The Zelevinsky involution, co-standard modules for quantum affine algebras and interval symmetric functions
Abstract
Let ℱn be the subcategory of finite-dimensional representations of the quantum loop algebra of sln+1 whose simple modules are indexed by multisegments consisting of intervals [a,b], where a,b are integers and 0<b−a≤ n. We define the co-standard module associated to a multisegment s to be the tensor product of the Kirillov--Reshetikhin modules, associated to the first node of the Dynkin diagram of sln+1, given by the segments of s. Under a suitable (and necessary) restriction on n, we show that its socle is isomorphic to the simple module given by the Zelevinsky dual s† of s. We then define an involution Φ of the Grothendieck ring 𝒦(ℱn) and prove that it maps the class of a simple module to the class of the simple module given by its Zelevinsky dual. We also show that on suitable subcategories Φ is induced by an exact autoequivalence reversing tensor products, and we give a t-analogue of the multiplicity identity in the quantum Grothendieck ring. Motivated by the determinantal formula for alternating snake modules, we realize 𝒦(ℱn) explicitly inside a polynomial ring in variables indexed by intervals, and, in the spirit of Macdonald's ninth variation of Schur functions, we show that the fundamental modules correspond to analogues of the elementary symmetric functions and the Kirillov--Reshetikhin modules to analogues of the complete homogeneous symmetric functions. This is compatible with Φ and the classical involution ω, and we prove that alternating snake modules give a family of Schur positive symmetric functions extending the skew Schur functions.
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BibTeX
@misc{brito2026zelevinsky,
title = {{The Zelevinsky involution, co-standard modules for quantum affine algebras and interval symmetric functions}},
author = {Matheus Brito and Vyjayanthi Chari},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05588},
eprint = {2611.05588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05588},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Zelevinsky involution, co-standard modules for quantum affine algebras and interval symmetric functions
AU - Matheus Brito
AU - Vyjayanthi Chari
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05588
UR - https://arxiv.org/abs/2611.05588
AB - Let ℱ_n be the subcategory of finite-dimensional representations of the quantum loop algebra of sl_n+1 whose simple modules are indexed by multisegments consisting of intervals [a,b], where a,b are integers and 0<b−a≤ n. We define the co-standard module associated to a multisegment s to be the tensor product of the Kirillov--Reshetikhin modules, associated to the first node of the Dynkin diagram of sl_n+1, given by the segments of s. Under a suitable (and necessary) restriction on n, we show that its socle is isomorphic to the simple module given by the Zelevinsky dual s^† of s. We then define an involution Φ of the Grothendieck ring 𝒦(ℱ_n) and prove that it maps the class of a simple module to the class of the simple module given by its Zelevinsky dual. We also show that on suitable subcategories Φ is induced by an exact autoequivalence reversing tensor products, and we give a t-analogue of the multiplicity identity in the quantum Grothendieck ring. Motivated by the determinantal formula for alternating snake modules, we realize 𝒦(ℱ_n) explicitly inside a polynomial ring in variables indexed by intervals, and, in the spirit of Macdonald's ninth variation of Schur functions, we show that the fundamental modules correspond to analogues of the elementary symmetric functions and the Kirillov--Reshetikhin modules to analogues of the complete homogeneous symmetric functions. This is compatible with Φ and the classical involution ω, and we prove that alternating snake modules give a family of Schur positive symmetric functions extending the skew Schur functions.
ER -