Gram Matrices and Stirling numbers of a class of Diagram Algebras

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.

54 pages, 14 figures