Decomposition results for Gram matrix determinants

TL;DR

This paper investigates the determinants of Gram matrices across classical, free, and half-liberated groups, providing comprehensive computations, proofs, and conjecturing a universal product decomposition involving partitions.

Contribution

It consolidates known results, offers simplified proofs, generalizes computations, and proposes a unifying conjecture for Gram matrix determinants across various group types.

Findings

Collected and simplified known determinant computations.

Proved several determinant formulas for different groups.

Conjectured a universal product decomposition involving partitions.

Abstract

We study the Gram matrix determinants for the groups $S_n,O_n,B_n,H_n$, for their free versions $S_n^+,O_n^+,B_n^+,H_n^+$, and for the half-liberated versions $O_n^*,H_n^*$. We first collect all the known computations of such determinants, along with complete and simplified proofs, and with generalizations where needed. We conjecture that all these determinants decompose as $D=\prod_\pi\phi(\pi)$, with product over all associated partitions.