W - graph versions of tensoring with the S n defining representation
نویسنده
چکیده
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 = Sn. Our main application is to give two W -graph versions of tensoring with the Sn defining representation V , one being H ⊗HJ − for H ,HJ the Hecke algebras of Sn,Sn−1 and the other ( ̂ H +⊗H −)1, where ̂ H + 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 ⊗−→ S2V ⊗−. 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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