Cocharacters of $U{T}_n(E)$

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 $U{T}_n(E)$ 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 $U{T}_n(E)$.

arXiv comment35 pages