Tensor and wreath products of symmetric groups

对称群的张量和花环积

• 批准号: EP/V00090X/1
• 负责人: Chris Bowman-Scargill
• 金额: $ 120.49万
• 依托单位: University of York
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV00090X%2F1
• 关键词: Tensor; wreath; products; symmetric; groups

The Kronecker and plethysm coefficients describe the decompositions of products of symmetric functions into their simple constituents. They are as ubiquitous across mathematics as the notion of symmetry itself. The Kronecker coefficients have been described as "perhaps the most challenging, deep and mysterious objects in algebraic combinatorics''. They play an important role in the theory of symmetric functions and in the representation theory of general linear and symmetric groups. Despite 80 years of study, "frustratingly little is known'' about these coefficients. I hope to understand the whole blueprint for these Kronecker and plethysm coefficients by first considering what they looks like "generically" or "up to a finite instability". I have recently pioneered a new approach to understanding the (stable) blueprints of these coefficients in the context of the partition algebra and hence completely described one half of the stable blueprint for Kronecker coefficients and inductively described the stable blueprint for plethysm coefficients. Building on this success, this proposal seeks to completely understand the stable Kronecker and plethysm coefficients and to solve old and new conjectures concerning the positivity properties of non-stable Kronecker and plethysm coefficients.

克罗内克系数和体积系数描述了对称函数的乘积分解为它们的简单成分。它们在数学中就像对称性本身一样无处不在。克罗内克系数被描述为“也许是代数组合学中最具挑战性、最深刻和最神秘的对象”。它们在对称函数理论以及一般线性和对称群的表示论中发挥着重要作用。尽管研究了80年，但“令人沮丧的是，对这些系数知之甚少”。我希望通过首先考虑它们“一般”或“有限不稳定性”的样子来理解这些克罗内克系数和体积系数的整个蓝图。我最近开创了一种新的方法来理解这些系数的（稳定）蓝图的背景下，分区代数，因此完全描述了一半的稳定蓝图克罗内克系数和归纳描述的稳定蓝图体积系数。在此成功的基础上，该建议旨在完全理解稳定的克罗内克和体积系数，并解决有关非稳定克罗内克和体积系数的正性性质的新旧问题。

项目成果

Kronecker positivity and 2-modular representation theory

克罗内克实证性和 2-模表示理论

• DOI: 10.1090/btran/70
• 发表时间: 2021
• 期刊: Transactions of the American Mathematical Society, Series B
• 作者: Bessenrodt C

The many integral graded cellular bases of Hecke algebras of complex reflection groups

• DOI: 10.1353/ajm.2022.0008
• 发表时间: 2017-02
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: C. Bowman

The classification of multiplicity-free plethysms of Schur functions

Schur函数的无多重性体积的分类

• DOI: 10.1090/tran/8642
• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Bessenrodt C

An integral second fundamental theorem of invariant theory for partition algebras

划分代数不变理论的积分第二基本定理

• DOI: 10.1090/ert/593
• 发表时间: 2022
• 期刊: Representation Theory of the American Mathematical Society
• 作者: Bowman C

The partition algebra and the plethysm coefficients I: Stability and Foulkes' conjecture

划分代数和体积系数 I：稳定性和福克斯猜想

• DOI: 10.1016/j.jalgebra.2023.08.042
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Bowman C