Cells and the reflection representation of Weyl groups and Hecke algebras

Douglass, J. Matthew

doi:10.1090/S0002-9947-1990-1035211-6

Cells and the reflection representation of Weyl groups and Hecke algebras
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by J. Matthew Douglass PDF

Trans. Amer. Math. Soc. 318 (1990), 373-399 Request permission

Abstract:

Let $\mathcal {H}$ be the generic algebra of the finite crystallographic Coxeter group $W$, defined over the ring $\mathbb {Q}[{u^{1/2}},{u^{ - 1/2}}]$. First, the two-sided cell corresponding to the reflection representation of $\mathcal {H}$ is shown to consist of the nonidentity elements of $W$ having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case $W$ is of type ${{\text {A}}_{l - 1}}$ or ${{\text {B}}_l}$.

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