Cyclage, catabolism, and the affine Hecke algebra

Jonah Blasiak
2011 Advances in Mathematics  
We identify a subalgebra H + n of the extended affine Hecke algebra H n of type A. The subalgebra H + n is a u-analogue of the monoid algebra of S n Z n 0 and inherits a canonical basis from that of H n . We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient R 1 n of H + n that is a u-analogue of the ring of coinvariants C[y 1 , . . .
more » ... y n ]/(e 1 , . . . , e n ) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element π ∈ H + n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that R 1 n has cellular quotients R λ that are u-analogues of the Garsia-Procesi modules R λ with left cells labeled by (a PAT version of) the λ-catabolizable tableaux. We give a conjectural description of a cellular filtration of H + n , the subquotients of which are isomorphic to dual versions of R λ under the perfect pairing on R 1 n . This turns out to be closely related to the combinatorics of the cells of H n worked out by Shi, Lusztig, and Xi, and we state explicit conjectures along these lines. We also conjecture that the k-atoms of Lapointe, Lascoux and Morse (2003) [9] and the R-catabolizable tableaux of Shimozono and Weyman (2000) [20] have cellular counterparts in H + n . We extend the idea of atom copies from Lapointe, Lascoux and Morse (2003) [9] to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.
doi:10.1016/j.aim.2011.07.006 fatcat:nswsy7dov5fcpkoyut3eh7ubxi