Generic representation theory of finite fields in nondescribing\n characteristic

2014/05/01 by Nicholas J. Kuhn, Kuhn, Nicholas J. · 4 citations
Mathematics · #16D90 (Secondary) #18A25 (Primary) Secondary 20G05 #20M25 #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1405.0318

openalex publication_date 2014/05/01 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

Let Rep(F;K) denote the category of functors from finite dimensional F-vector\nspaces to K-modules, where F is a field and K is a commutative ring. We prove\nthat, if F is a finite field, and Char F is invertible in K, then the K-linear\nabelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ...,\nof the categories of K[GL(k,F)]-modules.\n As a consequence, if K is also a field, then small projectives are also\ninjective in Rep(F;K), and will have finite length. Even more is true if Char K\n= 0: the category Rep(F;K) will be semisimple.\n In a last section, we briefly discuss "q=1" analogues and consider\nrepresentations of various categories of finite sets.\n The main result follows from a 1992 result by L.G.Kovacs about the semigroup\nring K[M_n( F)].\n

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