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Gram matrices and Stirling numbers of a class of diagram algebras. II. (English) Zbl 1401.16035

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)

Citations:

Zbl 1401.16034
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