Generalized roinsertive correspondence between multitableaux and multimonomials

Abstract

By generalizing the Robinson-Schensted-Knuth insertion procedure, we establish a bijective correspondence between the set of certain types of bitableaux over totally ordered sets and the set of indexed monomials satisfying certain constraints. As a consequence we show that the Straightening Law of Doubilet-Rota-Stein is not valid in the case of 'higher dimensional' matrices. In greater detail: In the classical two dimensional case, the said Law says that the standard monomials in the minors of a (rectangular) matrix X, which correspond to standard bitableaux, form a vector space basis of the polynomial ring K[X] in the indeterminate entries of 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 q > 2, the standard monomials in the multiminors of the multimatrix X do not span the polynomial ring K[X]; in a forthcoming paper it will be shown that they are linearly independent over K.