On the Hilbert series of the Grassmannian

TL;DR

This paper computes the Hilbert series of the complex Grassmannian, revealing its h-polynomial matches the k-Narayana polynomial, and provides simplified formulas for these polynomials using invariant theory and hypergeometric transforms.

Contribution

It introduces a new connection between the Hilbert series of Grassmannians and k-Narayana polynomials, with simplified formulas derived via hypergeometric transforms.

Findings

Hilbert series of the complex Grassmannian matches the k-Narayana polynomial.

Simplified formulas for h-polynomials of Schubert varieties are provided.

Hypergeometric Euler transform yields new formulas for k-Narayana numbers.

Abstract

We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods and show that its h-polynomial coincides with the k-Narayana polynomial. We give a simplified formula for the h-polynomial of Schubert varieties. 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.