THESIS
2022
1 online resource (viii, 156 pages) : illustrations
Abstract
A Gelfand model for an algebra is a module isomorphic to a direct sum of irreducible modules,
with every isomorphism class of irreducible modules represented exactly once. We introduce
and study the notion of a perfect model for a finite Coxeter group; such a model is a certain
set of discrete data parametrizing a Gelfand model for the associated Iwahori-Hecke algebra.
We classify which Coxeter groups have perfect models, and then describe explicit Gelfand
models for the Iwahori-Hecke algebras of classical finite Coxeter groups. This simultaneously
generalizes constructions of Adin, Postnikov, and Roichman (from type A to other classical
types) and of Araujo and Bratten (from group algebras to Iwahori-Hecke algebras). Our
Gelfand models have interesting “canonical bases” that give rise...[
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A Gelfand model for an algebra is a module isomorphic to a direct sum of irreducible modules,
with every isomorphism class of irreducible modules represented exactly once. We introduce
and study the notion of a perfect model for a finite Coxeter group; such a model is a certain
set of discrete data parametrizing a Gelfand model for the associated Iwahori-Hecke algebra.
We classify which Coxeter groups have perfect models, and then describe explicit Gelfand
models for the Iwahori-Hecke algebras of classical finite Coxeter groups. This simultaneously
generalizes constructions of Adin, Postnikov, and Roichman (from type A to other classical
types) and of Araujo and Bratten (from group algebras to Iwahori-Hecke algebras). Our
Gelfand models have interesting “canonical bases” that give rise to associated W-graphs,
which we call Gelfand W-graphs. A W-graph is a kind of directed graph encoding an
Iwahori-Hecke algebra module. For types B and D, we prove that these W-graphs are dual
to each other, a phenomenon which does not occur in type A. The strongly connected
components in a W-graph are called its cells, and the components connected by bidirected
edges are called its molecules. For type A, we classify the molecules in our Gelfand W-graphs,
and conjecture that in type A every molecule is a cell.
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