Gram matrices and Stirling numbers of a class of diagram algebras, II

In the paper [6], we introduced Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. $(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers of the second kind are also introduced and their identities are established. In this paper, we prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. As a consequence, the semisimplicity of a signed partition algebra is established.