The representation ring of SL₂(\mathbbF_p) and stable modular plethysms of its natural module in characteristic p

2022/10/17 by Turek, Pavel
#05E05 #05E10 #19A22 #20C20 (Primary) #20C33 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2210.08943

Abstract

Let p be an odd prime and let k be a field of characteristic p. We provide a practical algebraic description of the representation ring of kSL₂(\mathbbF_p) modulo projectives. We then investigate a family of modular plethysms of the natural kSL₂(\mathbbF_p)-module E of the form ∇^νSym^l E for a partition ν of size less than p and 0≤ l≤ p-2. Within this family we classify both the modular plethysms of E which are projective and the modular plethysms of E which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of E are replaced by kSL₂(\mathbbF_p)-modules of the form ∇^ν V, where V is a non-projective indecomposable kSL₂(\mathbbF_p)-module and |ν|

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