Lifting modular representations of finite groups

Abstract

Let (7r) be the maximal ideal of the ring R of P-integral elements of an algebrai'c number field K, where P is a prime of K dividing the rational prime p. The natural homomorphism from R to K=R/(7r) induces a map S->S from the set of representations by matrices with coefficients in R of a finite group G into the set of representations of G in K. The lifting problem in modular representation theory is to determine whether for a given representation T of G in K there exists a representation S of G by R-matrices such that S= T. In this paper we introduce a notion of lifting projective modular representations from characteristic p to characteristic zero and show how this concept may be applied to the lifting problem. NOTATION. Throughout this paper G denotes a finite group of order I GI and K denotes an algebraic number field which is a splitting field for G. Let p be a rational prime and let R be the ring of P-integral elements of K, where P is a prime of K dividing p. Let (r) be the maximal ideal of R and set K=R/(r). K is a finite field of characteristic p which is a splitting field for G. For aCR, set d=a+(ir)CK. If A = (aij) is a matrix with entries in R (R-matrix) we denote by A the matrix (aij). By a linear representation of G in a field L we shall understand a homomorphism from G into GL(m, L) for some m. By a projective integral representation (resp. projective modular representation) of G in R (resp. K) we mean a map T of G into GL(m, K) (resp. GL(m, K)) satisfying T(1)1=lm, T(g)T(h)=a(g, h) T(gh) where a(g, h) CR (resp. K) and T(g) has entries in R (resp. K) for all g, hCG. a is called the factor set associated with T. If a(g, h) =f(g, h) = 1 for all g, hC G, T is a linear integral representation (resp. linear modular representation). We identify linear representations with projective representations having trivial factor sets. We refer the reader to [3] and [7] for the relevant theory. DEFINITION. Let T be a projective modular representation of G in K and let a be the associated factor set. T is projectively liftable if there exists a projective integral representation S of G in R with factor set A such that S(g) = T(g) for all gCG. If a(g, h) =,3(g, h) = 1 for all g, hCtG (i.e. S and T are linear representations), we say that T is liftable.