Decomposition results for Gram matrix determinants

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 $O_n^{\ast },H_n^{\ast }$. 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 $D={\prod }_{\pi }\varphi (\pi )$, with product over all associated partitions.

18 pages

References in corpus (4)

• On some properties of orthogonal Weingarten functions
• Random matrices, free probability, planar algebras and subfactors
• Jucys-Murphy elements and Weingarten matrices
• Isomorphisms and Fusion Rules of Orthogonal Free Quantum Groups and their Complexifications

Cited by in corpus (1)

• Quantumness of pure-state ensembles via coherence of Gram matrix based on generalized $\alpha$-$z$-relative Rényi entropy