Generalized rodeletive correspondence between multitableaux and multimonomials

Abstract

In the third volume of his book on the art of computer programming, Knuth has refined a sorting procedure originated by Robinson and Schensted. By employing a modification of this procedure, in this paper 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, as a consequence of the Robinson-Schensted-Knuth correspondence, 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.