Young bitableaux, lattice paths and Hilbert functions
type: Item
instance of: scholarly article
title: Young bitableaux, lattice paths and Hilbert functions
zbMATH Open document ID: 0864.05088
DOI: 10.1016/0378-3758(95)00156-5
author: Sudhir R. Ghorpade
published in: Journal of Statistical Planning and Inference
publication date: 1997-06-09T00:00:00Z
review text: The straightening law of Doubilet-Rota-Stein tells that the standard bitableaux bounded by a pair \(m=(m(1),m(2))\) give a vector space basis of the polynomial algebra in \(m(1)m(2)\) variables. In an enumerative proof of the straightening law Abhyankar enumerated the set \(\text{stab}(2,m,p,a,V)\) of certain standard bitableaux. The Abhyankar formula gives also the Hilbert polynomial of a class of determinantal ideals \(I(p,a)\). In the paper under review, the author outlines an alternate proof of the Abhyankar formula for the cardinality of \(\text{stab}(2,m,p,a,V)\) using a recent result on nonintersecting lattice paths obtained independently by Modak, Kulkarni and Krattenthaler. The lattice path approach leads also to some other known results on the numerators of the Hilbert-Poincaré series of \(I(p,a)\), the so-called \(h\)-vector of the associated simplicial complex, and gives better bounds for the degree of the numerator of the Hilbert series of \(I(p,a)\) (and in some cases the exact value of the degree). The author also discusses some related problems concerning possible generalizations to higher dimensions. He indicates as well connections between the Hilbert function for the Schubert varieties in Grassmannians and the Abhyankar formula.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 932253
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: straightening law
zbMATH Keywords: standard bitableaux
zbMATH Keywords: Abhyankar formula
zbMATH Keywords: Hilbert polynomial
zbMATH Keywords: determinantal ideals
zbMATH Keywords: nonintersecting lattice paths
zbMATH Keywords: Hilbert series
zbMATH Keywords: Hilbert function
zbMATH Keywords: Schubert varieties
MaRDI profile type: Publication
full work available at URL: 0378-3758(95)00156-5
OpenAlex ID: W2043873939
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description: scientific article; zbMATH DE number 932253