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On the cocharacter sequence of some PI-algebras

Elitza Hristova

Abstract

Let $A$ be a unital associative PI-algebra over a field of characteristic zero. We study which partitions $λ$ appear with nonzero multiplicities in the cocharacter sequence of $A$ for several classes of algebras $A$. Berele defines the eventual arm width $ω_0(A)$ to be the maximal integer $d$ so that if $λ$ appears with nonzero multiplicity in the cocharacter sequence of $A$, then $λ$ can have at most $d$ parts arbitrarily large. Berele also shows that if $A$ is Lie nilpotent, then $ω_0(A) = 1$. In the first part of this paper, we show that if $A$ is unital, then $ω_0(A) = 1$ 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 $λ$ 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 $m_λ(A)$ correspond to partitions $λ$ 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 $\mathrm{SL}(n)$.

On the cocharacter sequence of some PI-algebras

Abstract

Let be a unital associative PI-algebra over a field of characteristic zero. We study which partitions appear with nonzero multiplicities in the cocharacter sequence of for several classes of algebras . Berele defines the eventual arm width to be the maximal integer so that if appears with nonzero multiplicity in the cocharacter sequence of , then can have at most parts arbitrarily large. Berele also shows that if is Lie nilpotent, then . In the first part of this paper, we show that if is unital, then if and only if is Lie nilpotent. To prove this statement, we show that the algebra of proper polynomials is finite dimensional if and only if is Lie nilpotent. In the second part, we give a bound on the nonzero multiplicities in the cocharacter sequence of , when the T-ideal of identities of is equal to a product of T-ideals generated by long commutators. As an application, we show that for a Lie nilpotent algebra , the nonzero multiplicities correspond to partitions 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 .
Paper Structure (5 sections, 16 theorems, 42 equations, 2 figures)

This paper contains 5 sections, 16 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. On the eventual arm width of PI-algebras
  3. On the cocharacter sequence of $A$ when $T(A) = I_{p_1+1} \cdots I_{p_k+1}$
  4. Applications to the cocharacters of Lie nilpotent associative algebras
  5. Applications to Noncommutative invariant theory

Key Result

Theorem 1.1

AR For any PI-algebra $A$ there exists $k$ and $l$ such that all nonzero multiplicities in the cocharacter sequence of $A$ lie in the $k$ by $l$ hook, i.e., if $m_{\lambda}(A) \neq 0$, then $\lambda \in H(k,l)$.

Figures (2)

  • Figure 1: Hook diagrams
  • Figure 2: Definitions of $\omega_0(A)$, $\omega_1(A)$, $s_1(A)$, and $s_2(A)$

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Corollary 3.3
  • ...and 13 more