Bound for the cocharacters of the identities of irreducible representations of ${\mathfrak{sl}}_2(\mathbb{C})$

Abstract

For each irreducible finite dimensional representation of the Lie algebra ${\mathfrak{sl}}_2(\mathbb{C})$ of $2\times 2$ traceless matrices, an explicit uniform upper bound is given for the multiplicities in the cocharacter sequence of the polynomial identities satisfied by the given representation.