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

Last updated: 26 March 2015

## 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 $β≠-αand ∣α(Hβ)∣ ≤∣β(Hα)∣.$ Let $l,r\in {ℤ}_{\ge 0}$ be maximal such that ${ β-lα,…, β-α,β,β+ α,…,β+rα } ⊆R.$ 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 $xβ(b) xα(a)= xα(a) xβ(b),$ $l=0$ and $r=1$ implies $xβ(b) xα(a)= xα(a) xβ(b) xα+β (±ab),$ $l=0$ and $r=2$ implies $xβ(b) xα(a)= xα(a) xβ(b) xα+β (±ab) x2α+β (±a2b),$ $l=0$ and $r=3$ implies $xβ(b) xα(a)= xα(a) xβ(b) xα+β(±ab) x2α+β (±a2b) x3α+β (±a3b) x3α+2β (±a3b2),$ $l=1$ and $r=1$ implies $xβ(b) xα(a)= xα(a) xβ(b) xα+β (±2ab),$ $l=1$ and $r=2$ implies $xβ(b) xα(a)= xα(a) xβ(b) xα+β (±2ab) x2α+β (±3a2b) xα+2β (±3ab2),$ $l=2$ and $r=1$ implies $xβ(b) xα(a)= xα(a) xβ(b) xα+β (±3ab),$ where $a,b\in {𝔽}_{q}$ and $±1$ 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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