Ordered Bell numbers, Hermite polynomials, Skew Young Tableaux, and Borel orbits

TL;DR

This paper explores three different interpretations of the number of Borel orbits on the variety of complete quadrics, linking combinatorics, special functions, and geometric decompositions, and provides asymptotic estimates for this count.

Contribution

It introduces three novel interpretations of the Borel orbit count, connecting combinatorics, Hermite polynomials, and geometric cell decompositions.

Findings

Number of Borel orbits equals the count of standard Young tableaux on skew-diagrams.

Borel orbit count relates to specific values of a modified Hermite polynomial.

Asymptotic estimates for the number of orbits as quadrics' dimension grows.

Abstract

We give three interpretations of the number $b$ of orbits of the Borel subgroup of upper triangular matrices on the variety $\ms{X}$ of complete quadrics. First, we show that $b$ is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate $b$ to certain values of a modified Hermite polynomial. Third, we relate $b$ to a certain cell decomposition on $\ms{X}$ previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for $b$ as the dimension of the quadrics increases.