Research in Algebraic Combinatorics

我的研究方向是代数组合学。第一部分是关于 几组变量中的多项式的重要子空间，自然地附着于有限的复反射群。更具体地说，我计划研究的情况下，几组变量的对角coinvariant空间的不可约组件的分级分解。更精确地说，对于一组作用在空间V上的矩阵，我们考虑该组在V的对称空间S（V）的张量幂上的对角作用。这个对称空间可以被认为是变量对应于V的给定基中的向量的多项式空间。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（如表示理论）产生了广泛的影响（对角调和多项式空间），代数几何（希尔伯特方案），对称函数（Macdonald polynomials），数学物理（仿射Hecke代数，Calogero-Sutherland模型），等等，许多人似乎认为，研究类似的空间超过两套变量将被证明是困难得多。我发现了一种令人兴奋的新方法，可以用任意数量的变量集来统一描述这种空间。这在几个方向上开辟了一个大的研究计划，每个方向都值得独立研究。这些方向从拟不变量的协不变空间的类似物，到Steenrod代数的扭曲版本，并包括广义调和多项式的空间。 我的另一个研究方向是关于八面体方程和簇代数的“frieze模式”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联系。我打算利用一种原创的方法来解决所有这些问题，这种方法利用了我最近在C中引入的“驯服属性”。你好

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.