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摘要翻译：
虽然作用于{1,……,n}的对称群S_n中的Young-Jucys-Murphya元素X_i=(1i)+...+(i-1i)，i=1，...，n的幂不在S_n的群代数的中心，但我们证明传递幂，即传递作用于{1,>...，n}的元素的贡献之和，是中心的。我们确定了在关于S_n中心的类基的传递幂的分解中出现的系数，我们称之为星分解数，并证明了它们具有多项式性质。这些中心性和多项式性质有着看似无关的后果。首先，他们回答了Pak提出的一个关于简化分解的问题；第二，他们解释和推广了欧文和藤条发现的美丽对称的结果；第三，我们将该多项式与球面分枝复盖的双Hurwitz数的已有多项式结果联系起来，从而表明可能存在一个与星因子分解数有关的Elsv型公式。
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英文标题：
《Transitive powers of Young-Jucys-Murphy elements are central》
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作者：
I. P. Goulden and D. M. Jackson
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Combinatorics 组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,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 centre of S_n, 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 about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan; 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 associated with the star factorization numbers.
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PDF链接：
https://arxiv.org/pdf/0704.1100
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