Hilbert Polynomial of a Certain Ladder-Determinantal Ideal
Devadatta M. Kulkarni
DOI: 10.1023/A:1022428731241
Abstract
A ladder-shaped array is a subset of a rectangular array which looks like a Ferrers diagram corresponding to a partition of a positive integer. The ideals generated by the p-by- p minors of a ladder-type array of indeterminates in the corresponding polynomial ring have been shown to be hilbertian (i.e., their Hilbert functions coincide with Hilbert polynomials for all nonnegative integers) by Abhyankar and Kulkarni [3, p 53-76]. We exhibit here an explicit expression for the Hilbert polynomial of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates in the corresponding polynomial ring. Counting the number of paths in the corresponding rectangular array having a fixed number of 
turning points 
above the path corresponding to the ladder is an essential ingredient of the combinatorial construction of the Hilbert polynomial. This gives a constructive proof of the hilbertianness of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates.
turning points above the path corresponding to the ladder is an essential ingredient of the combinatorial construction of the Hilbert polynomial. This gives a constructive proof of the hilbertianness of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates.Pages: 57–71
Keywords: Hilbert function; ladder-determinantal ideal; lattice path counting; nonintersecting $p$-tuple of paths with fixed number of turning points
Full Text: PDF
References
1. Abhyankar S., Combinatorie des Tableaux de Young, Varietes Determinantielles et Calcul de Fonctions de Hilbert, Nice Lecture Notes Rendiconti del Seminario Matematico 42, A. Galligo Universita E. Polytecnico di Torino (1984), 65-88.
2. Abhyankar S., Enumerative Combinatorics of Young Tableaux, Marcel Dekker, New York, 1988.
3. Abhyankar S. and Kulkarni D.M., "On Hilbertian ideals," Linear Algebra and Its Applications 116 (1989), 53-76.
4. Kulkarni D.M., thesis, Purdue University, West Lafayette, IN (1985).
5. Mulay S.B., "Determinantal loci and the flag variety," Advances in Mathematics 74 (1989), 1-30.
6. Musili C., "Applications of standard monomial theory," preprint.
2. Abhyankar S., Enumerative Combinatorics of Young Tableaux, Marcel Dekker, New York, 1988.
3. Abhyankar S. and Kulkarni D.M., "On Hilbertian ideals," Linear Algebra and Its Applications 116 (1989), 53-76.
4. Kulkarni D.M., thesis, Purdue University, West Lafayette, IN (1985).
5. Mulay S.B., "Determinantal loci and the flag variety," Advances in Mathematics 74 (1989), 1-30.
6. Musili C., "Applications of standard monomial theory," preprint.