2024/11/08 by Serge Bouc, Bouc, Serge, Denız Yılmaz +1
Computer Science · Mathematics · #16S50 #18B99 #20C20 #20J15 #Advanced Algebra and Geometry #Category Theory (math.CT) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2411.05700
openalex publication_date 2024/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a prime number. We consider diagonal p-permutation functors over a (commutative, unital) ring R in which all prime numbers different from p are invertible. We first determine the finite groups G for which the associated essential algebra ER(G) is non zero: These are groups of the form G=L\rtimes ⟨ u⟩, where (L,u) is a DΔ-pair. When R is an algebraically closed field \mathbbF of characteristic 0 or p, this yields a parametrization of the simple diagonal p-permutation functors over \mathbbF by triples (L,u,W), where (L,u) is a DΔ-pair, and W is a simple \mathbbFOut(L,u)-module. Finally, we describe the evaluations of the simple functor SL,u,W parametrized by the triple (L,u,W). We show in particular that if G is a finite group and \mathbbF has characteristic p, the dimension of S_L,1,\mathbbF(G) is equal to the number of conjugacy classes of p-regular elements of G with defect isomorphic to L.