RESEARCH ARTICLE

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

N. Karimilla Bi, M. Parvathi

Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai 600 005, Tamilnadu, India

Abstract: 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.

Keywords: Gram matrices, partition algebras, signed partition algebras, algebra of $\mathbb{Z}_2$-relations.

MSC: 16Z05

Received: 22.09.2015
Revised: 16.03.2018

Language: English