Kazhdan-Lusztig Immanants for k-Positive Matrices
Immanants are matrix functionals that generalize the determinant. One notable family of immanants are the Kazhdan-Lusztig immanants. These immanants are indexed by permutations and are defined as sums involving Kazhdan-Lusztig polynomials. One notable property of Kazhdan-Lustzig immanants is that they are nonnegative on totally positive matrices. We give a condition on permutations that allows us to extend this theorem to the setting of k-positive matrices. We also conjecture a larger class of permutations for which our theorem holds true.