Schubert Varieties Under a Microscope Sara Billey University of Washington http://www.math.washington.edu/ billey ∼ Summer Institute in Algebraic Geometry August 4, 2005 Famous Quotations Arnold Ross (and PROMYS). ”Think deeply of simple things.” Angela Gibney. Why do algebraic geometers love moduli spaces? “It is just like with people, if you want to get to know someone, go to their family reunion.” Goal. Focus our microscope on a particular family of varieties which are indexed by combinatorial data where lots is known about their structure and yet lots is still open. Schubert Varieties A Schubert variety is a member of a family of projective varieties which is defined as the closure of some orbit under a group action in a homogeneous space G/H. Typical properties: • They are all Cohen-Macaulay, some are “mildly” singular. • They have a nice torus action with isolated fixed points. • This family of varieties and their fixed points are indexed by combinatorial objects; e.g. partitions, permutations, or Weyl group elements. Schubert Varieties “Honey, Where are my Schubert varieties?” Typical contexts: • The Grassmannian Manifold, G(n,d) = GLn/P . • The Flag Manifold: Gln/B. • Symplectic and Orthogonal Homogeneous spaces: Sp2n/B, On/P • Homogeneous spaces for semisimple Lie Groups: G/P . • Homogeneous spaces for Kac-Moody Groups: G/P . • Goresky-MacPherson-Kotwitz spaces. The Flag Manifold n Defn. A complete flag F• = (F1,...,Fn) in C is a nested sequence of vector spaces such that dim(F ) = i for 1 i n. F is determined by an i ≤ ≤ • ordered basis f ,f ,...f where F = span f ,...,f . h 1 2 ni i h 1 ii Example. F = 6e + 3e , 4e + 2e , 9e + e + e , e • h 1 2 1 3 1 3 4 2i The Flag Manifold Canonical Form. F = 6e + 3e , 4e + 2e , 9e + e + e , e • h 1 2 1 3 1 3 4 2i 6 3 0 0 300 0 2 1 0 0 4 0 2 0 020 0 2 0 1 0 = ≈ 9 0 1 1 011 0 7 0 0 1 0 1 0 0 1 0 0 2 1 0 0 0 − 2e + e , 2e + e , 7e + e , e ≈h 1 2 1 3 1 4 1i l (C) := flag manifold over Cn n G(n,k) F n ⊂ k=1 = complete flags F Q { •} = B GL (C), B = lower triangular mats. \ n Flags and Permutations 2 1❤ 0 0 2 0 1❤ 0 Example. F• = 2e1+e2, 2e1+e3, 7e1+e4, e1 h i ≈ 7 0 0 1❤ 1❤ 0 0 0 Note. If a flag is written in canonical form, the positions of the leading 1’s form a permutation matrix. There are 0’s to the right and below each leading 1. This permutation determines the position of the flag F• with respect to the reference flag E = e , e , e , e . • h 1 2 3 4 i Many ways to represent a permutation 0 1 0 0 0 1 1 1 0 0 1 0 1 2 3 4 0 1 2 2 = = 2341 = 0 0 0 1 2 3 4 1 0 1 2 3 1 0 0 0 1 2 3 4 matrix two-line one-line rank notation notation notation table 1234 . ∗ . ∗ = = =(1, 2, 3) . ∗ . ... 2341 diagram of a reduced rc-graph string diagram permutation word The Schubert Cell Cw(E ) in ln(C) • F Defn. Cw(E•) = All flags F• with position(E•,F•) = w = F l dim(E F ) = rk(w[i,j]) { • ∈ F n | i ∩ j } 2 1❤ 0 0 1 0 0 ∗ 2 0 1❤ 0 0 1 0 Example. F• = C2341 = ∗ : C 7 0 0 1❤ ∈ 0 0 1 ∗ ∈ ∗ 1❤ 0 0 0 1 0 0 0 Easy Observations. • dimC(Cw) = l(w)=# inversions of w. • C = w B is a B-orbit using the right B action, e.g. w · 0 1 0 0 b1,1 0 0 0 b2,1 b2,2 0 0 0 0 1 0 b2,1 b2,2 0 0 b3,1 b3,2 b3,3 0 = 0 0 0 1 b3,1 b3,2 b3,3 0 b4,1 b4,2 b4,3 b4,4 1 0 0 0 b4,1 b4,2 b4,3 b4,4 b1,1 0 0 0 The Schubert Variety Xw(E ) in ln(C) • F Defn. Xw(E•) = Closure of Cw(E•) under the Zariski topology = F l dim(E F ) rk(w[i,j]) { • ∈ F n | i ∩ j ≥ } where E = e , e , e , e . • h 1 2 3 4 i 1❤ 0 0 0 1 0 0 ∗ 0 1❤ 0 0 1 0 Example. ∗ X2341(E•) = ∗ 0 0 1❤ ∈ 0 0 1 ∗ ∗ 0 1❤ 0 0 1 0 0 0 Why?. The Main Combinatorial Tool Bruhat Order. The closure relation on Schubert varieties defines a nice partial order. Xw = Cv = Xv v[≤w v[≤w Bruhat order (Ehresmann 1934, Chevalley 1958) is the transitive closure of w <wt w(i) < w(j). ij ⇐⇒ Equivalently, t w < w w(i) < w(j). ij ⇐⇒ Example. Bruhat order on permutations in S3. 321 ❅ 312 ❅ 231 ❅ ❅ 213 ❅ 132 ❅❅ 123 Bruhat order on S4 4 3 2 1 3 4 2 1 4 2 3 1 4 3 1 2 2 4 3 1 3 4 1 2 3 2 4 1 4 1 3 2 4 2 1 3 1 4 3 2 2 3 4 1 2 4 1 3 3 1 4 2 3 2 1 4 4 1 2 3 1 3 4 2 1 4 2 3 2 3 1 4 2 1 4 3 3 1 2 4 1 2 4 3 1 3 2 4 2 1 3 4 1 2 3 4 Bruhat order on S5 (5 4 3 2 1) (5 4 3 1 2) (5 4 2 3 1) (5 3 4 2 1) (4 5 3 2 1) (5 4 2 1 3) (5 3 4 1 2) (4 5 3 1 2) (5 4 1 3 2) (5 3 2 4 1) (4 5 2 3 1) (5 2 4 3 1) (4 3 5 2 1) (3 5 4 2 1) (5 3 2 1 4) (4 5 2 1 3) (5 4 1 2 3) (5 2 4 1 3) (4 3 5 1 2) (5 3 1 4 2) (3 5 4 1 2) (4 5 1 3 2) (5 1 4 3 2) (4 3 2 5 1) (5 2 3 4 1) (3 5 2 4 1) (4 2 5 3 1) (2 5 4 3 1) (3 4 5 2 1) (4 3 2 1 5) (5 3 1 2 4) (5 2 3 1 4) (3 5 2 1 4) (4 5 1 2 3) (4 2 5 1 3) (5 1 4 2 3) (5 2 1 4 3) (2 5 4 1 3) (4 3 1 5 2) (3 4 5 1 2) (5 1 3 4 2) (3 5 1 4 2) (4 1 5 3 2) (1 5 4 3 2) (4 2 3 5 1) (3 4 2 5 1) (2 5 3 4 1) (3 2 5 4 1) (2 4 5 3 1) (4 3 1 2 5) (4 2 3 1 5) (3 4 2 1 5) (5 2 1 3 4) (5 1 3 2 4) (3 5 1 2 4) (2 5 3 1 4) (3 2 5 1 4) (4 1 5 2 3) (4 2 1 5 3) (1 5 4 2 3) (5 1 2 4 3) (2 4 5 1 3) (2 5 1 4 3) (3 4 1 5 2) (4 1 3 5 2) (3 1 5 4 2) (1 5 3 4 2) (1 4 5 3 2) (3 2 4 5 1) (2 4 3 5 1) (2 3 5 4 1) (4 2 1 3 5) (3 4 1 2 5) (4 1 3 2 5) (3 2 4 1 5) (2 4 3 1 5) (5 1 2 3 4) (2 5 1 3 4) (3 1 5 2 4) (1 5 3 2 4) (3 2 1 5 4) (2 3 5 1 4) (4 1 2 5 3) (1 4 5 2 3) (1 5 2 4 3) (2 4 1 5 3) (2 1 5 4 3) (3 1 4 5 2) (1 4 3 5 2) (1 3 5 4 2) (2 3 4 5 1) (3 2 1 4 5) (4 1 2 3 5) (2 4 1 3 5) (3 1 4 2 5) (1 4 3 2 5) (2 3 4 1 5) (1 5 2 3 4) (2 1 5 3 4) (3 1 2 5 4) (1 3 5 2 4) (2 3 1 5 4) (1 4 2 5 3) (1 2 5 4 3) (2 1 4 5 3) (1 3 4 5 2) (3 1 2 4 5) (2 3 1 4 5) (1 4 2 3 5) (2 1 4 3 5) (1 3 4 2 5) (1 2 5 3 4) (2 1 3 5 4) (1 3 2 5 4) (1 2 4 5 3) (2 1 3 4 5) (1 3 2 4 5) (1 2 4 3 5) (1 2 3 5 4) (1 2 3 4 5) NIL Bruhat Order and the Geometry of Xw Xw = Cv = Xv v[≤w v[≤w Consequences. • The Schubert variety X = C = l where w = n... 21. w0 v F n 0 v≤[w0 ∗ • The cohomology ring of H (Xw) has linear basis [Xv] v w .