Transitive powers of Young-Jucys-Murphy elements are central
Transitive powers of Young-Jucys-Murphy elements are central
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Young-Jucys-Murphy 元素的传递能力是核心
DOI：
10.1016/j.jalgebra.2009.01.004
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
D. Jackson
中科院分区：
文献类型：
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作者：
I. Goulden;D. Jackson
尽管Young-Jucys-Murphy元素的幂xi=(1i)+(2i)+⋯+(i−1i)，i=1，…，n，在{1，…上作用的对称群中，n}不在群代数的中心，我们证明了传递幂，即作用在[n]上的元素的贡献之和是中心的。我们确定了传递幂关于S_n中心的类基的分解中出现的系数，并证明了它们具有多项式性质。这些中心性和多项式性质具有看似无关的结果。首先，他们回答了Pak提出的一个问题[I.Pak，置换的简化分解，关于星置换，广义加泰罗尼亚数和k元树，离散数学。第二，它们解释和推广了欧文和拉坦发现的美丽的对称结果[J.Irving，A.Ritan，极小因式分解为星形置换，离散数学，在Press，Math.CO/0610640]；第三，我们将多项式与一类与球面的分支覆盖有关的双Hurwitz数的现有多项式结果联系起来，从而暗示可能存在ELSV型公式(参见[T.Ekedahl，S.Lando，M.Shapiro，A.Vainshtein，Hurwitz数和曲线模空间上的交点，发明]。数学课。146(2001)297-327])。
Although powers of the Young–Jucys–Murphy elements Xi=(1i)+(2i)+⋯+(i−1i), i=1,…,n, in the symmetric group Snacting on {1,…,n} do not lie in the center of the group algebra of Sn, we show that transitive powers, namely the sum of the contributions from elements that act transitively on [n], are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the center of Sn, and show that they have a polynomiality property. These centrality and polynomiality properties have seemingly unrelated consequences. First, they answer a question raised by Pak [I. Pak, Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and k-ary trees, Discrete Math. 204 (1999) 329–335] about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan [J. Irving, A. Rattan, Minimal factorizations of permutations into star transpositions, Discrete Math., in press, math.CO/0610640]; and thirdly, we relate the polynomiality to an existing polynomiality result for a class of double Hurwitz numbers associated with branched covers of the sphere, which therefore suggests that there may be an ELSV-type formula (see [T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001) 297–327]) associated with the star factorization numbers.