A Short Proof of Gamas's Theorem

TL;DR

This paper provides a concise, self-contained proof of Gamas's theorem, which characterizes when symmetrizing a tensor with an irreducible character yields a non-zero result based on partitioning vectors into linearly independent sets.

Contribution

The paper offers a new, simplified proof of Gamas's theorem, clarifying the conditions under which tensor symmetrization is non-zero.

Findings

Proof confirms the if and only if condition for non-zero symmetrization

Clarifies the relationship between partitions and linear independence

Simplifies understanding of Gamas's theorem

Abstract

If \chi^\lambda is the irreducible character of the symmetric group S_n corresponding to the partition \lambda of n then we may symmetrize a tensor v_1 \otimes ... \otimes v_n by \chi^\lambda. Gamas's theorem states that the result is not zero if and only if we can partition the set {v_i} into linearly independent sets whose sizes are the parts of the transpose of \lambda. We give a short and self-contained proof of this fact.