On the enumeration and independence of standard Young tableaux of higher width
Abstract
Motivated by some recent work of Abhyankar, we consider in this thesis Young tableaux of higher width and discuss the extension of the Straightening Law of Doubilet-Rota-Stein to higher dimensional matrices. We obtain a polynomial formula for counting certain sets of standard Young tableaux of even width and as a consequence we are able to deduce that the analogue of the Straightening Law does not hold in dimensions higher than 2. The notion of higher dimensional determinant originated by Cayley plays a useful role in obtaining this formula. Either as a consequence or in the process of finding this formula, we are able to answer some of the problems posed by Abhyankar. We also show that the standard monomials in the multiminors of a miltimatrix with indeterminant entries do form a linearly independent subset of the corresponding ring of polynomials although they may fail to generate it. As a consequence of this result we can conclude that the polynomial formula that we obtain, also gives the Hilbert function of a certain determinantal module.
Degree
Ph.D.
Advisors
Abhyankar, Purdue University.
Subject Area
Mathematics
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