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1324- and 2143-avoiding Kazhdan–Lusztig immanants and k-positivity

Part of: Algebraic combinatorics Basic linear algebra Representation theory of groups

Published online by Cambridge University Press:  14 May 2021

Sunita Chepuri  and
Melissa Sherman-Bennett
Sunita Chepuri*
Affiliation:
University of Michigan, CMCC, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109, USA
Melissa Sherman-Bennett
Affiliation:
University of California at Berkeley, Evans Hall, Berkeley, CA, USA e-mail: m_shermanbennett@berkeley.edu
*
e-mail: chepuri@umich.edu
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Abstract

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan–Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan–Lusztig polynomials, and were defined by Rhoades and Skandera (2006, Journal of Algebra 304, 793–811). Using results of Haiman (1993, Journal of the American Mathematical Society 6, 569–595) and Stembridge (1991, Bulletin of the London Mathematical Society 23, 422–428), Rhoades and Skandera showed that Kazhdan–Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan–Lusztig immanants are positive on k-positive matrices (matrices whose minors of size $k \times k$ and smaller are positive). The Kazhdan–Lusztig immanant indexed by v is positive on k-positive matrices if v avoids 1324 and 2143 and for all noninversions $i< j$ of v, either $j-i \leq k$ or $v_j-v_i \leq k$ . Our main tool is Lewis Carroll’s identity.

Keywords

Immanantstotal positivity

MSC classification

Primary: 15A15: Determinants, permanents, other special matrix functions
Secondary: 05E10: Combinatorial aspects of representation theory 20C30: Representations of finite symmetric groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 52 - 82
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author was partially supported by NSF RTG grant DMS-1745638. The second author was supported by NSF grant DGE-1752814.

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