The polytabloid basis expands positively into the web basis

TL;DR

This paper proves that the transition matrix between the polytabloid basis and the web basis for a specific symmetric group representation has nonnegative integer entries, confirming a conjecture by Russell and Tymoczko.

Contribution

It establishes the positivity of the transition matrix between two important bases in representation theory, confirming a previously conjectured property.

Findings

Transition matrix has nonnegative integer entries.

Confirms a conjecture of Russell and Tymoczko.

Advances understanding of basis relationships in symmetric group representations.

Abstract

We show that the transition matrix from the polytabloid basis to the web basis of the irreducible $\mathfrak{S}_{2n}$-representation of shape $(n,n)$ has nonnegative integer entries. This proves a conjecture of Russell and Tymoczko.