Cyclage, catabolism, and the affine Hecke algebra

This thesis consists of four papers and an introduction. All four papers use canonical bases to attack difficult conjectures in algebraic combinatorics surrounding the subtle tableau combinatorics of cyclage and catabolism. Our main result is to exhibit q-analogues Rl of the Garsia-Procesi modules, endowed with canonical bases coming from the extended affine Hecke algebra. We show that Rl has cells naturally in bijection with the set of lambda-catabolizable tableaux.;Paper I expands on the theory of inducing W-graphs began by Howlett and Yin in [17, 18], carefully working out the combinatorics of cells in type A. It then applies this to give two W-graph versions of tensoring with the Sn defining representation V. The corresponding W-graph versions of the projection V ⊗ V ⊗ - → S2 V ⊗ - are worked out and determined combinatorially in terms of cells.;Papers II--IV relate to the extended affine Hecke algebra H&d14; of type A. Paper II gives an algorithm for computing catabolizability of standard tableaux that was motivated by the cellular picture. This algorithm leads to two new characterizations of catabolizability and strengthens and simplifies proofs of some of its known properties.;In Paper III we show that any canonical basis element in the lowest two-sided cell of an extended affine Hecke algebra factors as the product of a symmetric function in the Bernstein generators and the canonical basis elements of what we call primitive elements, which are in bijection with elements of the associated finite Weyl group. The type A case of this result is used in Paper IV and was our starting point for using the canonical basis of H&d14; to better understand Garsia-Procesi modules. Paper IV identifies the cellular Garsia-Procesi modules Rl . Important for this cellular picture is a subalgebra H&d14;+ of H&d14; . Its cells are labeled by positive affine tableaux, tableaux filled with positive integer entries having distinct residues mod n. We show how these are ideal objects for studying cyclage posets. We present an array of conjectures that realize k-atoms and atom copies as cellular subquotients of Rl and identify the positive affine tableaux comprising their cells.