id: 00932253
dt: j
an: 00932253
au: Ghorpade, Sudhir R.
ti: Young bitableaux, lattice paths and Hilbert functions.
so: J. Stat. Plann. Inference 54, No.1, 55-66 (1996).
py: 1996
pu: Elsevier Science B.V. (North-Holland), Amsterdam
la: EN
cc: 
ut: Stanley-Reisner ring; straightening law; standard bitableaux; Abhyankar
    formula; Hilbert polynomial; determinantal ideals; nonintersecting
    lattice paths; Hilbert series; Hilbert function; Schubert varieties
ci: 
li: doi:10.1016/0378-3758(95)00156-5
ab: 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.
rv: V.Drensky (Sofia)