Group and algebra hyperdeterminant

TL;DR

This paper solves the problem of factoring the group hyperdeterminant, reducing it to the matrix multiplication tensor, and extends the concept to associative algebra tensors, linking non-vanishing to semisimplicity.

Contribution

It provides a closed-form formula for the group hyperdeterminant and extends the concept to associative algebras, establishing a connection with semisimplicity.

Findings

Factoring the group hyperdeterminant reduces to the matrix multiplication tensor.

Derived a closed-form formula for the group hyperdeterminant.

Showed the hyperdeterminant is nonzero iff the algebra is semisimple.

Abstract

In 1896, Dedekind posed the problem of factoring the group determinant in the non-abelian case to Frobenius, whose solution sparked the birth of finite-group representation theory. Several decades earlier, Cayley introduced the notion of the combinatorial hyperdeterminant of a $d$-way tensor, which is the most natural generalization of an ordinary determinant. In this note, we solve the problem of factoring the group hyperdeterminant. We reduce the computation of the group hyperdeterminant to the computation of the hyperdeterminant at the matrix multiplication tensor and derive a nice closed formula. Further, we extend this notion to associative algebra tensors and show that this polynomial is nonzero if and only if the algebra is semisimple.