Hilbert functions of points on Schubert varieties in Grassmannians. (Q1419000)

The purpose of this article is to determine the Hilbert function of the tangent cone to points in a Schubert subvariety of a Grassmannian. \textit{S. S. Abhyankar} [``Enumerative combinatorics of Young tableaux'', Pure and Applied Mathematics 115 (1988; Zbl 0643.05001)] and \textit{J. Herzog} and \textit{N. V. Trung} [Adv. Math. 96, No. 1, 1--37 (1992; Zbl 0778.13022)] gave formulas for the multiplicity and the Hilbert function of the quotient ring of the determinantal ideal of the tangent cone at the identity coset, in the ring of the tangent space of the Grassmann variety. \textit{V. Lakshmibai} and \textit{J. Weyman} [Adv. Math. 84, No. 2, 179--208 (1990; Zbl 0729.14037)] gave a recursive formula for the multiplicity at any point, and from this formula \textit{J. Rosenthal} and \textit{A. Zelevinsky} [J. Algebr. Comb. 13, No. 2, 213--218 (2001; Zbl 1015.14025)] obtained a closed form for the multiplicity at any point. \textit{V. Kreiman} and \textit{V. Lakshmibai} [in: Algebra and algebraic geometry with applications, 553--563 (2002; Zbl 1092.14060)] obtained an expression for the Hilbert function at the identity coset in terms of the combinatorics of the Weyl group, and they recovered the interpretation of the multiplicity due to Herzog and Trung. They also reformulated their main result in terms of generic determinantal minors. In addition they conjectured an expression for the Hilbert function and the multiplicity when \(x\) is any point. In the present article the approach of Kreiman and Lakshmibai is clarified, and their expression for the Hilbert function and the multiplicity is extended to all points, as is their combinatorial interpretation of the multiplicity. \textit{C. Krattenthaler} [Sémin. Lothar. Comb. 45, B45c, 11 p. (2000; Zbl 0965.14023)] has given an interpretation of the Rosenthal-Zelevinsky formula using combinatorial techniques, and he proves the multiplicity formula of Kreiman and Lakshmibai and shows that their conjecture about the Hilbert function is equivalent to a certain finite problem. The approach of Krattenthaler is completely different from that of the present article. While Krattenthaler explores the combinatorial interpretations of the formula, the present article, in the spirit of Lakshmibai-Weyman and Kreiman-Lakshmibai, uses standard monomial theory to translate the problems from geometry to combinatorics.