On the Dimension of Some Modular Irreducible Representations of the Symmetric Group

We compute the dimension of some irreducible representations of the sym­ metric groups in characteristic ρ (Theorem 2). The representations considered here are associated with Young diagrams m : πΐχ > m 2 > ... > m/ such that πΐχ — mi < (jp — I). The formula is based on a variant of Verlinde's formula which computes some tensor product multiplicities of indecomposable modules for GLi(¥p), as it is proved in [7] [8]. Mathematics Subject Classification (1991): 20 C 30 Introduction: In this paper we will compute the dimension of some modular irreducible representations of the symmetric group EJV, (see Theorem 2 below for a precise statement). By a classical formula of Frobenius, the dimension of a characteristic zero irreducible Σ^γrepresentation is given as the number of standard tableaux of a given shape. However in the modular case, it is not very convenient to use the standard tableaux to describe these dimensions. Instead, we will use a combinatorial description based on paths in the set of Young diagrams. For this reason, we will first "translate" the classical Frobenius formula in terms of paths. Recall that a Young diagram of height < / is a sequence of non-negative integers m : mi > m 2 > ... > mi. Pictorially one represents a Young diagram as follows, namely a set of boxes with m\ boxes on the first line, m2 boxes on the second line and so on.... The total number m\ + m2 + ··. of boxes will be called the size of the Young diagram m. In order to give a completely rigorous definition, we also require that two Young diagrams which can be obtained one from the other one by adding or removing empty lines are considered as identical. For example the Young diagrams 3 > 1 and 3 > 1 > 0 are viewed as the same. Let Yi be the set of all Young diagrams of height < /. We consider Y*/ as an oriented * Institut de Recherches Mathématiques Avancées, Université Louis Pasteur, 7 rue René Descartes, F-67084 Strasbourg Cedex