Eigenvalues of Gram Matrices of a class of Diagram Algebras

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.

32 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:1504.01241