Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov

科克塞特群是对称的群，它们构成了关于我们生活的世界的许多物理理论的一部分。此外，这些群经常被发现在深刻的数学理论的核心，最著名的是谎言理论。这个项目涉及两个看似不相关的代数，Orlik-所罗门代数和下降代数，这两个代数都描述了潜在Coxeter群的某些几何和组合方面。实验证据表明，这两个代数之间存在令人惊讶的联系。它们被表示为关于这些代数的某些自然表示的分解的两个精确猜想。证明，即使是这些猜想的特殊情况，也将大大加强我们对Coxeter群和与它们相关的代数的理解，并潜在地影响涉及这些对称群的其他理论。

Coxeter groups are groups of symmetries which form part of many physical theories about the world we live in. Moreover, these groups are frequently found at the heart of deep mathematical theories, most notably Lie theory. This project is concerned with two seemingly unrelated algebras, the Orlik-Solomon algebra and the Descent algebra, both of which describe certain geometric and combinatorial aspects of the underlying Coxeter group. Experimental evidence suggests the existence of surprising connections between these two algebras. These have been formulated as two precise conjectures about decompositions of certain natural representations of these algebras. A proof, even of special cases of these conjectures, will greatly enhance our understanding of Coxeter groups and the algebras associated to them and potentially impact on other theories involving these groups of symmetries.