ALGEBRAIC COMBINATORICS

no. 1
k-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan–Lusztig immanants
Chepuri, Sunita1; Sherman-Bennett, Melissa2
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.

In this note, we show that certain dual canonical basis elements of ℂ[SL m ] 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[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).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.257
Classification: 15A15, 05E10, 20C30
Keywords: immanants, total positivity, dual canonical basis
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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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/

[1] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | MR

[2] Brosowsky, Anna; Chepuri, Sunita; Mason, Alex Parametrizations of k-nonnegative matrices: cluster algebras and k-positivity tests, J. Combin. Theory Ser. A, Volume 174 (2020), p. 105217, 25 | DOI | MR | Zbl

[3] Chepuri, Sunita Generalizations of Total Positivity, ProQuest LLC, Ann Arbor, MI, 2020, 172 pages Thesis (Ph.D.)–University of Minnesota | MR

[4] Chepuri, Sunita; Kulkarni, Neeraja; Suk, Joe; Tang, Ewin Factorizations of k-nonnegative matrices, J. Comb., Volume 13 (2022) no. 2, pp. 201-250 | DOI | MR

[5] Chepuri, Sunita; Sherman-Bennett, Melissa 1324- and 2143-avoiding Kazhdan-Lusztig immanants and k-positivity, Canadian Journal of Mathematics (2021), pp. 1-33 | DOI

[6] Chmutov, Michael; Jiradilok, Pakawut; Stevens, James Double Rim Hook Cluster Algebras, 2021 | arXiv

[7] Choudhury, Projesh Nath Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution, 2021 | arXiv

[8] Choudhury, Projesh Nath; Kannan, M. Rajesh; Khare, Apoorva Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull, Bull. Lond. Math. Soc., Volume 53 (2021) no. 4, pp. 981-990 | DOI | MR | Zbl

[9] Du, Jie Canonical bases for irreducible representations of quantum GL n , Bull. London Math. Soc., Volume 24 (1992) no. 4, pp. 325-334 | DOI | MR | Zbl

[10] Fomin, Sergey; Zelevinsky, Andrei Totally nonnegative and oscillatory elements in semisimple groups, Proc. Amer. Math. Soc., Volume 128 (2000) no. 12, pp. 3749-3759 | DOI | MR | Zbl

[11] Goulden, I. P.; Jackson, D. M. Immanants of combinatorial matrices, J. Algebra, Volume 148 (1992) no. 2, pp. 305-324 | DOI | MR | Zbl

[12] Greene, Curtis Proof of a conjecture on immanants of the Jacobi-Trudi matrix, Linear Algebra Appl., Volume 171 (1992), pp. 65-79 | DOI | MR | Zbl

[13] Haiman, Mark Hecke algebra characters and immanant conjectures, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 569-595 | DOI | MR | Zbl

[14] Lusztig, G. Total positivity in reductive groups, Lie theory and geometry (Progr. Math.), Volume 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531-568 | DOI | MR | Zbl

[15] Pylyavskyy, Pavlo personal communication, 2018

[16] Rhoades, Brendon; Skandera, Mark Kazhdan-Lusztig immanants and products of matrix minors, J. Algebra, Volume 304 (2006) no. 2, pp. 793-811 | DOI | MR | Zbl

[17] Sjöstrand, Jonas Bruhat intervals as rooks on skew Ferrers boards, J. Combin. Theory Ser. A, Volume 114 (2007) no. 7, pp. 1182-1198 | DOI | MR | Zbl

[18] Skandera, Mark On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations, J. Pure Appl. Algebra, Volume 212 (2008) no. 5, pp. 1086-1104 | DOI | MR | Zbl

[19] Stembridge, J. R. Immanants of totally positive matrices are nonnegative, Bull. London Math. Soc., Volume 23 (1991) no. 5, pp. 422-428 | DOI | MR | Zbl

[20] Stembridge, J. R. Some conjectures for immanants, Canad. J. Math., Volume 44 (1992) no. 5, pp. 1079-1099 | DOI | MR | Zbl

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