Hilbert series of the Grassmannian and k-Narayana numbers

Lukas Braun 1
  • 1 Mathematisches Institut, Universität Tübingen, 72076, Tübingen, Germany

Abstract

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.

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