UC Berkeley

Let k be an integral domain, n a positive integer, X a generic n x n matrix over k (i.e., the matrix (x_ij) over a polynomial ring k[x_ij] in n² indeterminates x_ij): and adj(X) its classical adjoint. For char k = 0, it is shown that if n is odd, adj(X) is not the product of two noninvertible n x n matrices over k[x_ij], while for n even, only one special sort of factorization occurs. Whether the corresponding results hold in positive characteristic is not known. The operation adj on matrices arises from the (n-1)st exterior power functor on modules; the analogous factorization question for matrix constructions arising from other functors is raised, as are several other questions.