On stability and nonvanishing of homomorphism spaces between Weyl modules

Evangelou, Charalambos; Maliakas, Mihalis; Stergiopoulou, Dimitra-Dionysia

doi:10.5802/alco.397

Evangelou, Charalambos ¹; Maliakas, Mihalis ¹; Stergiopoulou, Dimitra-Dionysia ¹

¹ University of Athens Department of Mathematics Athens Greece

Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1761-1792.

Abstract

Consider the general linear group $G = G L_{n} (K)$ defined over an infinite field $K$ of positive characteristic $p$ . We denote by $Δ (λ)$ the Weyl module of $G$ which corresponds to a partition $λ$ . Let $λ, μ$ be partitions of $r$ and let $γ$ be a partition with parts divisible by $p$ . In the first main result of this paper, we find sufficient conditions on $λ, μ$ and $γ$ so that ${Hom}_{G} (Δ (λ), Δ (μ)) ≃ {Hom}_{G} (Δ (λ + γ), Δ (μ + γ))$ , thus providing an answer to a question of D. Hemmer. As corollaries we obtain stability and periodicity results for homomorphism spaces. In the second main result we find related sufficient conditions on $λ, μ$ and $p$ so that ${Hom}_{G} (Δ (λ), Δ (μ))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $μ$ and weight $λ$ .

Received: 2023-10-24
Revised: 2024-09-02
Accepted: 2024-09-02
Published online: 2025-01-07

Zbl

DOI: 10.5802/alco.397

Classification: 20G05, 05E10
Mots-clés : Weyl modules, general linear group, homomorphism spaces, stability, nonvanishing

Author's affiliations:

Evangelou, Charalambos ¹; Maliakas, Mihalis ¹; Stergiopoulou, Dimitra-Dionysia ¹

¹ University of Athens Department of Mathematics Athens Greece

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Evangelou, Charalambos and Maliakas, Mihalis and Stergiopoulou, Dimitra-Dionysia},
     title = {On stability and nonvanishing of homomorphism spaces between {Weyl} modules},
     journal = {Algebraic Combinatorics},
     pages = {1761--1792},
     publisher = {The Combinatorics Consortium},
     volume = {7},
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     year = {2024},
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Evangelou, Charalambos; Maliakas, Mihalis; Stergiopoulou, Dimitra-Dionysia. On stability and nonvanishing of homomorphism spaces between Weyl modules. Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1761-1792. doi : 10.5802/alco.397. https://alco.centre-mersenne.org/articles/10.5802/alco.397/

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