Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $W$-graph versions of tensoring with the $\S_n$ defining representation   We further develop the theory of inducing $W$-graphs worked out by Howlett
and Yin in \cite{HY1}, \cite{HY2}, focusing on the case $W = \S_n$. Our main
application is to give two $W$-graph versions of tensoring with the $\S_n$
defining representation $V$, one being $\H \tsr_{\H_J} -$ for $\H, \H_J$ the
Hecke algebras of $\S_n, \S_{n-1}$ and the other $(\pH \tsr_{\H} -)_1$, where
$\pH$ 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 $V \tsr V \tsr - \to S^2 V \tsr -$. This does not
send canonical basis elements to canonical basis elements, but we show that it
approximates doing so as the Hecke algebra parameter $\u \to 0$. We make this
approximation combinatorially explicit by determining it on cells. Also of
interest is a combinatorial conjecture stating the restriction of $\H$ to
$\H_J$ is "weakly multiplicity-free" for $|J| = n-1$, and a partial
determination of the map $\H \tsr_{\H_J} \H \xrightarrow{\counit} \H$ on
canonical basis elements, where $\counit$ is the counit of adjunction.

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The categories of this paper are: math-ph
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin