Combinatorial Constructions of Weight Bases: The GelfandTsetlin Basis This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of mathfrak sl_n, the celebrated GelfandTsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a new interpretation and a simple combinatorial proof of the GelfandTsetlin formulas based on a rational function identity (all the known proofs use more sophisticated algebraic tools). A constructive approach to the GelfandTsetlin formulas is then given, based on a simple algorithm for solving certain equations on the lattice of semistandard Young tableaux. This algorithm also implies certain extremal properties of the GelfandTsetlin basis.

Coloured Generalised Young Diagrams for Affine WeylCoxeter Groups Coloured generalised Young diagrams T(w) are introduced that are in bijective correspondence with the elements w of the WeylCoxeter group W of mathfrakg, where mathfrakg is any one of the classical affine Lie algebras mathfrakgA(1)_ell, B(1)_ell, C(1)_ell, D(1)_ell, A(2)_2ell, A(2)_2ell1 or D(2)_ell1.  These diagrams are coloured by means of periodic coloured grids, one for each mathfrakg, which enable T(w) to be constructed from any expression ws_i_1s_i_2cdots 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 mathfrakg.   With overlinemathfrakg a certain maximal finite dimensional simple Lie subalgebra of mathfrakg, we examine the set W_s of minimal right coset representatives of overlineW in W, where overlineW is the WeylCoxeter group of overlinemathfrakg.  For win W_s, we show that T(w) has the shape of a partition (or a slight variation thereof) whose rcore takes a particularly simple form, where r or r2 is the dual Coxeter number of mathfrakg.  Indeed, it is shown that W_s is in bijection with such partitions.