Authors: Lukas Braun ¹

• 1 Mathematik Universität Tübingen

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.

• 05A30 - \(q\)-calculus and related topics
• 13A50 - Actions of groups on commutative rings; invariant theory
• 13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
• 14M15 - Grassmannians, Schubert varieties, flag manifolds
• 33C90 - Applications of hypergeometric functions