Symmetric Functions and Caps

TL;DR

This paper derives an explicit formula for moments related to symmetric functions over finite fields, using Schur-Weyl duality and elementary symmetric function properties, with implications for the study of maximal caps.

Contribution

It provides a novel explicit matrix-based formula for moments of symmetric functions over finite fields, connecting representation theory and combinatorics.

Findings

Derived an explicit matrix formula for moments F(m,n)

Connected symmetric functions to the problem of maximal caps

Presented polynomial expressions in q for these moments

Abstract

Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a product of two matrices, ultimately yielding a polynomial in q=p^d. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric functions. This problem is closely to the study of maximal caps.