Coloured generalised Young diagrams for affine Weyl-Coxeter groups

King, R.C. and Welsh, T.A. (2007) Coloured generalised Young diagrams for affine Weyl-Coxeter groups. The Electronic Journal of Combinatorics, 14 (1), 1-64.

Abstract

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)}_\ell, B^{(1)}_\ell, C^{(1)}_\ell, D^{(1)}_\ell, A^{(2)}_{2\ell}, A^{(2)}_{2\ell-1} or $D^{(2)}_{\ell+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 = s_{i_1}s_{i_2}\c ... s_{i_t} in terms of generators s_k of W, and any (reduced) expression for w to be obtained from T(w). The diagram T(w) is especially useful because w(\Lambda)-\Lambda may be readily obtained from T(w) for all \Lambda in the weight space of \g. With \ov{\g} a certain maximal finite dimensional simple Lie subalgebra of \g, we examine the set W_s of minimal right coset representatives
of \ov{W} in W, where \ov{W} is the Weyl-Coxeter group of \ov{\g}. For w\in W_s, 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 W_s is in bijection with such partitions.

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