The immanant of a matrix \(M=(m_{i,j})\in\mathrm{M}_{n\times n}(\mathbb{C})\) associated with a function \(f:S_n\longrightarrow \mathbb{C}\) is defined as \begin{align*} \mathrm{Imm}_f:& \, \mathrm{M}_{n\times n}(\mathbb{C})\longrightarrow \mathbb{C}\\ & \, (m_{i,j})\mapsto \sum_{w \in S_n}f(w) \prod_{i=1}^n m_{i,w(i)}, \end{align*} where \(n\in \mathbb{N}\) and \(S_n\) is the symmetric group on \(n\) elements. The notable examples of immanants are the determinant and the permanent of a matrix. Here the corresponding functions \(f\) are the sign function of the permutation and a constant function, i.e., \(f(w)=\mathrm{sgn}(w)\) and \(f(w)\equiv 1\), respectively.
Given a \(k\)-positive matrix, i.e., a matrix with all minors of size at most \(k\) being positive, an immanant is said to be \(k\)-positive if it is positive on all \(k\)-positive matrices. Let \(v\in S_n.\) The Kazhdan-Lusztig immanant is defined by \[\mathrm{Imm}_v(M)=\sum_{w \in S_n}(-1)^{l(w)-l(v)}P_{w_0w,w_0v}(1)\prod_{i=1}^n m_{i,w(i)},\] where \(l(w)\) is the length of \(w\), \(w_0\) is the longest permutation in \(S_n,\) and \(P_{x,y}(q)\) is the Kazhdan-Lusztig polynomial in \(q\), for \(x,y \in S_n.\)
Motivated by a conjecture of P. Pylyavskyy [personal communication, 2018] about the \(k\)-positivity of Kazhdan-Lusztig immanants for avoiding pattern permutations, the authors provide a description of certain \(k\)-positive Kazhdan-Lusztig immanants. Theorem 1.3 shows that \(\mathrm{Imm}_v\) is \(k\)-positive if \(v\in S_n\) is avoiding \(1324\) and \(2143\), and such that if \(i<j\) and \(v(i)<v(j),\) then \(j-i\le k\) or \(v(j)-v(i)\le k\). The first half of the paper proves this result by relating permutations to graphs and making use of Dodgson’s condensation. Later parts of the paper discuss the conditions of the main result in relation to the conjecture. As a byproduct, the paper expands on the motivations for the following question: For which \(v\in S_n\) do we have \(\mathrm{Imm}_v\) \(k\)-positive? Generalisations to semisimple Lie groups and cluster algebras are discussed as well.