Coloured Generalised Young Diagrams for Affine Weyl-Coxeter Groups

Coloured generalised Young diagrams T (w) are introduced that are in bijective correspondence with the elements w of the Weyl-Coxeter group W of g, where g is any one of the classical affine Lie algebras g = A (1) ` , B (1) ` , C (1) ` , D (1) ` , A (2) 2` , A (2) 2`−1 or D (2) `+1. These diagrams are coloured by means of periodic coloured grids, one for each g, which enable T (w) to be constructed from any expression w = si1si2 · · · sit in terms of generators sk of W , and any (reduced) expression for w to be obtained from T (w). The diagram T (w) is especially useful because w(Λ)−Λ may be readily obtained from T (w) for all Λ in the weight space of g. With g a certain maximal finite dimensional simple Lie subalgebra of g, we examine the set Ws of minimal right coset representatives of W in W , where W is the Weyl-Coxeter group of g. For w ∈ Ws, we show that T (w) has the shape of a partition (or a slight variation thereof) whose r-core takes a particularly simple form, where r or r/2 is the dual Coxeter number of g. Indeed, it is shown that Ws is in bijection with such partitions.