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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.

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
Full Text: DOI
[1] Bourbaki, N, ()
[2] Demazure, M, Désingularisation des variétés de Schubert, Ann. E.N.S., 6, 53-88, (1974) · Zbl 0312.14009
[3] Deodhar, V, On some geometric aspects of Bruhat orderings, I: a finer decompositions of Bruhat cells, Invent. math., 79, 499-511, (1985) · Zbl 0563.14023
[4] Edelman, P; Greene, C, Combinatorial correspondences for Young tableaux, balanced tableaux, and maximal chains in the weak Bruhat order, Contemp. math., 34, 155-162, (1984) · Zbl 0562.05008
[5] Edelman, P; Greene, C, Balanced tableaux, Adv. math., 63, 1, 42-99, (1987) · Zbl 0616.05005
[6] Farahat, H.K; Peel, M.H, J. algebra, 67, 280-304, (1980)
[7] Güemes, J.J, On the homology classes for the components of some fibers of Springer’s resolution, Asterisque, 173-174, 257-269, (1989) · Zbl 0704.20038
[8] Haiman, M.D, Dual equivalence with applications including a conjecture of proctor, (1989), preprint MIT · Zbl 0760.05093
[9] Hiller, H, ()
[10] James, G.D, (), LN 682
[11] James, G.D; Peel, M.H, J. algebra, 59, 343-364, (1979)
[12] Klachko, A.A, Orbits of maximal torus on the flag variety, Funct. anal., 19, 77-78, (1985), (in Russian)
[13] Kraśkiewicz, W, C.R. acad. sci. Paris, 309, 903-907, (1989), Serie I
[14] Lascoux, A; Schützenberger, M.P, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck de la variete de drapeaux, C.R. acad. sci. Paris, 295, 629-633, (1982), Serie I · Zbl 0542.14030
[15] Macdonald, I.G, Symmetric functions and Hall polynomials, (1979), Oxford University Press Oxford · Zbl 0487.20007
[16] Stanley, R, Europ. J. combin., 5, 359-372, (1984)
[17] Zhelobenko, D.P, Funct. anal., 21, 3, 11-21, (1987), (in Russian)
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