$k$-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan-Lusztig immanants

In this note, we show that certain dual canonical basis elements of C[SLm] are positive when evaluated on k-positive matrices, matrices whose minors of size k × k and smaller are positive. Skandera showed that all dual canonical basis elements of C[SLm] can be written in terms ofKazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324and 2143-avoiding permutations. This extends previous work of the authors on KazhdanLusztig immanants and uses similar tools, namely Lewis Carroll’s identity (also known as the Desnanot-Jacobi identity).