A homological study of Green polynomials
A homological study of Green polynomials
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DOI：
10.24033/asens.2265
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发表时间：
2011-11
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影响因子：
--
通讯作者：
Syu Kato
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文献类型：
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作者：
Syu Kato
我们解释了由复反射群（c.f. [昌次，发明。Math.74（1983），J. Algebra 245（2001）]和[Lusztig，Adv.Math.61（1986）]）。这使我们得到了Kostka系统的概念，它可以被看作是Kostka多项式的范畴对应部分。然后，我们表明，每一个广义施普林格对应[Lusztig，发明。数学75（1984）]（在良好的特性）产生一个Kostka系统。这使我们能够看到广义Springer纤维的（扭曲）同调的顶项生成性质，以及两个$\mathsf{BC}$型广义Springer对应之间的Kostka多项式的转换公式。后者通过升级[Ciubotaru-Kato-K，Invent. 178（2012）] \S 3到其分级版本。在附录中，我们给出了Kostka系统在$\mathsf{A}$和渐近$\mathsf{BC}$情况下存在的纯代数证明，因此人们可以跳过几何部分来查看关键思想和基本示例/技术。
We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups (c.f. [Shoji, Invent. Math. 74 (1983), J. Algebra 245 (2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counter-part of Kostka polynomials. Then, we show that every generalized Springer correspondence [Lusztig, Invent. Math. 75 (1984)] (in good characteristic) gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type $\mathsf{BC}$. The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [Ciubotaru-Kato-K, Invent. Math. 178 (2012)] \S 3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type $\mathsf{A}$ and asymptotic type $\mathsf{BC}$ cases, and therefore one can skip geometric sections \S 3--5 to see the key ideas and basic examples/techniques.