The probability of positivity in symmetric and quasisymmetric functions

TL;DR

This paper calculates the probability that a nonnegative combination of basis vectors in a finite-dimensional space is also a nonnegative combination in another basis, with applications to symmetric and quasisymmetric functions.

Contribution

It provides a general method to compute positivity probabilities for basis transformations and applies it to various classes of symmetric and quasisymmetric functions.

Findings

Probability tends to zero as degree increases

Recovered recent results on Schur-positivity

Extended to e- and h-positivity and quasisymmetric functions

Abstract

Given an element in a finite-dimensional real vector space, $V$, that is a nonnegative linear combination of basis vectors for some basis $B$, we compute the probability that it is furthermore a nonnegative linear combination of basis vectors for a second basis, $A$. We then apply this general result to combinatorially compute the probability that a symmetric function is Schur-positive (recovering the recent result of Bergeron--Patrias--Reiner), $e$-positive or $h$-positive. Similarly we compute the probability that a quasisymmetric function is quasisymmetric Schur-positive or fundamental-positive. In every case we conclude that the probability tends to zero as the degree of a function tends to infinity.