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Coloured generalised Young diagrams for affine Weyl-Coxeter groups. (English) Zbl 1161.05069

Summary: Coloured generalised Young diagrams \(T(w)\) are introduced that are in bijective correspondence with the elements \(w\) of the Weyl-Coxeter group \(W\) of \(\mathfrak g\), where \(\mathfrak g\) is any one of the classical affine Lie algebras \(\mathfrak g=A^{(1)}_\ell\), \(B^{(1)}_\ell\), \(C^{(1)}_\ell\), \(D^{(1)}_\ell\), \(A^{(2)}_{2\ell}\), \(A^{(2)}_{2\ell-1}\) or \(D^{(2)}_{\ell+1}\). These diagrams are coloured by means of periodic coloured grids, one for each \(\mathfrak g\), 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 \(\mathfrak g\).
With \(\overline{\mathfrak g}\) a certain maximal finite dimensional simple Lie subalgebra of \(\mathfrak g\), 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 \(\overline {\mathfrak g}\). 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 \(\mathfrak g\). Indeed, it is shown that \(W_s\) is in bijection with such partitions.

MSC:

05E10 Combinatorial aspects of representation theory
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05A17 Combinatorial aspects of partitions of integers
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