| Journal of Statistical Planning and Inference | ||
| Volume 54, Issue 1 |
|
Abstract |
| 2 September 1996 | ||
| Pages 55-66 |
0378-3758(95)00156-5
Young bitableaux, lattice paths and Hilbert functions
a Department of Mathematics, Indian Institute of Technology, Powai Bombay 400076 India
Received 28 April 1994; revised 6 March 1995. Available online 20 February 1999.
A recent result on the enumeration of p-tuples of nonintersecting lattice paths in an integral rectangle is used to deduce a formula of Abhyankar for standard Young bitableaux of certain type, which gives the Hilbert function of a class of determinantal ideals. The lattice path formula is also shown to yield the numerator of the Hilbert series of these determinantal ideals and the h-vectors of the associated simplicial complexes. As a consequence, the a-invariant of these determinantal ideals is obtained in some cases, extending an earlier result of Gräbe. Some problems concerning generalizations of these results to `higher dimensions' are also discussed. In an appendix, the equivalence of Abhyankar's formula for unitableaux of a given shape and a formula of Hodge, obtained in connection with his determination of Hilbert functions of Schubert varieties in Grassmannians, is outlined.
Author Keywords: Standard tableaux; Nonintersecting lattice paths; Determinantal ideals; Hilbert functions; Stanley-Reisner ring; a-invariant
*Corresponding author.
| Journal of Statistical Planning and Inference |
|
Abstract |
| Volume 54, Issue 1 | ||
| 2 September 1996 | ||
| Pages 55-66 |