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

Chepuri, Sunita; Sherman-Bennett, Melissa

doi:10.5802/alco.257

k

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

Chepuri, Sunita ¹; Sherman-Bennett, Melissa ²

¹ Lafayette College Pardee Hall Easton PA 18042 (USA)
² University of Michigan 2074 East Hall Ann Arbor MI 48109 (USA)

Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 95-108.

Abstract

In this note, we show that certain dual canonical basis elements of $ℂ [S L_{m}]$ 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 $ℂ [S L_{m}]$ 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).

Received: 2022-01-12
Revised: 2022-07-07
Accepted: 2022-07-10
Published online: 2023-02-24

DOI: 10.5802/alco.257

Classification: 15A15, 05E10, 20C30
Keywords: immanants, total positivity, dual canonical basis

Author's affiliations:

Chepuri, Sunita ¹; Sherman-Bennett, Melissa ²

¹ Lafayette College Pardee Hall Easton PA 18042 (USA)
² University of Michigan 2074 East Hall Ann Arbor MI 48109 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Chepuri, Sunita and Sherman-Bennett, Melissa},
     title = {$k$-positivity of dual canonical basis elements from 1324- and 2143-avoiding {Kazhdan{\textendash}Lusztig} immanants},
     journal = {Algebraic Combinatorics},
     pages = {95--108},
     publisher = {The Combinatorics Consortium},
     volume = {6},
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Chepuri, Sunita; Sherman-Bennett, Melissa. $k$-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan–Lusztig immanants. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 95-108. doi : 10.5802/alco.257. https://alco.centre-mersenne.org/articles/10.5802/alco.257/

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