On the Decomposition of Modular Tensors (I)

Abstract

The present paper is the second part of [8] (brackets refer to bibliography) referred to below as Part I. We number sections, formulae, theorems, etc. consecutively from those in Part I, and use the same notation. For instance, f is a field of characteristic p with q (?_ cc ) elements; Zi is a t-vector space of dimension n; (5 = 65(n, f) is the full linear (modular) group of all f-automorphisms of 318 ; 93m is the space of all tensors of rank m over QI ; HIm is the Kronecker mth power representation of 6; 21m is the enveloping algebra of H1m. Superscripts zero indicate the analogous quantities defined over a field to of characteristic zero. The decomposition of non-modular tensors, or equivalently the determination of the reduced form of Ho is now a classical part of algebra. The objective of the present sequence of papers is to obtain a similar theory for modular tensors, or equivalently to study the reduced form of Hm . Any representation of 65 whose space is a subspace or factor space of 93m , or a direct sum of such spaces we call a tensor representation of 6. If q (and therefore 6) is finite we denote the group ring by r. One of the main results of the present paper is that there exists a faithful tensor representation of F (Th. X ?14). From this it follows from an unpublished theorem of Nesbitt that every representation of 6) is equivalent to a tensor representation, but we do not establish or apply this last result below. An important feature of the study of modular representations of groups has been the use of induced representations. One starts with a finite group and a non-modular representation of it, and after suitable number theoretic preparations take residue classes and obtain a modular representation. A generalization of this process would be to induce both the group and the representation. We leave for future investigation the determination of the general theory of such a process, and content ourselves here with the application of the idea to obtain from each irreducible representation of the non-modular full linear group a representation of the modular full linear group. This is done in ?9 below. In ?10 a character theory is developed for tensor representations of 6; this is applied in ?11 to prove that every irreducible representation of (M is equivalent to a tensor representation,1 and in ??12, 13 to obtain specific values of the irreducible and indecomposable modular characters for the representations H.m having m m. In ??15-20 the situation for m < 2p is cleaned up by re-