Proper permutations, Schubert geometry, and randomness

TL;DR

This paper introduces proper permutations as a geometric criterion for Schubert varieties to be Levi-spherical and analyzes the probability of a permutation being proper, showing it diminishes as permutations grow large.

Contribution

It defines proper permutations and establishes their significance in Schubert geometry, providing probabilistic results about their prevalence among all permutations.

Findings

Probability that a random permutation is proper tends to zero as permutation size increases.

Properness is a necessary geometric condition for Levi-sphericity of Schubert varieties.

Abstract

We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation is proper goes to zero in the limit.