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Keywords:
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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