Representations on real toric spaces

Transcription

1 Representations on real toric spaces Soojin Cho Ajou University, Korea joint with Suyoung Choi (Ajou Univ.) & Shizuo Kaji (Yamaguchi Univ.) JMM Special Session in honor of Dennis Stanton January 11, 2018

2 S. Cho, S. Choi and S. Kaji Geometric representations of finite groups on real toric spaces arxiv:

3 Goal To understand the Weyl group representation on the (co)homology of real toric variety associated to Weyl chambers

4 Toric variety A toric variety is a normal variety X that contains a torus T = (C ) n as a dense open subset together with an action T X X of T on X that extends the natural action of T on itself. toric varieties fans CP 2 ( K = {, 1, 2, 3, 12, 13, 23}, Λ = ( 1 0 ))

5 Real toric variety For a toric variety X, real toric variety X R of X is the real locus of X ; that is, X R is the set of points that are stable under the canonical involution. Example (CP 2 ) R = RP 2 For X = (K, Λ), if K a simplicial complex on [m] let RZ K = σ K { (x1,..., x m ) (D 1 ) m x i S 0 when i / σ }, where D 1 = [ 1, 1], S 0 = { 1, 1}, and Λ 2 be the 0, 1 matrix obtained from Λ by taking mod 2 values. Then X R = RZ K / ker Λ 2

6 Toric variety associated to Weyl chambers [1990 Procesi] Φ: (irreducible) root system of type R and rank n W R : Weyl group of type R K R : Coxeter complex of type R R = A 2 K A2 = {, ( 1, 2, 3, 4, 5, 6, 12, 23, 34, 45, ) 56, 16} Λ = ( ) Λ 2 =

7 (Co)homology X = (K, Λ) with K a simplicial complex on [m] X R : real toric variety of X Theorem H (X ; Q) = Q[x 1,..., x m ]/(I + J) Theorem [Suciu 2012, Choi-Park 2017] H (X R ; Q) = H 1 (K ω ; Q) ω rowλ 2 H (X R ; Q) = H 1 (K ω ; Q) ω rowλ 2 as Q vector spaces.

8 Weyl group action on the cohomology of X R W -action on H (X R ; Q) induced from the action on the Coxeter complex 1990 Procesi 1992, 1994 Stembridge Theorem [Betti number] dim(h 2k (X An ; Q)) = A(n, k + 1)

9 Weyl group action on homology of X R R W -action on H (XR R ; Q) induced from the action on the Coxeter complex Remark Since irreducible representations of Weyl groups are self-dual and the cohomology and the homology of XR R are dual representations, they are isomorphic representations. Theorem [Suciu 2012, Choi-Park 2017] H (XR R ; Q) = H 1 ((K R ) ω ; Q) ω rowλ 2 as Q vector spaces. ( )

10 Weyl group action on homology of X R R Theorem [C-Choi-Kaji, 2017] Let X = (K, Λ) and suppose a finite group G acts on the simplicial complex K. Then, the following are equivalent: 1. G preserves ker Λ 2 ; 2. the action induces one on X R = RZ K / ker Λ 2 ; 3. each element of G permutes columns of Λ 2 without changing its row space; 4. each element g G permutes columns of Λ 2 in such a way that there exists an n n-matrix A g such that gλ 2 = A g Λ 2 ; 5. there exists a G-action on H (X R ) which is compatible with the isomorphism ( ), where the action of g G on the right hand side is induced by g : K ω K gω.

11 Weyl group action on homology of X R R Theorem [C-Choi-Kaji 2017] The Weyl group W R acts on K R and preserves ker Λ 2. More precisely, let Λ j 2 Zm 2 be the jth row of Λ 2, which corresponds the ω j coordinates of the rays. Then, we have (s i (Λ 2 )) j = Λ j 2 c ijλ 2 R i, where c ij = (αi, α j ) are the entries of the Cartan matrix of R. Hence we have H (XR R ; Q) = H 1 ((K R ) ω ; Q) ( ) ω rowλ 2 as W R modules.

12 Type A representation W An = S n+1. ω Row((Λ An ) 2 ) S ω [n] Theorem [Choi-Park, 2015] For S [n], let I S = S if S is even and I S = S {n + 1} otherwise. Then (K An ) ω is homotopically equivalent to the odd rank-selected Boolean algebra B odd I Sω. Theorem [Solomon 1968, Stanley 1982] Let Q [m 1]. Then the homology of the Q-rank-selected poset B Q [m] is given, as an S m-module, by H (B Q [m] ) = { ν c Q,νS ν ( = Q 1) 0 ( Q 1), where c Q,ν is the number of standard tableaux of shape ν with descent set Q.

13 Type A representation Theorem [C-Choi-Kaji 2017] Let Q = {1, 3,..., 2r 1} and c Q,ν be the number of standard tableaux of shape ν with descent set Q. Then, we have H r (XA R n ) = ( ) c Q,ν S η, η (n+1) where ν runs over all partitions of 2r that is contained in η, and η/ν has at most one box in each column. Proof ( ) H r (XA R n ) = Ind Sn+1 S {1,...,2r} S {2r+1,...,n+1} c Q,ν S ν S (n 2r) ν ( = c Q,ν (Ind Sn+1 S {1,...,2r} S {2r+1,...,n+1} S ν S (n 2r))) ν = ν ( ) c Q,ν S η = ν η ν η (n+1) ( ) c Q,ν ν η S η

14 Type A representation Let a n be the number of alternating permutations(snakes) in S n+1. Theorem [Hendersen 2012] The rth Betti number of XA R n is ( ) n+1 2r a2r. Corollary [C-Choi-Kaji 2017] The rth Betti number of XA R n, which is known to be ( ) n+1 2r a2r, is η (n+1) ( ν c Q,ν) f η. Example H 3 (X R A 5 ) = S (3,3) 2S (3,2,1) S (3,1,1,1) S (2,2,2) S (2,2,1,1)

15 Type B representation W Bn is the group of signed permutations on [n]. ω Row((Λ Bn ) 2 ) S ω [n] Theorem [Choi-Park-Park, 2017] (K Bn ) ω is homotopically equivalent to the odd rank-selected lattice C odd S ω of faces of the cross-polytope over S ω. Theorem [Stanley 1982] When S ω = r, H (C odd S ω ) = { (λ,µ) r b(λ, µ)s (λ,µ) ( = r 1 2 ) 0 ( r 1 2 ), where b(λ, µ) is the number of double standard Young tableaux of shape (λ, µ) whose descent set is the set of odd numbers less than or equal to r = λ + µ

16 Type B representation Theorem [C-Choi-Kaji 2017] The kth homology H k (XB R n ) of XB R n with the natural action of W Bn is isomorphic to the sum of two induced representations Ind W Bn W Br W Bn r b(λ, µ)s (λ,µ) S ( ;(n r)) r {2k 1,2k} (λ,µ) r of W Bn, where S ( ;(n r)) is the trivial representation of W Bn r. Corollary [C-Choi-Kaji 2017] H k (XB R n ) = (λ,ν) n r {2k 1,2k} (λ,µ) r,µ ν b(λ, µ) S (λ,ν), where µ in the inside summation, runs over all partitions that is contained in ν, and ν/µ has (n r) boxes with at most one box in each column.

17 Type B representation Let b n be the number of alternating signed permutations(snakes) in W Bn. Theorem [Choi-Park-Park 2017] The rth Betti number of XB R n is ( ) n 2r b2r + ( n 2r 1) b2r 1. Corollary [C-Choi-Kaji 2017] The rth Betti number of XA R n, which is known to be ( n 2r) b2r + ( n 2r 1) b2r 1, is (λ,ν) n ( r {2k 1,2k} Example (λ,µ) r,µ ν b(λ, µ) ) ( n λ ) f λ f ν. H 2 (X R B 3 ) = S ((1),(1,1)) S ((2),(1)) S ((1,1),(1)) S ((2),(1)). ( ) 1 3, 2 is the double standard tableau of shape ((1, 1), (1)) with descent set {1, 3}.

18 Thank You!

19 Thank You! Congratulations!!