2025/09/01 by Álvaro Gutiérrez, Gutiérrez, Álvaro, Álvaro L. Martínez +5
Mathematics · #05A30 #05E05 (Secondary) #20C20 (Primary) 05E10 #20G05 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2509.01490
openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Δλ be the Weyl functor for the partition λ and let E be the natural 2-dimensional representation of SL2(\mathbbF), where \mathbbF is an arbitrary field. We give an explicit isomorphism showing that any SL2(\mathbbF)-plethysm Δ(M,1N)Symd E factors as a tensor product of two simpler SL2(\mathbbF)-plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity \binomab\binombc=\binomac\binoma-cb-c, obtaining a new family of SL2(\mathbbF)-isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of SL2(\mathbbF).