The Binet-Cauchy functional equation and nonsingular multiindexed matrices

In a first theorem it is shown that a multiindexed matrix M = (Mσ,τ) is nonsingular where Mσ,τ is a multivariate polynomial in q-tuples of nonnegative integers σ and τ. In a second theorem a uniqueness relation of multinomial type is established. Finally, it is shown that, up to isomorphism, a nonzero function f:Mn(K)→K must be the determinant function if f(E) = 0, where E is the n × n matrix with all entries 1 n, and f satisfies the Binet-Cauchy function equation f(AB) = 1 n! ∑ |s| = n n sf(As)f{hook}(Bs) for square matrices A, B∈Mn(K) and for rectangular matrices A∈Mn×(n+1)(K) and B∈M(n+1)×n(K). © 1990