On the cocharacter sequence of some PI-algebras

Abstract

Let $A$ be a unital associative PI-algebra over a field of characteristic zero. We study which partitions $\lambda$ appear with nonzero multiplicities in the cocharacter sequence of $A$ for several classes of algebras $A$. Berele defines the eventual arm width $\omega_0(A)$ to be the maximal integer $d$ so that if $\lambda$ appears with nonzero multiplicity in the cocharacter sequence of $A$, then $\lambda$ can have at most $d$ parts arbitrarily large. Berele also shows that if $A$ is Lie nilpotent, then $\omega_0(A) = 1$. In the first part of this paper, we show that if $A$ is unital, then $\omega_0(A) = 1$ if and only if $A$ is Lie nilpotent. To prove this statement, we show that the algebra of proper polynomials $B_n(A)$ is finite dimensional if and only if $A$ is Lie nilpotent. In the second part, we give a bound on the nonzero multiplicities $\lambda$ in the cocharacter sequence of $A$, when the T-ideal of identities of $A$ is equal to a product of T-ideals generated by long commutators. As an application, we show that for a Lie nilpotent algebra $A$, the nonzero multiplicities $m_{\lambda}(A)$ correspond to partitions $\lambda$ which are supported in step-like diagrams in which the number of steps grows with the index of Lie nilpotency. Finally, we give also some applications to the noncommutative invariant theory of the special linear group $\mathrm{SL}(n)$.