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

Last updated: 26 March 2015

## Abstract

Unipotent Hecke algebras

The Iwahori-Hecke algebra and Gelfand-Graev Hecke algebras have been critical in developing the representation theory of finite groups and its implications in algebraic combinatorics. This thesis lays the foundations for a theory of unipotent Hecke algebras, a family of Hecke algebras that includes both the classical Gelfand-Graev Hecke algebra and a generalization of the Iwahori-Hecke algebra (the Yokonuma Hecke algebra). In particular, this includes a combinatorial analysis of their structure and representation theory. My main results are (a) a braid-like multiplication algorithm for a natural basis; (b) a canonical commutative subalgebra that may be used for weight space decompositions of irreducible modules. When the underlying group is the general linear group ${GL}_{n}\left({𝔽}_{q}\right)$ over a finite field ${𝔽}_{q}$ with $q$ elements, I also supply (c) an explicit construction of the natural basis; (d) a generalization of a Robinson-Schensted-Knuth correspondence (a bijective proof of representation theoretic identities); and (e) an explicit construction of the irreducible Yokonuma algebra modules using the known irreducible module structure of the Iwahori-Hecke algebra.

## 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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