On the table of marks of a direct product of finite groups

Masterson, Brendan ORCID: https://orcid.org/0000-0003-1929-1349 and Pfeiffer, Götz (2018) On the table of marks of a direct product of finite groups. Journal of Algebra, 499 . pp. 610-644. ISSN 0021-8693 [Article] (doi:10.1016/j.jalgebra.2017.12.019)

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Abstract

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S_3.

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