Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers Let (mathscr F) stand for the flag complex associated to the lattice of proper subspaces of a finitedimensional vector space V. This paper aims at giving a (discrete) Morse theoretical proof of the fact that the kth skeleton of (mathscr F) is homotopy equivalent to a wedge of spheres of dimension (min k,dim (mathscr F)). The tight control provided by Morse theoretic methods (through an explicit discrete gradient field) allows us to give a formula for the number of spheres appearing in each of these wedge summands. As an application, we derive an explicit formula for the number of flags on V of a given dimension, i.e., the number of simplices in (mathscr F) of the given dimension. Rather than depending on generalized Galois numbers, our formula for flags is given in terms of weighted inversion statistics of the symmetric group.

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.