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Group and algebra hyperdeterminant

Alimzhan Amanov

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.

Group and algebra hyperdeterminant

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 -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.
Paper Structure (13 sections, 14 theorems, 64 equations)

This paper contains 13 sections, 14 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Matrix algebra
  3. Semisimple algebras
  4. Nonsemisimple algebras
  5. Preliminaries
  6. Tensors and hypermatrices
  7. Hyperdeterminants
  8. Tensor block-diagonalization
  9. Matrix algebra tensor
  10. Algebra tensor
  11. Invariance
  12. Semisimple algebras
  13. Nonsemisimple algebras

Key Result

Theorem 1.1

For any finite group $G$, the group determinant factors as follows: where $\widehat{G}$ is a complete set of inequivalent irreducible unitary representations, and each polynomial factor is irreducible of total degree $n_{\rho}:=\dim \rho$ and occurs with multiplicity $n_{\rho}$.

Theorems & Definitions (30)

  • Theorem 1.1: Frobenius frob
  • Theorem 1.2: Main theorem
  • Example 1.3: Circulant hyperdeterminant
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1: SL-invariance
  • Proposition 2.2
  • proof
  • Theorem 3.1: Tensor block-diagonalization
  • ...and 20 more