Research in Algebraic Combinatorics

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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.