A Combinatorial Determinant Dual to the Group Determinant

We define the commuting algebra determinant of a finite group action on a finite set, a notion dual to the group determinant of Dedekind. We show that the following combinatorial example is a commuting algebra determinant. Let Bq(n) denote the set of all subspaces of an n-dimensional vector space over Fq. The type of an ordered pair (U, V ) of subspaces, where U, V ∈ Bq(n), is the ordered triple (dim U, dim V,dim U∩V ) of nonnegative integers. Assume that there are independent indeterminates corresponding to each type. Let Xq(n) be the Bq(n) × Bq(n) matrix whose entry in row U , column V is the indeterminate corresponding to the type of (U, V ). We factorize the determinant of Xq(n) into irreducible polynomials.