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.
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Blasiak, J. W-graph versions of tensoring with the \(\mathcal{S}_{n}\) defining representation. J Algebr Comb 34, 545–585 (2011). https://doi.org/10.1007/s10801-011-0281-9
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DOI: https://doi.org/10.1007/s10801-011-0281-9