arXiv subjects

John M. O'Brien

Publications and source records attributed to John M. O'Brien.

2 recordsLinked to original sources

arXiv2020

Characters of Renner Monoids and Their Hecke Algebras

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e., the Cartan type $A$ Renner monoid) was computed earlier by Dieng, Halverson, and Poladian using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type $C$ and use them to obtain an explicit description of the character table for the type $C$ Renner monoid Hecke algebra.

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arXiv2018

Convolution Algebras for Finite Reductive Monoids

For an arbitrary finite monoid $M$ and subgroup $K$ of the unit group of $M$, we prove that there is a bijection between irreducible representations of $M$ with nontrivial $K$-fixed space and irreducible representations of $\mathcal{H}_K$, the convolution algebra of $K\times K$-invariant functions from $M$ to $F$, where $F$ is a field of characteristic not dividing $|K|$. When $M$ is reductive and $K = B$ is a Borel subgroup of the group of units, this indirectly provides a connection between irreducible representations of $M$ and those of $F[R]$, where $R$ is the Renner monoid of $M$. We conclude with a quick proof of Frobenius Reciprocity for monoids for reference in future papers.

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