On the Spectrum of Exterior Algebra, and Generalized Exponents of Small Representations

TL;DR

This paper investigates the irreducible representations in the exterior algebra of simple Lie algebras, providing new results for types B, C, D, and proposing conjectures and formulas for generalized exponents of small representations.

Contribution

It identifies specific irreducible components in the exterior algebra for types B, C, D, and introduces conjectures and formulas related to generalized exponents and covariants.

Findings

Irreducible representations appear in the exterior algebra for types B, C, D.

Proposes an analogue of Kostant's conjecture for the little adjoint representation.

Provides closed-form expressions for generalized exponents of small fundamental representations.

Abstract

We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $ \mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that certain irreducible representations, associated to weights characterized in a combinatorial way, appear as irreducible components of $\Lambda \mathfrak{g}$. Moreover, we propose an analogue of a conjecture of Kostant, about irreducibles appearing in the exterior algebra of the little adjoint representation. Finally, we give some closed expressions, in type $B$, $C$ and $D$, for generalized exponents of small representations that are fundamental representations and we propose a generalization of some results of De Concini, M\"oseneder Frajria, Procesi and Papi about the module of special covariants of adjoint and little adjoint type.