The dimension problem for Hecke algebras

Hecke 代数的维数问题

• 批准号: DP150103431
• 负责人: Prof Andrew Mathas
• 金额: $ 27.56万
• 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: 2015
• 资助国家: 澳大利亚
• 起止时间: 2015-06-20 至 2019-12-31
• 项目状态: 已结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/DP150103431
• 关键词: dimension; problem; Hecke; algebras

This project aims to give important new information about the graded Specht modules and the irreducible graded modules of the cyclotomic Hecke algebras. Experts have long considered that computing the dimensions of the irreducible representations to be completely intractable, however, the powerful new tools provided by the recently discovered KLR-grading gives rise to the combinatorics for solving this problem and for describing the graded decomposition numbers of these algebras. Even in characteristic zero this is incredibly interesting because, as a special case, it would give explicit combinatorial formulas for parabolic Kazhdan-Lusztig polynomials, a problem that has been studied intensely (without solution) for over thirty years.

本项目旨在给出分圆Hecke代数的分次Specht模和不可约分次模的重要新信息。专家们一直认为计算不可约表示的维数是完全困难的，然而，最近发现的KLR-分次提供了强大的新工具，产生了组合数学来解决这个问题，并描述这些代数的分次分解数。即使在特征为零的情况下，这也是非常有趣的，因为作为一种特殊情况，它会给出抛物型Kazhdan-Lusztig多项式的显式组合公式，这个问题已经被深入研究了30多年（没有解决方案）。