The Group Determinant

Group use ← → to switch groups

Presentation

Multiplication table row $g$ , column $h$ , entry $gh$

The matrix $(x_{gh})_{g,h\in G}$ rows and columns in the order above

Factorization

Definition

Let $G=\{g_1,\dots,g_n\}$ be a finite group, and let $x_g$ be an independent variable for each element $g\in G$ . The group determinant of $G$ is the determinant of the $n\times n$ matrix whose $(i,j)$ entry is the variable indexed by the product $g_i g_j$ :

\Delta_G \;=\; \det\bigl(x_{g_i g_j}\bigr)_{1\le i,j\le n}.

It is a homogeneous polynomial of degree $n$ in $n$ variables with integer coefficients, independent of how the elements are ordered. Frobenius originally used $\det(x_{g_i g_j^{-1}})$ , which differs from $\Delta_G$ only by the sign of the permutation $g\mapsto g^{-1}$ of $G$ .