Column symmetric polynomials

TL;DR

This paper investigates a specialized polynomial algebra with column-based relations, revealing its structure as a quotient of a simpler algebra and exploring applications in synthetic differential geometry.

Contribution

It characterizes the invariant sub-algebra under column permutations as a quotient of a polynomial algebra in fewer variables.

Findings

Invariant sub-algebra is a quotient of polynomial algebra in m variables

The quotient map relates variables to row sums of the matrix

Application to synthetic differential geometry is proposed

Abstract

We study the polynomial algebra (over a ring containing the rationals) in an n by m matrix of variables, and subject to the relation that says that the product of any two variables in the same column is zero. We show that the sub-algebra of polynomials, which are invariant under n! permutations of the columns, is a quotient of the polynomial algebra in m variables; the quotient map sends the ith variable to the sum of the entries in the ith row of the matrix. - An application in synthetic differential geometry is sketched