Geodesic
Hilbert series of the Grassmannian and $k$-Narayana numbers
Communications in Mathematics, Tome 27 (2019) no. 1, pp. 27-41

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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.
Classification : 13D40, 14M15, 33C90
Keywords: Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform 
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     author = {Braun, Lukas},
     title = {Hilbert series of the {Grassmannian} and $k${-Narayana} numbers},
     journal = {Communications in Mathematics},
     pages = {27--41},
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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/