1324- and 2143-avoiding Kazhdan-Lusztig immanants and k-positivity

Abstract

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan-Lusztig polynomials, and were defined in (Rhoades-Skandera, 2006). Using results of (Haiman, 1993) and (Stembridge, 1991), Rhoades and Skandera showed that Kazhdan-Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan-Lusztig immanants are positive on $k$-positive matrices (matrices whose minors of size $k\times k$ and smaller are positive). We show that the Kazhdan-Lusztig immanant indexed by $v$ is positive on $k$-positive matrices when $v$ avoids 1324 and 2143 and for all non-inversions $i<j$ of $v$, either $j-i\le k$ or $v_j-v_i\le k$. Our main tool is Lewis Carroll's identity.

arXiv comment28 pages, 9 figures. v2: Rephrased assumption of main theorem in terms of non-inversions, expanded Section 5, various minor organizational and expository changes

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• Parametrizations of $k$-Nonnegative Matrices: Cluster Algebras and $k$-Positivity Tests