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{\bf R.C. King and T.A. Welsh}
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{\bf Coloured Generalised Young Diagrams for Affine Weyl-Coxeter Groups}
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Coloured generalised Young diagrams $T(w)$ are introduced that are in
\font\fraktur=cmfrak10 \def\gfrak{\hbox{\fraktur g}}
bijective correspondence with the elements $w$ of the Weyl-Coxeter
group $W$ of $\gfrak$, where $\gfrak$ is any one of the classical
affine Lie algebras $\gfrak=A^{(1)}_\ell$, $B^{(1)}_\ell$,
$C^{(1)}_\ell$, $D^{(1)}_\ell$, $A^{(2)}_{2\ell}$, $A^{(2)}_{2\ell-1}$
or $D^{(2)}_{\ell+1}$. These diagrams are coloured by means of
periodic coloured grids, one for each $\gfrak$, which enable $T(w)$ to
be constructed from any expression $w=s_{i_1}s_{i_2}\cdots s_{i_t}$ 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 $\gfrak$.
With $\overline{\gfrak}$ a certain maximal finite dimensional simple
Lie subalgebra of $\gfrak$, we examine the set $W_s$ of minimal right
coset representatives of $\overline{W}$ in $W$, where $\overline{W}$
is the Weyl-Coxeter group of $\overline{\gfrak}$. 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 $\gfrak$. Indeed, it is shown
that $W_s$ is in bijection with such partitions.
\bye