Littlewood Richardson coefficients for reflection groups

Transcription

1 Littlewood Richardson coefficients for reflection groups Arkady Berenstein and Edward Richmond* University of British Columbia Joint Mathematical Meetings Boston January 7, 2012 Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

2 1 Preliminaries on flag varieties 2 Algebraic approach to Schubert Calculus 3 Statement of results 4 Examples 5 The Nil-Hecke ring Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

3 Preliminaries on flag varieties Preliminaries: Schubert Calculus of G/B Let G be a Kac-Moody group over C (or a simple Lie group). Fix T B G a maximal torus and Borel subgroup of G. Let W := N(T )/T denote the Weyl group G. Let G/B be the flag variety (projective ind-variety). For any w W, we have the Schubert variety X w = BwB/B G/B. Denote the cohomology class of X w by σ w H 2l(w) (G/B). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

4 Preliminaries on flag varieties Additively, we have that H (G/B) w W Z σ w. Goal (Schubert Calculus) Compute the structure (Littlewood-Richardson) coefficients c w u,v with respect to the Schubert basis defined by the product σ u σ v = w W c w u,vσ w. Note that if l(w) l(u) + l(v), then c w u,v = 0. For any w, u, v W, we have that c w u,v 0. (proofs are geometric) For example, if G is a finite Lie group, then the cardinality for a generic choice of g 1, g 2, g 3 G. g 1 X u g 2 X v g 3 X w0w = c w u,v Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

5 Algebraic approach to Schubert Calculus Algebraic approach to Schubert calculus Let A = A(G) denote the Cartan matrix of G. Alternatively, fix a finite index set I and let A = {a ij } be an I I matrix such that a ii = 2 a ij Z 0 if i j a ij = 0 a ji = 0. The matrix A defines an action of a Coxeter group W generated by reflections {s i } i I on the vector space V := Span C {α i } i I given by where α i, α j := a ij. s i (v) := v v, α i α i In particular, Coxeter groups of this type are crystallographic. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

6 Algebraic approach to Schubert Calculus Algebraic approach to Schubert calculus If we abandon the group G, we can consider a matrix A as follows: Fix a finite index set I and let A = {a ij } be an I I matrix such that a ii = 2 a ij R 0 if i j a ij = 0 a ji = 0. The matrix A defines an action of a Coxeter group W generated by reflections {s i } i I on the vector space V := Span C {α i } i I given by where α i, α j := a ij. s i (v) := v v, α i α i Every Coxeter group can be represented as above. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

7 Algebraic approach to Schubert Calculus Some examples G = SL(4) (type A 3 ) A = and W = S 4 (symmetric group) G = Sp(4) (type C 2 ) [ ] 2 2 A = and W = I (4) (dihedral group of 8 elements) G = ŜL(2) (affine type A) [ ] 2 2 A = and W = I ( ) (free dihedral group) Let ρ = 2 cos(π/5) [ ] 2 ρ A = and W = I ρ 2 2 (5) (dihedral group of 10 elements) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

8 Algebraic approach to Schubert Calculus Notation on sequences and subsets Any sequence i := (i 1,..., i m ) I m has a corresponding element s i1 s im W. If s i1 s im is a reduced word of some w W, then we say that i R(w), the collection of reduced words. For each i I m and subset K = {k 1 < k 2 < < k n } of the interval [m] := {1, 2,..., m} let the subsequence i K := (i k1,..., i kn ) I n. We say a sequence i is admissible if i j i j+1 for all j [m 1]. Observe that any reduced sequence is admissible. (In general, the converse is false.) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

9 Algebraic approach to Schubert Calculus Definition Let m > 0 and let K, L be subsets of [m] := {1, 2,..., m} such that K + L = m. We say that a bijection is bounded if φ(k) < k for each k K. φ : K [m] \ L Definition Given a reduced sequence i = (i 1,..., i m ) I m, we say that a bounded bijection φ : K [m] \ L is i-admissible if the sequence i L and the sequences i L(k) are admissible for all k K where L(k) := L φ(k k ). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

10 Statement of results Recall that the Littlewood-Richardson coefficients c w u,v are defined by the product σ u σ v = w W c w u,v σ w. Theorem: Berenstein-R, 2010 Let u, v, w W such that l(w) = l(u) + l(v) and let i = (i 1,..., i m ) R(w). Then c w u,v = p φ where the summation is over all triples (û, ˆv, φ), where û, ˆv [m] such that iû R(u), iˆv R(v). φ : û ˆv [m] \ (û ˆv) is an i-admissible bounded bijection. The Theorem is still true without i-admissible. The Theorem generalizes to structure coefficients of T -equivariant cohomology H T (G/B). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

11 Statement of results Definition For any k [m] and i = (i 1,..., i m ), denote α k := α ik bounded bijection φ : K [m] \ L we define the monomial p φ Z by the formula and s k := s ik. For any p φ := ( 1) K k K w k (α k ), α φ(k) where w k := r L(k) φ(k)<r<k where the product is taken in the natural order induced by the sequence [m] and if the product is empty, we set w k = 1. Also, if K =, then p φ = 1. Theorem: Berenstein-R, 2010 If the Cartan matrix A = (a ij ) satisfies a ij a ji 4 i j, then p φ 0 when φ is an i-admissible bounded bijection. s r Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

12 Examples Example: G = ŜL(2) Examples where G = ŜL(2), i = (1, 2, 1, 2) [ A = Compute c w u,v where w = s 1 s 2 s 1 s 2, u = v = s 1 s 2. Find û, ˆv [4] = {1, 2, 3, 4} For , we have with bounded bijections NOTE: φ 1 is not i-admissible û ˆv = {3, 4} and [4] \ (û ˆv) = {1, 2} φ 1 : (3, 4) (1, 2) φ 2 : (3, 4) (2, 1). ] Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

13 Examples Example: G = ŜL(2) Examples where G = ŜL(2), i = (1, 2, 1, 2) For , we have bounded bijections φ 1 : (3, 4) (1, 2) φ 2 : (3, 4) (2, 1). p φ1 = α 3, α 1 s 3 (α 4 ), α 2 p φ2 = α 3, α 2 s 2 s 3 (α 4 ), α 1 Totaling over all bounded bijections, we have c w u,v = α 3, α1 s 3 (α 4 ), α2 + α 3, α2 s 2 s 3 (α 4 ), α1 s 3 (α 4 ), α 2 s 3 (α 4 ), α = = 6 With only i-admissible terms. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

14 Examples Example: G = ŜL(2) Examples where G = ŜL(2), i = (1, 2, 1, 2, 1) Compute c w u,v where w = s 1 s 2 s 1 s 2 s 1, u = s 1 s 2 s 1, v = s 2 s 1. Find û, ˆv [5] = {1, 2, 3, 4, 5} Totaling over all bounded bijections, we have c w u,v = α 4, α2 s 4 (α 5 ), α3 + α 4, α3 s 3 s 4 (α 5 ), α2 + s 3 (α 4 ), α1 s 3 s 4 (α 5 ), α2 + s 3 (α 4 ), α2 s 2 s 3 s 4 (α 5 ), α1 s 2 (α 3 ), α1 s 4 (α 5 ), α 3 s 4 (α 5 ), α3 s 2 s 3 s 4 (α 5 ), α = = 10 With only i-admissible terms. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

15 Examples where G = Examples Example: G = ŜL(2) ŜL(2), i = (2, 1,..., 2, 1) }{{} 2n Question: Bounded bijections vs. i-admissible bounded bijections? c(n) := c w u,v, w = s 2 s 1 s 2 s }{{} 1, u = v = s 2 s }{{} 1 2n n #{Bounded bijections} #{i-admissible bounded bijections} c(2) 6 3 c(3) 20 7 c(4) c(5) c(6) c(n)?? largest coeff of (1 + x + x 2 ) n?? ( ) 2n Remark: c(n) = n Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

16 Examples Example: G is rank 2. General rank 2 group G. Let a, b 0, Define Chebyshev sequences [ A = 2 a b 2 ]. A k := ab k 1 A k 2 and B k := ba k 1 B k 2 where A 0 = B 0 = 0 and A 1 = B 1 = 1. Let u k = s 2 s 1 }{{} k Corollary: Binomial formula (Kitchloo, 2008) The rank 2 Littlewood-Richardson coefficients and v k = s 1 s }{{} 2. k A c un u k,u n k = c vn+1 n A 2 A 1 v k+1,u n k = (A k A 2 A 1 )(A n k A 2 A 1 ) B c vn v k,v n k = c un+1 n B 2 B 1 u k+1,v n k = (B k B 2 B 1 )(B n k B 2 B 1 ) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

17 Examples Example: G = SL(4) Examples where G = SL(4), i = (3, 2, 1, 3, 2) Compute c w u,v where A = w = s 3 s 2 s 1 s 3 s 2, u = s 3 s 1 s 2 = s 1 s 3 s 2, v = s 3 s 1 = s 1 s 3. Totaling over all bounded bijections, we have c w u,v = α 3, α1 s 3 (α 4 ), α2 + α 3, α2 s 2 s 3 (α 4 ), α1 α 3, α2 α 3, α2 = = 1 In this case: {Bounded bijections}={i-admissible bounded bijections} In general, the i-admissible formula is not nonnegative. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

18 The Nil-Hecke ring Construction of Kostant and Kumar Recall the the matrix A gives an action of the Coxeter group W on V and thus W acts on the algebras S := S(V ) and Q := Q(V ) (polynomials and rational functions). Define with product structure Q W := Q C[W ] and a Q-linear coproduct (q 1 w 1 )(q 2 w 2 ) := q 1 w 1 (q 1 )w 1 w 2 : Q W Q W Q Q W by (qw) := qw w = w qw. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

19 The Nil-Hecke ring For any i I, define x i := 1 α i (s i 1). If i = (i 1,..., i m ) R(w), then define x w := x i1 x im. If A is a Cartan matrix of some Kac-Moody group G, then (by Kostant-Kumar 1986) x w is independent of i R(w) x 2 i = 0 i I. Define the Nil-Hecke ring H W := w W S x w Q W. We have that (H W ) H W S H W (Kostant-Kumar, 1986). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20

20 The Nil-Hecke ring Define the coproduct structure constants p w u,v S by (x w ) = p w u,v x u x v. u,v W Consider the T -equivariant cohomology ring HT (G/B) S σ w and L-R w W coefficients c w u,v S defined by the cup product σ u σ v = c w u,vσ w. w W Theorem: Kostant-Kumar 1986 The coefficients c w u,v = p w u,v. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20