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Reduced decompositions in Weyl groups. (English) Zbl 0828.05064
Let \(R\) be a root system with fixed basis \(\Sigma\) and let \(W\) be the Weyl group of it. For any element \(w\in W\), let \(\Gamma_w\) be the set of inversions of \(w\), i.e. the set of positive roots \(\alpha\) such that \(w(\alpha)\) is negative. It is known that if \(w= s_1 s_2\dots s_k\) is a reduced decomposition in \(W\), with \(s_i\) being reflection with respect to \(\alpha_i\in \Sigma\), then \(\Gamma_w\) is equal to the set of \(\theta_i= s_k ssbk- 1\dots s_{i+ 1}(\alpha_i)\), \(i= 1, 2,\dots, k\). In this way parameterization of reduced decompositions of \(w\) by some linear orders of \(\Gamma_w\) is obtained.
In the paper, linear orders of \(\Gamma_w\) are replaced by an equivalent notion of \(w\)-tableau and combinatorial characterization of standard \(w\)-tableaux, i.e. tableaux corresponding to reduced decompositions are given. For some special elements \(w\), \(w\)-tableaux are in natural one-to-one correspondence with standard Young tableaux which leads to explicit formulae for the number of reduced decompositions of \(w\). If \(W\) is the symmetric group \(S_n\) and \(w\) is any element of \(W\) then slight modification of tableau constructions of irreducible representations of the symmetric groups allows to define a linear action of the symmetric group \(S_{\ell(w)}\) on the space spanned by standard \(w\)-tableaux. It is an interpretation of Stanley’s result [R. P. Stanley, Eur. J. Comb. 5, 359-372 (1984; Zbl 0587.20002)] on the number of reduced decompositions of permutations in the framework of representation theory.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
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