# On the Table of Marks of a Direct Product of Finite Groups

## TL;DR

This paper introduces a novel method to compute the table of marks for direct products of finite groups, revealing a complex structure that differs from the simple Kronecker product of individual tables.

## Contribution

It provides a new approach using matrix products and partial order decompositions to compute the table of marks for direct products of finite groups.

## Key findings

- The table of marks can be expressed as a matrix product of three incidence matrices.
- The method applies to construct a ghost ring and mark homomorphism for the rational double Burnside algebra of S_3.
- The approach highlights the non-trivial structure of the table of marks for direct products.

## 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~$S_3$.