Research in Algebraic Combinatorics

我的研究项目属于代数组合学领域。第一部分是关于 多变量多项式的重要的子空间，自然地依附于有限复反射群。更具体地说，我计划研究这种群的对角余不变空间的分次分解为不可约分量，在多个变量集的情况下。更确切地说，对于作用在空间V上的一组矩阵，我们考虑了该群对V的对称空间S(V)的张量幂的对角作用。这个对称空间可以被认为是由变量中的多项式组成的空间，该多项式对应于给定的V基上的向量。对角协不变空间就是这个张量幂与它的对角不变元所产生的理想的商。在过去的20年里，两组变量的情况一直是几位作者大量工作的起源，在几个领域产生了广泛的影响，如表示论(对角调和多项式空间)、代数几何(希尔伯特格式)、对称函数(Macdonald多项式)、数学物理(仿射Hecke代数、Calogero-Sutherland模型)等。对许多人来说，研究两组以上变量的相似空间似乎要困难得多。我发现了一种令人兴奋的新方法，它允许在任意数量的变量集中对这种空间进行统一描述。这开启了一个多个方向的大型研究项目，每个方向都值得独立研究。这些方向从拟不变量的余不变空间的类似，到Steenrod代数的扭曲形式，并包括广义调和多项式的空间。 我的另一项研究涉及到与八面体方程和簇代数有关的“Flaeze Patterns”概念的推广。这也与数学的几个领域和数学物理的离散广田方程有关。我打算在所有这些问题上采用一种独创的方法，即利用我最近与C·鲁特诺介绍的“驯服属性”。

My program of research lies within the area of algebraic combinatorics. A first part concerns the study of important subspaces of polynomials in several sets of variables, naturally attached to finite complex reflection groups. More specifically, I plan to study the graded decomposition into irreducible components of diagonal coinvariant space of such groups, in the case of several sets of variables. More precisely, for a group of matrices acting on a space V, one considers the diagonal action of the group on a tensor power of the symmetric space S(V) of V. This symmetric space can be consider as the space of polynomials in variables that correspond to vectors in a given basis of V. The diagonal coinvariant space is simply the quotient of this tensor power by the ideal generated by its diagonally invariant elements. The case of two sets of variables has been at the origin of a large body of work by several authors in the last 20 years, with broad impacts in several areas such as representation theory (spaces of diagonal harmonic polynomials), algebraic geometry (Hilbert schemes), symmetric functions (Macdonald polynomials), mathematical physics (affine Hecke algebras, Calogero-Sutherland models), etc. To many it seemed that the study of similar spaces for more than two sets of variables would prove to be much harder. I have found an exciting new approach that allows for a uniform description of such spaces in any number of sets of variables. This opens up a large program of research in several directions, each worthy of independent study. These directions go from analogs of coinvariant spaces for quasi-invariants, to twisted versions of Steenrod algebras, and include spaces of generalized harmonic polynomials. Another of my line of research concerns a generalization of the notion of "frieze patterns", in relation to the octahedron equation and cluster algebras. This also has ties with several areas of mathematics and the discrete Hirota equation of mathematical physics. I intend to exploit an original approach on all of this that exploits a "tameness property" that I have recently introduced with C. Reutenauer.