Even Partitions in Plethysms

TL;DR

This paper proves a conjecture by Weintraub, showing that certain irreducible representations appear in plethysms, with implications for geometric complexity theory and quantum information.

Contribution

It establishes the existence of specific irreducible GL(d,C)-representations in plethysms for all relevant parameters, confirming Weintraub's conjecture.

Findings

Confirmed Weintraub's conjecture for all natural numbers k,n,d with k <= d.

Connected representation theory with quantum information theory.

Provides new tools for geometric complexity theory.

Abstract

We prove that for all natural numbers k,n,d with k <= d and every partition lambda of size kn with at most k parts there exists an irreducible GL(d, C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n) (C^d)). This gives an affirmative answer to a conjecture by Weintraub (J. Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory.