Gram Matrices and Stirling numbers of a class of Diagram Algebras

TL;DR

This paper introduces Gram matrices for certain diagram algebras, establishes their block structure, and proves the semisimplicity of signed partition algebras using Stirling numbers of the second kind.

Contribution

It introduces new Gram matrices for signed partition and related algebras, and connects their properties to Stirling numbers, proving semisimplicity.

Findings

Gram matrices are similar to block diagonal matrices.

$(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers are defined and related.

Semisimplicity of signed partition algebra is established.

Abstract

In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. We prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. In this connection, $(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers of the second kind are introduced and their identities are established. As a consequence, the semisimplicity of a signed partition algebra is established.