On the Existence of Elementwise Invariant Vectors in Representations of Symmetric Groups

TL;DR

This paper characterizes when irreducible representations of symmetric groups have non-zero invariant vectors under permutations, revealing that most pairs of representations and permutations possess this property with few exceptions.

Contribution

It provides a complete characterization of the pairs (, ) for which invariant vectors exist in irreducible symmetric group representations, identifying the conditions and exceptions.

Findings

Most pairs (, ) have invariant vectors in irreducible representations.

A small number of simple exceptions where invariant vectors do not exist.

The results clarify the structure of symmetric group representations under permutation actions.

Abstract

We determine when a permutation with cycle type $\mu$ admits a non-zero invariant vector in the irreducible representation $V_\lambda$ of the symmetric group. We find that a majority of pairs $(\lambda,\mu)$ have this property, with only a few simple exceptions.