Seaweed Algebras and the Index Statistic for Partitions

TL;DR

This paper proves a conjecture linking a specific q-series to a parity statistic for seaweed algebras, confirming a long-standing hypothesis through advanced analytic methods.

Contribution

It establishes the non-negativity of coefficients in a key q-series, confirming the conjecture about the generating function for a parity statistic in seaweed algebras.

Findings

Confirmed the Coll-Mayers-Mayers Conjecture.

Established non-negativity of coefficients in the q-series.

Applied a variant of the circle method with Euler-Maclaurin summation.

Abstract

In 2018 Coll, Mayers, and Mayers conjectured that the $q$-series $( q, -q^3; q^4 )_\infty^{-1}$ is the generating function for a certain parity statistic related to the index of seaweed algebras. We prove this conjecture. Thanks to earlier work by Seo and Yee, the conjecture would follow from the non-negativity of the coefficients of this infinite product. Using a variant of the circle method along with Euler-Maclaurin summation, we establish this non-negativity, thereby confirming the Coll-Mayers-Mayers Conjecture.