Bi, N. Karimilla; Parvathi, M. Gram matrices and Stirling numbers of a class of diagram algebras. II. (English) Zbl 1401.16035 Algebra Discrete Math. 25, No. 2, 215-256 (2018). Summary: In Part I [ibid. 25, No. 1, 73–97 (2018; Zbl 1401.16034)], 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. MSC: 16S99 Associative rings and algebras arising under various constructions 20C15 Ordinary representations and characters 05E10 Combinatorial aspects of representation theory 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) Keywords:Gram matrices; partition algebras; signed partition algebras and the algebra of \(\mathbb Z_2\)-relations Citations:Zbl 1401.16034 PDFBibTeX XMLCite \textit{N. K. Bi} and \textit{M. Parvathi}, Algebra Discrete Math. 25, No. 2, 215--256 (2018; Zbl 1401.16035) Full Text: arXiv