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Dean Alvis

Publications and source records attributed to Dean Alvis.

7 recordsLinked to original sources
arXiv2021

A class of representations of Hecke algebras II

Let $W$ be a Coxeter group whose proper parabolic subgroups are finite. According to Theorem~1.12 of [1], if the module of a finite $W$-digraph $Γ$ is isomorphic to the module of a $W$-graph over $Q$, then $Γ$ is acyclic. We extend this result to Coxeter groups with finite dihedral parabolic subgroups and $W$-graphs over arbitrary fields $F$ of $C$. Also, an example is provided showing the converse of this theorem is false. That is, there is an example of a finite, acyclic $W$-digraph whose module does not afford a $W$-graph.

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arXiv2021

An extension of a theorem and errata for "A Class of Representations of Hecke Algebras"

By Theorem~1.12 of the paper "A Class of Representations of Hecke Algebras", if $W$ is a Coxeter group whose proper parabolic subgroups are finite, and if the module of a finite $W$-digraph $Γ$ is isomorphic to the module of a $W$-graph, then $Γ$ must be acyclic. Here we extend this result to Coxeter groups with finite dihedral parabolic subgroups and $W$-graphs with arbitrary scalar edge labels. Also, errata for the paper are listed in the last section.

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arXiv2015

A class of representations of Hecke algebras

A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of Lusztig, a complete characterization of W-digraphs is given in terms of subdigraphs for dihedral parabolic subgroups. Graph-theoretic properties of W-digraphs are established including, under certain assumptions, acyclicity. In case the Coxeter system is finite, a bound on the number of vertices of a connected W-digraph is obtained, and a graph-theoretic version of the usual duality operation is obtained.

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arXiv2007

Subrings of the asymptotic Hecke algebra of type $H_4$

The structure of subring $J^{Γ\cap Γ^{-1}}$ of the asymptotic Hecke algebra is described for $Γ$ a left cell of the Coxeter group of type $H_4$. A small set of generators is produced. The subalgebras spanned by a subset of the basis ${t_x}_{x\in Γ\capΓ^{-1}}$ are determined.

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arXiv2005

Induce/restrict matrices for exceptional Weyl groups

This manuscript contains tables giving the multiplicities with which irreducible characters of exceptional Weyl groups appear in characters induced from certain reflection subgroups containing maximal parabolic subgroups.

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arXiv2002

A family of polynomials related to generic degrees

A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are described completely for classes that meet a parabolic subgroup whose components are of type A or are dihedral, and for the class of Coxeter elements.

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