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Keywords:
Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform
Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform
Summary:
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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