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 ${<em>{(s+r)}}C_s$ whose entries are ${x_i}</em>{min{s,r}}$. 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.