Algebraic Schubert geometry and unitary reflection groups

代数舒伯特几何和酉反射群

• 批准号: DP160103897
• 负责人: Prof Gustav Lehrer
• 金额: $ 36.55万
• 依托单位: The University of Sydney
• 依托单位国家: 澳大利亚
• 项目类别: Discovery Projects
• 财政年份: 2016
• 资助国家: 澳大利亚
• 起止时间: 2016-01-01 至 2024-01-01
• 项目状态: 已结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/DP160103897
• 关键词: Algebraic; Schubert; geometry; unitary; reflection

This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that this geometry may be simulated algebraically for any Coxeter group, so positivity for Kazhdan-Lusztig polynomials holds for all Coxeter groups. This result has explosive consequences in many areas of geometry and algebra. This project is designed to extend these results to complex unitary reflection groups, with potentially dramatic consequences in number theory, representation theory and topology.

这个项目的目的是概括埃利亚斯和威廉姆森的复杂情况下，最近的工作。对称性研究的基础是普遍存在的考克斯特群，它有一组非常重要的“Kazhdan-Lusztig多项式”。对于某些Coxeter群，这些可以用经典几何来解释，导致它们的系数具有深正性。埃利亚斯和威廉姆森最近证明，这种几何可以用代数方法模拟任何考克斯特群，因此卡日丹-卢斯蒂希多项式的正性对所有考克斯特群都成立。这个结果在几何和代数的许多领域都有爆炸性的影响。该项目旨在将这些结果扩展到复杂的酉反射群，在数论，表示论和拓扑学中具有潜在的戏剧性后果。