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

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). For $v$ a permutation avoiding 1324 and 2143, we give a sufficient condition on $k$ so that the Kazhdan-Lusztig immanant indexed by $v$ is positive on $k$-positive matrices. Our main tool is Lewis Carroll\u0027s identity.