## 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

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

## 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
${GL}_{n}\left({\mathbb{F}}_{q}\right)$ over
a finite field ${\mathbb{F}}_{q}$ 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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