$W$-graph versions of tensoring with the $§_n$ defining representation
| dc.creator | Blasiak, Jonah | |
| dc.date | 2008-09-28 | |
| dc.date.accessioned | 2026-07-07T10:06:02Z | |
| dc.date.available | 2026-07-07T10:06:02Z | |
| dc.description | 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. | |
| dc.description | 43 pages, 2 figures, youngtab.sty for Young tableaux | |
| dc.identifier | https://arxiv.org/abs/0809.4810 | |
| dc.identifier | http://arxiv.org/abs/0809.4810 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/170193 | |
| dc.subject | Representation Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 20C08 | |
| dc.title | $W$-graph versions of tensoring with the $§_n$ defining representation | |
| dc.type | text |