Total trades, intersection matrices and Specht modules

TL;DR

This paper explores the algebraic structure of total trades in combinatorial design theory, revealing their connection to Specht modules and symmetric group representations, and extends results to intersection matrices.

Contribution

It demonstrates that the span of permutations of a total trade forms an irreducible symmetric group representation and provides a Specht polynomial basis, generalizing previous decomposition results.

Findings

Total trades span an irreducible symmetric group module.

A Specht polynomial basis for total trades is constructed.

Decomposition of intersection matrix images generalizes known rank results.

Abstract

Trades are important objects in combinatorial design theory that may be realized as certain elements of kernels of inclusion matrices. Total trades were introduced recently by Ghorbani, Kamali and Khosravshahi, who showed that over a field of characteristic zero the vector space of trades decomposes into a direct sum of spaces of total trades. In this paper, we show that the vector space spanned by the permutations of a total trade is an irreducible representation of the symmetric group. As a corollary, the previous decomposition theorem is recovered. Also, a basis is obtained for the module of total trades in the spirit of Specht polynomials. More generally, in the second part of the paper we consider intersection matrices and determine the irreducible decompositions of their images. This generalizes previously known results concerning ranks of special cases.