Various correspondences 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 generalizing this procedure in various ways, we establish several correspondences between the sets of certain types of tableaux and the sets of 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: Using the Robinson-Schensted-Knuth procedure, we establish generalized rodeletive and codeletive correspondences from the sets of certain types of tableaux of any width q over integers to the sets of monomials of some specific kinds; it will turn out that these correspondences are surjective but not injective for q = 1, bijective for q = 2, and injective but not surjective for q $>$ 2. We extend the said procedure for tableaux over any totally ordered sets and obtain generalized roinsertive and coinsertive correspondences from the sets of monomials of some particular types to the sets of certain kinds of tableaux of any width q $>$ 1 over totally ordered sets; it will turn out that these correspondences are bijective for q = 2, and injective for q $>$ 2. In the classical two dimensional case, the Straightening 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. We prove that for q $>$ 2, the standard monomials in the multiminors of the multimatrix X, corresponding to standard multitableaux, do not span the polynomial ring K (X).
Degree
Ph.D.
Advisors
Abhyankar, Purdue University.
Subject Area
Mathematics
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