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

TL;DR

This paper provides an explicit uniform upper bound for the multiplicities in the cocharacter sequence of polynomial identities satisfied by each irreducible finite-dimensional representation of (), advancing understanding of their algebraic structure.

Contribution

It introduces a new explicit bound for cocharacter multiplicities in polynomial identities of irreducible () representations, a previously unquantified aspect.

Findings

Established a uniform upper bound for multiplicities

Applied the bound to all irreducible finite-dimensional representations

Enhanced understanding of polynomial identities in Lie algebra representations

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.