La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson)

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2017-11-07 DOI:10.24033/ast.1083

S. Riche

引用次数: 5

Abstract

Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is algebraic and rather elementary, some of their crucial properties were known until recently only in the case of crystallographic Coxeter groups, where these bimodules can be interpreted in terms of equivariant cohomology of Schubert varieties. In recent work Elias and Williamson have proved these properties in full generality by showing that these bimodules possess "Hodge type" properties. These results imply positivity of Kazhdan-Lusztig polynomials in full generality, and provide an algebraic proof of the Kazhdan-Lusztig conjecture.

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Soergel的Hodge双模理论(基于Soergel和Elias-Williamson)

Soergel双模是多项式代数上的某些双模，与Coxeter群有关，由Soergel在20世纪90年代研究复半单李代数的O类时引入。尽管它们的定义是代数的，而且相当初级，但直到最近，人们才知道它们的一些关键性质是在晶体Coxeter群的情况下才知道的，在这种情况下，这些双模可以用Schubert变异的等变上同调来解释。在最近的工作中，Elias和Williamson通过证明这些双模具有“Hodge型”性质，证明了这些性质的全面性。这些结果暗示了Kazhdan-Lusztig多项式的完全一般正性，并提供了Kazhdan-Lusztig猜想的代数证明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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