Eigenvalues of Gram Matrices of a class of Diagram Algebras

TL;DR

This paper computes eigenvalues of symmetric diagram matrices and Gram matrices for various diagram algebras, providing a systematic approach to understanding their spectral properties.

Contribution

It introduces a method to compute eigenvalues of symmetric diagram matrices and extends this to Gram matrices of several diagram algebras, including signed partition and partition algebras.

Findings

Eigenvalues of symmetric diagram matrices are computed explicitly.

Eigenvalues of Gram matrices for multiple diagram algebras are derived.

Method uses elementary row and column operations inductively.

Abstract

In this paper, we introduce symmetric diagram matrices $A_{s+r,s}$ of size ${_{(s+r)}}C_s$ whose entries are $\{x_i\}_{min\{s,r\}}$. We compute the eigenvalues of symmetric diagram matrices using elementary row and column operations inductively. As a byproduct, we obtain the eigenvalues of Gram matrices of a larger class of diagram algebras like the signed partition algebras, algebra of $\mathbb{Z}_2$ relations and partition algebras.