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

TL;DR

This paper demonstrates that specific dual canonical basis elements of the coordinate ring of SL_m are positive on matrices with positive minors up to size k, focusing on those associated with 1324- and 2143-avoiding permutations.

Contribution

It extends previous work by showing positivity of dual canonical basis elements related to particular Kazhdan-Lusztig immanants indexed by pattern-avoiding permutations.

Findings

Positivity of certain basis elements on k-positive matrices.

Extension of previous results to 1324- and 2143-avoiding permutations.

Use of Lewis Carroll's identity in the analysis.

Abstract

In this note, we show that certain dual canonical basis elements of $\mathbb{C}[SL_m]$ are positive when evaluated on $k$-positive matrices, matrices whose minors of size $k \times k$ and smaller are positive. Skandera showed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can be written in terms of Kazhdan-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 1324- and 2143-avoiding permutations. This extends previous work of the authors on Kazhdan-Lusztig immanants and uses similar tools, namely Lewis Carroll's identity (also known as the Desnanot-Jacobi identity).