The probability of positivity in symmetric and quasisymmetric functions

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 the function tends to infinity.

Keywords

composition tableau, $e$-positive, fundamental-positive, Kostka number, quasisymmetric Schur-positive, Schur-positive, Young tableau

2010 Mathematics Subject Classification

Primary 05E05. Secondary 05E45, 60C05.

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The first-named author was supported in part by the National Sciences and Engineering Research Council of Canada, CRM-ISM, and the Canada Research Chairs Program.

The second-named author was supported in part by the National Sciences and Engineering Research Council of Canada, the Simons Foundation, and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program.

Received 10 March 2019

Published 11 May 2020