Symmetric polynomials over finite fields

TL;DR

The paper characterizes when two vectors over a finite field are in the same orbit under symmetric group action using elementary symmetric polynomials, providing a near-minimal separating set of invariants.

Contribution

It introduces a new set of polynomial invariants that effectively distinguish orbits over finite fields, especially when the field size equals the characteristic.

Findings

Identifies conditions for vectors to be in the same orbit using elementary symmetric polynomials.

Provides a near-minimal separating set of invariants for large dimensions.

Derives a small set of multisymmetric polynomial invariants over finite fields.

Abstract

It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree $p^k,2p^k,\dots,(q-1)p^k$, $k=0,1,2,\dots$ has the same value on them. This separating set of polynomial invariants for the natural permutation representation of the symmetric group is not far from being minimal when $q=p$ and the dimension is large compared to $p$. A relatively small separating set of multisymmetric polynomials over the field of $q$ elements is derived.