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
S. Cho, S. Choi and S. Kaji Geometric representations of finite groups on real toric spaces arxiv:1704.08591
Goal To understand the Weyl group representation on the (co)homology of real toric variety associated to Weyl chambers
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 )) 1 0 1 1
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
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} 1 0 1 1 0 1 Λ = 0 1 1 0 1 1 ( ) 1 0 1 1 0 1 Λ 2 = 0 1 1 0 1 1
(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.
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)
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. ( )
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.
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.
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.
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
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). 1 3 5 2 4 6 1 3 5 2 4 6 1 3 5 2 6 4 1 3 5 2 4 6 1 3 2 5 4 6 1 3 2 5 4 6
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
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.
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}.
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