Geodesic
Hilbert series of the Grassmannian and $k$-Narayana numbers
Communications in Mathematics, Tome 27 (2019) no. 1, pp. 27-41 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$-Hilbert series is a Vandermonde-like determinant. We show that the $h$-polynomial of the Grassmannian coincides with the $k$-Narayana polynomial. A simplified formula for the $h$-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$-Narayana numbers, i.e.~the $h$-polynomial of the Grassmannian.
We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$-Hilbert series is a Vandermonde-like determinant. We show that the $h$-polynomial of the Grassmannian coincides with the $k$-Narayana polynomial. A simplified formula for the $h$-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$-Narayana numbers, i.e.~the $h$-polynomial of the Grassmannian.
Classification : 13D40, 14M15, 33C90
Keywords: Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform 
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Braun, Lukas. Hilbert series of the Grassmannian and $k$-Narayana numbers. Communications in Mathematics, Tome 27 (2019) no. 1, pp. 27-41. http://geodesic.mathdoc.fr/item/COMIM_2019_27_1_a2/

[1] Berman, J., Köhler, P.: Cardinalities of finite distributive lattices. Mitt. Math. Sem. Giessen, 121, 1976, 103-124, | MR

[2] Chipalkatti, J.V.: Notes on Grassmannians and Schubert varieties. Queen's Papers in Pure and Applied Mathematics, 13, 119, 2000,

[3] Derksen, H., Kemper, G.: Computational invariant theory. 2015, Springer, | MR

[4] Ghorpade, S.R.: Note on Hodge's postulation formula for Schubert varieties. Lecture Notes in Pure and Applied Mathematics, 217, 2001, 211-220, Marcel Dekker, New York, | MR

[5] Gross, B.H., Wallach, N.R.: On the Hilbert polynomials and Hilbert series of homogeneous projective varieties. Arithmetic geometry and automorphic forms, 19, 2011, 253-263, | MR

[6] Hodge, W.V.D.: A note on $k$-connexes. Mathematical Proceedings of the Cambridge Philosophical Society, 38, 2, 1942, 129-143, Cambridge University Press, | DOI | MR

[7] Hodge, W.V.D: Some enumerative results in the theory of forms. Mathematical Proceedings of the Cambridge Philosophical Society, 39, 1, 1943, 22-30, Cambridge University Press, | DOI | MR

[8] Hodge, W.V.D., Pedoe, D.: Methods of algebraic geometry, Volume 2. 1952, Cambridge University Press, | MR

[9] Lakshmibai, V., Brown, J.: The Grassmannian Variety: Geometric and Representation-theoretic Aspects. 42, 2015, Springer, | MR

[10] Littlewood, D.E.: On the number of terms in a simple algebraic form. Mathematical Proceedings of the Cambridge Philosophical Society, 38, 4, 1942, 394-396, Cambridge University Press, | DOI | MR

[11] MacMahon, P.A.: Combinatory analysis. 1916, Cambridge University Press, | MR

[12] Maier, R.: A generalization of Euler's hypergeometric transformation. Transactions of the American Mathematical Society, 358, 1, 2006, 39-57, | DOI | MR

[13] Miller, A.R., Paris, R.B.: Euler-type transformations for the generalized hypergeometric function ${}_{r+2}F_{r+1}(x)$. Zeitschrift für angewandte Mathematik und Physik, 62, 1, 2011, 31-45, | MR

[14] Miller, A.R., Paris, R.B.: Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mountain Journal of Mathematics, 43, 1, 2013, 291-327, | DOI | MR

[15] Mukai, S.: An introduction to invariants and moduli. 2003, Cambridge University Press, Cambridge Studies in Advanced Mathematics 81. | MR

[16] Nanduri, R.: Hilbert coefficients of Schubert varieties in Grassmannians. Journal of Algebra and Its Applications, 14, 3, 2015, 1550036, | DOI | MR

[17] Narayana, T.V.: A partial order and its applications to probability theory. Sankhya: The Indian Journal of Statistics, 1959, 91-98, | MR

[18] Santos, F., Stump, C., Welker, V.: Noncrossing sets and a Gra\IL2\ss mann associahedron. Forum of Mathematics, Sigma, 5, 2017, 49p, Cambridge University Press, | MR

[19] Stanley, R.P.: Theory and application of plane partitions. Part 2. Studies in Applied Mathematics, 50, 3, 1971, 259-279, | DOI | MR

[20] Sturmfels, B.: Gröbner bases and convex polytopes. 1996, American Mathematical Society, University Lecture Series Vol. 8.. | MR

[21] Sulanke, R.A.: Generalizing Narayana and Schröder numbers to higher dimensions. The Electronic Journal of Combinatorics, 11, 1, 2004, 54p, | DOI | MR

[22] Sulanke, R.A.: Three dimensional Narayana and Schröder numbers. Theoretical computer science, 346, 2--3, 2005, 455-468, | DOI | MR