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Combinatorial aspects of the Lascoux-Schützenberger tree. (English) Zbl 1018.05102
For a permutation $$\sigma$$ of the symmetric group $$S_n$$, let $$\text{Red}(\sigma)$$ be the set of all reduced decompositions $$\sigma=s_{a_1}\cdots s_{a_l}$$, i.e. presentations of $$\sigma$$ as a product of minimal length with respect to the generators $$s_i=(i,i+1)$$, $$i=1,\dots,n-1$$. As an approach to the fundamental problem to determine the cardinality of $$\text{Red}(\sigma)$$ for a fixed $$\sigma$$, R. P. Stanley [Eur. J. Comb. 5, 359-372 (1984; Zbl 0587.20002)] introduced a function $$F_{\sigma}(X)$$. He showed that $$F_{\sigma}(X)$$ is symmetric and for the permutation $$\sigma=(n,n-1,\dots,1)$$, the element of longest length, the number of reduced words is equal to the number of standard Young tableaux of staircase shape $$(n-1,n-2,\dots,1)$$. Stanley also conjectured that the symmetric function $$F_{\sigma}(X)$$ is Schur positive. The conjecture was confirmed by P. Edelman and C. Green [in: Combinatorics and algebra, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 34, 155-162 (1984; Zbl 0562.05008) and Adv. Math. 63, 42-99 (1987; Zbl 0616.05005)] using the technique of balanced tableaux.
The main contribution of the paper under review is the construction of a correspondence $$\Theta_{\sigma}$$ which sends the reduced decomposition $$w\in \text{Red}(\sigma)$$ to a pair $$(\alpha(w),T(w))$$, where $$\alpha(w)$$ is a Grassmanian permutation (a permutation with only one descent) and $$T(w)$$ is a standard tableau of shape $$\lambda'(\alpha(w))$$. The main idea is to associate a line diagram to each word $$w$$ which illustrates the trajectories of the numbers $$1,2,\dots,n$$ as they are rearranged by successive simple transpositions. The proof that $$\Theta_{\sigma}$$ is a bijection is quite simple and its properties can be established in a straightforward manner. This gives an elementary proof of the Schur positivity of the Stanley symmetric functions. The author also obtains a simple and purely combinatorial proof of the version of the Littlewood-Richardson rule given by A. Lascoux and M.-P. Schützenberger [Lett. Math. Phys. 10, 111-124 (1985; Zbl 0586.20007)].

##### MSC:
 500000 Symmetric functions and generalizations 5e+15 Combinatorial aspects of groups and algebras (MSC2010)
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##### References:
 [1] P. Edelman, C. Greene, Combinatorial correspondences for Young tableaux, balanced tableaux, and maximal chains in the weak Bruhat order of Sn, in: Combinatorics and Algebra (Boulder, CO, 1983), Amer. Math. Soc., Providence, RI, 1984, pp. 155-162. [2] Edelman, P.; Greene, C., Balanced tableaux, Adv. in math., 63, 1, 42-99, (1987) · Zbl 0616.05005 [3] A.M. Garsia, The saga of reduced factorizations of elements of the symmetric group, Publications du LaCIM, Université du Québec á Montréal, Canada, Vol. 29, 2002. [4] Lascoux, A.; Schützenberger, M.-P., Schubert polynomials and the Littlewood-Richardson rule, Lett. math. phys., 10, 2-3, 111-124, (1985) · Zbl 0586.20007 [5] Stanley, R.P., On the number of reduced decompositions of elements of Coxeter groups, European J. combin., 5, 4, 359-372, (1984) · Zbl 0587.20002
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