Young bitableaux, lattice paths and Hilbert functions
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.
scientific article; zbMATH DE number 932253