Showing papers on "Cartan matrix published in 1980"

A hyperbolic GCM Lie algebra and the Fibonacci numbers

Abstract: Introduction. Since their independent introduction by Kac [2], [3] and Moody [8],[9] there has been increasing interest in the Generalized Cartan Matrix (GCM) Liealgebras because of their connections with other parts of mathematics (see [4]). Forinstance, the Macdonald identities [7] were shown to be the "denominatorformula" for the Euclidean GCM Lie algebras [3], [10]. We are concerned here withan algebra which falls into the class of algebras called "hyperbolic", whose rootsystems were studied by Moody in [11]. An exciting connection was recently foundby Lepowsky and Moody in [6] between the rank 2 hyperbolic GCM Lie algebrasand Hubert modular surfaces with respect to a real quadratic field. For one ofthese algebras the quadratic field is Q(Vs ). In this paper we show that theFibonacci numbers are intimately related to this algebra by using them to expressthe Weyl-Macdonald-Kac denominator formula (§2). We do not compute the rootmultiplicities, but see §3.1. Preliminaries. The following preliminaries may be found in [6]. A 2 X 2integral matrix (aiy) of the form

The distribution of modular representations into blocks

Abstract: The p-modular representations of a finite group that are induced from a p-subgroup are investigated. A series of three results describing how these representations are distributed into p-blocks are presented. Several applications are discussed, including the result that there are a finite number of indecomposable p-modular representations (up to equivalence) in a p-block of a group if and only if its defect group is cyclic. The notation and point of view will be that of Dornhoff [2]. Fix a finite group G, a prime integer p, and a characteristic p field F. All modules are assumed to be finitely generated. The author wishes to thank Professor L. L. Scott for his contributions to the development of the results in this article. THEOREM 1. Suppose P is a p-subgroup of G, and L is an irreducible FG-module. Then there is a positive integer n = n(P, L) with the property that for any FP-module M, MG has L as a composition factor n(dim M) times. PROOF. Let M = Mn D D Ml D Mo = f0) be a composition series of FP-modules. Since P is a p-group, each factor is isomorphic to T, the trivial FP-module. Then MG = MG ** MG D MOG = fO) is a series of FGmodules with each factor isomorphic to TG. This gives the theorem except for the possibility of n being zero. To show that n is positive, it is enough to show L is a composition factor of TG. Again since P is a p-group, Homp(Lp, T) # f0). Thus by Frobenius reciprocity (see [5, Remark 2 in 7.1]), 0 = dim(Homp(Lp, T)) = dim(HomG(L, TG)). Hence L is a composition factor of TG, as required. THEOREM 2. Suppose P is a p-subgroup of G, and M is an FP-module. Then MG has direct summands in every RG-block. PROOF. By Theorem 1, every irreducible FG-module is a composition factor of MG. Hence by the structure of the Cartan matrix (see [2, 46.2]), the theorem holds. Received by the editors January 24, 1979. AMS (MOS) subject classifications (1970). Primary 20C05, 20C20.