Classification of skew multiplicity free modules

TL;DR

This paper classifies all finite-dimensional representations of connected reductive groups over complex numbers where the exterior algebra decomposes into irreducible representations with multiplicity at most one, known as skew multiplicity-free modules.

Contribution

It provides a complete classification of skew multiplicity-free modules for connected reductive groups, a problem previously unaddressed.

Findings

Identified all skew multiplicity-free representations of connected reductive groups.

Established criteria characterizing skew multiplicity-free modules.

Provided a comprehensive list of such modules.

Abstract

Let $G$ be a connected reductive group defined over $\CC$ with a finite dimensional representation $V$. The action of $G$ is said to be skew multiplicity-free (SMF) if the exterior algebra $\bigwedge V$ contains no irreducible representation of $G$ with multiplicity $> 1$. In this paper we classify all such representations.