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W$-graph versions of tensoring with the $ _n$ defining representation
ArXiv.org
01 Jan 2008
Abstract
We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin in \cite{HY1}, \cite{HY2}, focusing on the case $W = _n$. Our main application is to give two $W$-graph versions of tensoring with the $ _n$ defining representation $V$, one being $\H \tsr_{\H_J} -$ for $\H, \H_J$ the Hecke algebras of $ _n, _{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.
43 pages, 2 figures, youngtab.sty for Young tableaux
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Details
- Title
- W$-graph versions of tensoring with the $ _n$ defining representation
- Creators
- Jonah Blasiak
- Publication Details
- ArXiv.org
- Publisher
- arXiv
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862386304721