 1.1 Flag and Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Infinite flags 16 2.1 The infinite Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 The affine Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Heralded spaces 28 3.1 Lattices as heralded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Parametrization of heralded spaces . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Stable closure and heralded closure . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Lattice-respecting orbit decompositions . . . . . . . . . . . . . . . . . . . . . 41 4 Ideal generators 46 4.1 Matrix varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Matrix shuffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Abacus slides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Set-theoretic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 Schubert classes 71 iii 5.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Multidegrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Compatibility of rectangle complements . . . . . . . . . . . . . . . . . . . . . 83 5.4 Gröbner geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5 GL-stable Gröbner geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A Root systems and Lie groups 110 A.1 Lie algebras and root systems . . . . . . . . . . . . . . . . . . . . . . . . . . 110 A.2 Kac–Moody groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 B Schubert calculus 118 B.1 Flags and Schubert polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.2 Grassmannians and Schur polynomials . . . . . . . . . . . . . . . . . . . . . 121 B.3 The infinite Grassmannian and symmetric functions . . . . . . . . . . . . . . 124 B.4 The affine Grassmannian and affine Schur functions . . . . . . . . . . . . . . 127 C Example calculations 132 C.1 Slide thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 C.2 Gröbner bases, multidegrees, and punchcards . . . . . . . . . . . . . . . . . . 135 iv Introduction This paper is the start of a project to exhibit a natural geometric construction of the sym- metric function representatives of Schubert classes in the homology of the (Type A) affine Grassmannian. The project will proceed as follows, using n ≥ 2 and w an affine Grassman- nian permutation: (1) Embed the affine Schubert variety X (cid:44)→ Gr into a finite Grassmannian. This is w h,m accomplished via an embedding Gr (cid:44)→ Gr0 of the affine Grassmannian Gr = SLn ∞ SLn SL (C((t)))/SL (C[[t]]) into the infinite Grassmannian Gr0 = lim Gr . n n ∞ ∞←h,m−h h,m (2) Construct the matrix affine Schubert variety Yh,m = π−1(X ) ⊆ M , where π : w w m×h M◦ → Gr isthecolumnspanmapfromtheStiefelmanifoldtotheGrassmannian, m×h h,m and describe its ideal i(Yh,m) ⊆ C[M ]. (Note that Yh,m is stable under the column w m×h w action of the maximal torus T < GL .) h (3) Computethetorus-equivariantcohomologyclass[Yh,m]T ∈ H∗(M ) ∼= Z[x ,...,x ]Sh, w T m×h 1 h identify a torus-equivariant homology representative via the rectangle complement bi- jection ω on (the partition diagrams that index) the basis of Schur polynomials h×(m−h) and verify that each representative is the localization of some common symmetric func- tion f(n−1) = lim ω [Yh,m]T ∈ Λ. We verify the existence of f(n−1) using w ∞←h,m−h h×(m−h) w w the multidegree representation of equivariant cohomology. (4) Perform a Gröbner degeneration of Yh,m into a union of coordinate subspaces and w describe this union in combinatorial terms. We represent these subspaces as m × h arrays with ideal generators marked. We prove part of the claim, and conjecture the rest,thatthecollectionofthesearraysbehaveliketheplanarhistoriesofapermutation. (5) Use the Gröbner combinatorics to explicitly describe f(n−1) and its relationship to the w known homological Schubert class representative, the k-Schur function s(n−1), using w k = n − 1. We conjecture, having several cases empirically validated, that f(n−1) = w s(n−1). If true, this provides a purely geometric, monomial-by-monomial construction w of k-Schur functions. The proof of (2) involves the introduction of a class of heralded spaces in Gr0 that help us ∞ interface between Gr0 and Gr . In particular, we use heraldedness to identify canonical ∞ SLn 1 2 bases in Gr0 for V ∈ Gr from which much of the combinatorics of the affine Weyl group ∞ SLn can be recovered. Meanwhile, we observe in our examples, and clarify into a conjecture, a partial Gröbner degeneration of Yh,m into a union of GL -stable matrix varieties whose w h equivariantclassesareStanleysymmetricpolynomials. Uptotheseconjectures, thisprovides a geometrically natural expansion of s(n−1) into Stanley symmetric polynomials. w Notational conventions vary from chapter to chapter but in most cases remain steady within each chapter. Results are categorized as follows: Most sections contain a theorem, which represents the central result of that section. Propositions are borrowed from other sources and not proved. Lemmata and corollaries precede (respectively, follow) theorems and propo- sitions (and corollaries lemmata) but range from highly technical miscellany to important results. The document is organized as follows: Chapter 1 provides an overview of the classical finite flag varieties, with an emphasis on Grassmannians, upon which foundation Chapter 2 builds an introduction to the affine Grassmannian and its realization as an ind-subvariety of an infinite Grassmannian. Chapter 3 develops machinery (the heralded spaces) that interfaces between these infinite and affine Grassmannians and facilitates several proofs in Chapter 4. That chapter introduces matrix affine Schubert varieties, our main objects of interest, and proposes polynomial generators for their ideal. Finally, Chapter 5 constructs homological polynomial invariants of these varieties and examines their combinatorics. Chapter 1 Finite flags In this chapter we introduce two classical collections X—full flag varieties and Grassman- nians—of subspaces of a finite-dimensional vector space satisfying certain intersectional criteria. Parametrized as a subspace of projective space, X inherits the topology of a man- ifold. We exhibit the embedding of the Grassmannian specifically into the projectivization of a certain vector space, the relations governing which grant it the structure of a projective variety. We then recover X as a quotient of the general linear group of the vector space, conferring smoothness on X as a manifold and leading to its stratification into orbits of the Borel subgroup called Schubert cells. The associated Schubert classes are central to the cohomology (alternatively, the intersection theory) of the Grassmannian, and to that of flags in general. We briefly recall the polynomial ring presentation of the cohomology of X and identify canonical polynomial representatives of the Schubert classes. 1.1 Flag and Grassmannian manifolds Begin with an n-dimensional complex vector space V. Definition 1.1.1. A (k-step) partial flag in V is a filtration of strict inclusions F : ({0} = F ⊂ F ⊂ ··· ⊂ F ⊂ F = V), • 0 i1 ik n of subspaces of dimensions dimF = i . F is a full flag if each i = j and k = n−1. Given ij j • j a dimension sequence I = (i < i < ··· < i ) ⊆ {1,2,...,n − 1}, denote by Fl (V) the 1 2 k I collection of flags arising from that sequence, which we call a flag variety because (as will be seen in the next section) it admits the structure of an algebraic variety. For integers a < b, write [a,b] for the sequence of integers from a to b, inclusive, and [a] := [1,a]. Write Fl (V) := Fl (V) and Gr (V) = Fl (V). The latter collection is a n [n−1] k,n (k) Grassmannian. 3 Jason Cory Brunson Chapter 1. Finite flags 4 Pick a basis (e ,...,e ) for V. Then the standard flag F = Fstd is defined by 1 n • • F = Span(e ,...,e ). i 1 i We will also make use of the opposite flag Fopp defined by • Fopp = Span(e ,...,e ). i n+1−i n Forthesakeoffollowingconvention, inlaterchapterswewilltransitiontoafocusonopposite flags. (cid:5) The general linear group G = GL(V) exactly parametrizes the (ordered) bases of V by the assignment g (cid:55)→ (ge ,...,ge ). The action of G on V induces an action on the ordered bases 1 n of V, and in turn on any flag variety Fl (V). This provides the projection I G → Fl (V). (1.1) I g (cid:55)→ (Span(ge ,...,ge )k ) 1 ij j=1 Write g = ge and denote the full flag F(g) = (F(g) = Span(g ,...,g )). i i • i 1 i Thenextfewparagraphscoverseveralbasicfactsaboutlinearalgebraicgroupsthatestablish a standard framework for what follows. Definition 1.1.2. Given a finite linear algebraic group, any maximal connected solvable subgroup is Borel, and any subgroup containing a Borel subgroup is parabolic. Explicitly, write P = {g ∈ G | ∀i, g ,...,g ∈ Fstd} I ij−1+1 ij ij forthestandard parabolic subgroups of G,includingthestandard Borel subgroup B = P < [n−1] G and the maximal parabolic subgroups P = P < G. Designate the corresponding k (k) opposite subgroups B− and P−. (cid:5) k By this definition, a Borel subgroup is a minimum parabolic subgroup. Proposition 1.1.3 (?). The stabilizer in G of a partial flag is a parabolic subgroup. In particular, the stabilizer of a full flag is a Borel subgroup. Moreover, the parabolic subgroups of G are precisely these stabilizers. Since the action of G is transitive, we have an equivalence of congruence classes ∼ Fl (V) = G/P . (1.2) I I Corollary 1.1.4. Every parabolic subgroup of G is conjugate to a unique P . I Jason Cory Brunson Chapter 1. Finite flags 5 Without loss of generality, we may then focus only on the standard parabolic subgroups, which are the stabilizers of the standard partial flags. Proof. Take a parabolic subgroup P. By Proposition 1.1.3, P = Stab (F ) for some F ∈ G • • Fl (V) (for some I). Find a basis v ,...,v for V that recovers the partial flag Fl (V) under I 1 n I (1.1), and express this basis as a matrix g ∈ G of column vectors. Thus, for all j ∈ [n], v = ge . This provides the expansion j j (cid:88) pv = a v . j jj(cid:48) j(cid:48) j(cid:48)≤j for any p ∈ P and for each j ∈ [n]. Replacing each v with ge and multiplying on the left j(cid:48) j(cid:48) by g−1 yields (cid:88) g−1pge = a e . j jj(cid:48) j(cid:48) j(cid:48)≤j Since g−1pg ranges over the elements of g−1P g, and conversely since any element g(cid:48) ∈ Fstd I • can be obtained this way from p = gg(cid:48)g−1, we have P = g−1P g. I Definition 1.1.5. An algebraic torus (over C) is an algebraic group isomorphic to the direct product of some number of copies of the multiplicative group C∗. The unipotent radical of an algebraic group is the set of unipotent elements u in the radical of G; u ∈ G is unipotent if 1−u is nilpotent. (cid:5) Proposition 1.1.6 ([Ros02]). Let G be any (complex) linear algebraic group. The stardard Borel subgroup B < G contains a unique maximal torus T < B. The quotient N (T)/T G of the normalizer of T in G is a discrete group isomorphic to the Weyl group W associated with G. ∼ InoursettingW = S (seeAppendixA),andwetakeW toacton[n]. Thechoiceofstandard n basis defines an inner product (cid:104)e ,e (cid:105) = δ and provides the faithful matrix representation i j ij G → M . (1.3) n×n g (cid:55)→ ((cid:104)g ,e (cid:105)) j i ij This representation sends B to the subgroup of upper triangular matrices, its unipotent radical U to the subgroup having 1s along the diagonal, and each P to the block–upper I triangular matrices with blocks of sizes i ,i −i ,...,i −i ,n−i . In particular, 1 2 1 k k−1 k (cid:18) (cid:19) GL ∗ P = k . k 0 GL n−k The action of W on Cn given by w(e ) = e defines a canonical embedding W (cid:44)→ G. j w(j) Thought of as a point in M , w = (δ ) is called a permutation matrix. n×n w(j),j Jason Cory Brunson Chapter 1. Finite flags 6 Lemma 1.1.7. Each two-sided Borel orbit of G contains a unique permutation w. This delivers the Bruhat decomposition (cid:71) G = BwB. w∈W See [Eme] for a basis-free proof. We will prove the stronger decomposition (cid:71) G = UwwB, (1.4) − w∈W where Uw = U ∩wU w−1 < B. The proof will make use of one-parameter subgroups, which − − we define here in terms of a general group and will generalize in Section 2.2. Definition 1.1.8. LetGbeany(complex)linearalgebraicgroupandθ : C → Ganalgebraic endomorphism, sothatθ(C)isanontrivialsubgroupofG. Werefertoθ andtoitsimageinG as a one-parameter subgroup. The subgroup of G generated by the images of one-parameter subgroups θ ,...,θ is said to be generated by θ ,...,θ . (cid:5) 1 m 1 m Returning to G = GL , it is straightforward that the one-parameter subgroups n θ : C → GL (1.5) ij n a (cid:55)→ I +aE ij across 1 ≤ i < j ≤ n generate the unipotent radical U < G, and that those across 1 ≤ j < i ≤ n generate U . − Proof. For existence, pick g ∈ G. We shall identify u ∈ U and b ∈ B so that u−1gb−1 = w is a permutation matrix, then verify that u ∈ wU w−1. − Weconstructuandbiteratively, usingone-parametersubgroups, overtheentries(i,j) ∈ [n]2, proceeding from the bottom row to the top row and within each row from left to right. For the general step, suppose that all (i(cid:48),j(cid:48))th entries of g with i(cid:48) > i, or with i(cid:48) = i and j(cid:48) < j, are either 0 or 1, and no two such entries 1 lie in the same row or column. • If some g = 1, j(cid:48) < j, then take θ = θ (−g ) ∈ B to get (gθ) = 0. Now g = 0. ij(cid:48) j(cid:48)j ij ij ij • If all entries of g leftward of (i,j) are zero but some g = 1, i(cid:48) > i, then take i(cid:48)j θ = θ (−g ) ∈ U to get (θg) = 0. Now g = 0. ii(cid:48) ij ij ij • If all entries of g leftward of or below (i,j) are zero, then take θ = θ (−g ) ∈ B to jj ij get (gθ) = 1. Now g = 1. ij ij