Cocharacters of $UT_n(E)$

TL;DR

This paper develops algorithms to compute the cocharacter sequence and multiplicity series of the algebra of upper triangular matrices over the infinite Grassmann algebra, advancing understanding of polynomial identities in this context.

Contribution

It introduces an algorithm for calculating the generating function of the cocharacter sequence of $UT_n(E)$ and computes the double Hilbert series of $E$, enabling easier determination of multiplicity series.

Findings

Algorithm for cocharacter generating function of $UT_n(E)$

Explicit computation of the double Hilbert series of $E$

Efficient method to find the multiplicity series of $UT_n(E)$

Abstract

Let $F$ be a field of characteristic $0$ and let $E$ be the infinite dimensional Grassmann algebra over $F$. In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the $n\times n$ upper triangular matrix algebra $UT_n(E)$ with entries in $E$, lying in a strip of a fixed size. In the second part we compute the double Hilbert series $H(E;\mathrm{T}_k,\mathrm{Y}_l)$ of $E$, then we define the $(k,l)$-multiplicity series of any PI-algebra. As an application, we derive from $H(E;\mathrm{T}_k,\mathrm{Y}_l)$ an easy algorithm determining the $(k,l)$-multiplicity series of $UT_n(E)$.