Standard Young tableaux for nite root systems

The study of representations of aane Hecke algebras has led to a new notion of shapes and standard Young tableaux which works for the root system of any nite Coxeter group. This paper is completely independent of aane Hecke algebra theory and is purely combinatorial. We deene generalized shapes and standard Young tableaux and show that these new objects coincide with the classical ones for root systems of Type A. The classical notions of conjugation of shapes, ribbon shapes, axial distances, and the row reading and column reading standard tableaux, have natural generalizations to the root system case. In the nal section we give an interpretation of the shapes and standard tableaux for root systems of Type C which is in a form similar to classical theory of shapes and standard tableaux. 0. Introduction In my recent work on representations of aane Hecke algebras Ra1] I have been led to a generalization of standard Young tableaux. These generalized tableaux are important in the context of representation theory because the standard tableaux model the internal structure of irreducible representations of the aane Hecke algebra. In fact, most of the time the number of tableaux of a given shape is the same as the dimension of the corresponding irreducible representation of the aane Hecke algebra. In this paper I introduce and study generalized shapes and standard tableaux purely combina-torially. The main theorem is that the generalized standard tableaux of a given shape describe the connected components of a certain graph, the calibration graph. It is this graph which is intimately connected to the structure of representations of aane Hecke algebras. In the Type A case the generalized shapes can be converted into \placed conngurations of boxes". This conversion is nontrivial and is the subject of Section 3. In the cases where this placed connguration of boxes is a placed skew shape the generalized standard tableaux coincide with the classical standard tableaux of a skew shape. The generalized skew shapes play a major role in the results on representations of aane Hecke algebras which are obtained in Ra1]. In Section 1 I give deenitions of (a) skew shapes, (b) ribbon shapes, (c) axial distances, (d) conjugation of shapes, and (e) row reading and column reading tableaux, in the generalized setting. In Section 4 it is shown that these deenitions yield the classical versions of these objects in the Type A case. The last …