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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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