The Stabilizer of Elementary Symmetric Polynomials

TL;DR

This paper characterizes the complete set of linear transformations that leave the r-th elementary symmetric polynomial invariant, showing that only permutations and specific scalings preserve it.

Contribution

It proves that the only linear transformations leaving the elementary symmetric polynomial invariant are permutations and scalings by r-th roots of unity.

Findings

Only permutations and r-th roots of unity scalings preserve the polynomial.

No other linear transformations leave the elementary symmetric polynomial invariant.

The result fully characterizes the symmetry group of the polynomial.

Abstract

We study the r-th elementary symmetric polynomial in $n$ variables with 2<r<n. There are two kinds of linear transformations on the parameter space that leave this polynomial invariant: Namely, any permutation of the variables and simultaneous scaling by any r-th root of unity. We prove that there are no other linear transformations with this property.