Young tableaux and linear independence of standard monomials in multiminors of a multimatrix

Abstract

As a culmination of the efforts of the invariant theorists from Clebsch, Gordan, Young, to Rota, in 1972 Doublet-Rota-Stein proved the Straightening Law which says that the standard monomials in the minors of a matrix X, which correspond to standard bitableaux, form a vector space basis of the polynomials ring K[X] in the indeterminate entries X over the coefficient field K. Now we may ask what happens to this when we consider ‘higher dimensional’ matrices by using cubical, 4-way,…,q-way determinants which were already introduced by Cayley in 1843. In the present paper we show that, for every q>2, the standard monomials in the multiminors of the multimatrix X are linearly independent over K. In a forthcoming paper it will be shown that they do not span the polynomial ring K[X]. The proof of linear independence given in this paper also applies to the classical case of q=2.