W-graph versions of tensoring with the $\mathcal{S}_{n}$ defining representation

Abstract

We further develop the theory of inducing W-graphs worked out by Howlett and Yin (Math. Z. 244(2):415---431, 2003 and Manuscr. Math. 115(4):495---511, 2004), focusing on the case $W = \mathcal{S}_{n}$ . Our main application is to give two W-graph versions of tensoring with the $\mathcal{S}_{n}$ defining representation V, one being for the Hecke algebras of $\mathcal{S}_{n}, \mathcal{S}_{n-1}$ and the other , where is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding W-graph versions of the projection V?V???S 2 V??. This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter u?0. We make this approximation combinatorially explicit by determining it on cells and relate this to RSK growth diagrams.