Combinatorial aspects of the Lascoux–Schützenberger tree

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• 作者：Little ; David P.
• 关键词：05E05 ; 05E15 ; Stanley symmetric functions ; Lascoux&ndash ; Schü ; tzenberger tree ; Reduced factorizations of symmetric group ; Littlewood&ndash ; Richardson rule
• 刊名：Advances in Mathematics
• 出版年：2003
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：March 25, 2003
• 卷：174
• 期：2
• 页码：236-253
• 文件大小：277 K

摘要

In 1982, Richard Stanley introduced the formal series F_σ(X) in order to enumerate reduced decompositions of a given permutation σ. Stanley (European J. Combin. 5(4) (1984) 359) not only showed F_σ(X) to be symmetric, but in certain cases, F_σ(X) was a Schur function. Stanley conjectured that for arbitrary σ,F_σ(X) was always Schur positive. Edelman and Greene subsequently proved this fact (Combinatories and Algebra (Boulder, CO, 1983), Amer. Math. Soc., Providence RI, 1984, pp. 155–162; Adv. in Math. 63(1) (1987) 42). Using the techniques of Lascoux and Schützenberger (Lett. Math. Phys. 10(2–3) (1985) 111) for computing Littlewood–Richardson coefficients, we will exhibit a new bijective proof of the Schur positivity of F_σ(X).