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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. Skanderashowed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can bewritten in terms of Kazhdan-Lusztig immanants, which were introduced by Rhoadesand Skandera. We focus on the basis elements which are expressed in terms ofKazhdan-Lusztig immanants indexed by 1324- and 2143-avoiding permutations. Thisextends previous work of the authors on Kazhdan-Lusztig immanants and usessimilar tools, namely Lewis Carroll's identity (also known as theDesnanot-Jacobi identity).

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