Unipotent Hecke algebras: the structure, representation theory, and combinatorics

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 26 March 2015

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

Appendix A

Commutation Relations

The following relations are lifted directly from [Dem1965, Proposition 5.4.3]. Let $G$ be a finite Chevalley group over a finite field ${𝔽}_q$ with $q$ elements, defined as in Section 2.2.2. Let $R=R^{+}\cap R^{-}$ be as in Section 2.2.1. Let $\alpha ,\beta \in R$ such that $$\beta \ne -\alpha \text{and}\mid \alpha \left(H_{\beta }\right)\mid \le \mid \beta \left(H_{\alpha }\right)\mid \text{.}$$ Let $l,r\in {\mathbb{Z}}_{\ge 0}$ be maximal such that $$\{\beta -l\alpha ,\dots ,\beta -\alpha ,\beta ,\beta +\alpha ,\dots ,\beta +r\alpha \}\subseteq R\text{.}$$ Note that $l+r\le 3$ [Hum1972, Section 9.4]. Thus, the following analysis includes all the possible $l$ and $r$ values for $\alpha$ and $\beta \text{.}$ • $r=0$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right),$$ • $l=0$ and $r=1$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha +\beta }(\pm ab),$$ • $l=0$ and $r=2$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha +\beta }(\pm ab)x_{2\alpha +\beta }(\pm a^{2}b),$$ • $l=0$ and $r=3$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha +\beta }(\pm ab)x_{2\alpha +\beta }(\pm a^{2}b)x_{3\alpha +\beta }(\pm a^{3}b)x_{3\alpha +2\beta }(\pm a^3{b}^2),$$ • $l=1$ and $r=1$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha +\beta }(\pm 2ab),$$ • $l=1$ and $r=2$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha +\beta }(\pm 2ab)x_{2\alpha +\beta }(\pm 3a^{2}b)x_{\alpha +2\beta }(\pm 3a{b}^2),$$ • $l=2$ and $r=1$ implies $$x_{\beta }\left(b\right)x_{\alpha }\left(a\right)=x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha +\beta }(\pm 3ab),$$ where $a,b\in {𝔽}_q$ and $\pm 1$ depends in part on the original choice of Chevalley basis made in Section 2.2.2.

Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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