The partition algebra: a new approach to the symmetric group and applications to P vs NP.

克罗内克问题是数学中的一个经典问题，它已经被提出了一百多年。它要求描述对称群的表示理论中出现的一些系数，称为Kronecker系数。最近，这个问题已经被证明在几何复杂性理论中发挥了关键作用，几何复杂性理论是一种寻求解决著名的P对NP问题的方法。我们最近提出了一种利用对称群与划分代数之间的对偶性来解决Kronecker问题的全新方法。早期的结果表明，这很可能导致一个完整的解决方案。本提案旨在进一步发展此工具（并将其扩展到正特征），以系统地研究对称群的普通和模表示理论。与此同时，我们计划与计算机科学界联系，调查我们的工作对著名的P / NP问题的影响。

The Kronecker problem is a classical problem in Mathematics, which has been open for over a hundred years. It asks for a description of some coefficients, called the Kronecker coefficients, appearing in the representation theory of the symmetric group. More recently, this problem has been shown to play a key role in Geometric Complexity theory, an approach that seeks to settle the famous P vs NP problem.We have recently proposed a completely new approach to the Kronecker problem using the duality between the symmetric group and the partition algebra. Early results suggest that this may well lead to a complete solution. This proposal aims at developing this tool further (and extending it to positive characteristics) to give a systematic study of the ordinary and modular representation theory of the symmetric group.In parallel, we plan to liaise with the Computer Science community to investigate the implications of our work to the celebrated P vs NP problem.