Schur Polynomials and Pl\"ucker degree of Schubert Varieties

TL;DR

This paper explains how derivatives of Schur polynomials relate to the Pl"ucker degree of Schubert varieties, connecting combinatorics of Young tableaux with geometric degrees.

Contribution

It provides a new interpretation of the Pl"ucker degree of Schubert varieties via derivatives of Schur polynomials, linking algebraic and geometric perspectives.

Findings

Partial derivatives of Schur polynomials give Pl"ucker degrees

The generating function for degrees of Schubert varieties is determined

Corollaries relating to Young tableaux counts are discussed

Abstract

The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the following fact. If $\blamb$ is a partition of weight $d$, then the partial derivative of order $d$ with respect to $x_1$ of the Schur polynomial $S_\blamb(\bfx)$ coincides with the Pl\"ucker degree of the Schubert variety of dimension $d$ associated to $\blamb$, equal to the number of standard Young tableaux of shape $\blamb$. The generating function encoding all the degree of Schuberte varieties is determined and some (known) corollaries are also discussed. (The following is an informal report on the contributed talk given by the author during the INPANGA 2020[+1] meeting on Schubert Varieties.)