JOURNAL OF
ALGEBRAIC
COMBINATORICS

Home > Vol 22, No 3 (2005)

Christian F. Krattenthaler

Institut für Mathematik der Universität Wien Nordbergstraße 15 A-1090 Wien Austria

We prove a conjecture by Kreiman and Lakshmibai on a combinatorial description of multiplicities of points on Schubert varieties in Graßmannian in terms of certain sets of reflections in the corresponding Weyl group. The proof is accomplished by relating these sets of reflections to the author's previous combinatorial interpretation of these multiplicities in terms of non-intersecting lattice paths (Séminaire Lotharingien Combin. 45 (2001), Article B45c). Moreover, we provide a compact formula for the Hilbert series of the tangent cone to a Schubert variety in a Graßmannian assuming the truth of another conjecture of Kreiman and Lakshmibai.

1. S.C. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Birkh\ddot auser, Boston, 2000.
2. A. Conca and J. Herzog, “On the Hilbert function of determinantal rings and their canonical module,” Proc. Amer. Math. Soc. 122 (1994), 677-681.
3. M. Desainte-Catherine, X. Viennot, Enumeration of certain Young tableaux with bounded height, in Combinatoire énumérative, G. Labelle, P. Leroux (Eds.), Springer-Verlag, Berlin, Heidelberg, New York, 1986, pp. 58-67.
4. W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.
5. I.M. Gessel and X. Viennot, Determinants, Paths, and Plane Partitions, preprint, 1989; available at