The Stabilizer Of Immanants

TL;DR

This paper characterizes the stabilizer groups of immanants, showing that for most cases, the stabilizer is as expected, with specific results for non-symmetric partitions and exceptions like the determinant and permanent.

Contribution

It provides a detailed description of the stabilizer groups of immanants, including new results for non-symmetric partitions and clarifies the structure of these groups.

Findings

The identity component of the stabilizer of most immanants is as expected.

For n>5, the stabilizer of non-symmetric immanants matches the expected group.

The paper identifies exceptions like the determinant and permanent where the stabilizer differs.

Abstract

We describe immanants as trivial modules of the symmetric group and show that any homogeneous polynomial of degree n on the space of n by n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by permutation matrices is a linear combination of immanants. we prove that the identity component of the stabilizer of any immanant (except determinant, permanent) is the expected one. We also prove that for n>5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is again the expected one.