# Quasi-invariants of finite Coxeter groups and integrable systems

To each finite Coxeter group W, Chalykh and Veselov associated a family of algebras consisting of what they called quasi-invariant polynomials, which interpolate between the algebra of all polynomials and the algebra of W-invariant polynomials. Despite being natural generalisations of invariants, quasi-invariants were introduced as late as 1990 and in the context of integrable systems.

Even though quasi-invariants are known have very nice algebraic properties, effective and explicit descriptions in terms of some form of generators are missing. The main aim of the project is to bridge this important gap, in particular by studying so-called m-harmonic-polynomials, conjectured by Feigin and Veselov (in 2002) to provide sets of free generators of the quasi-invariants over the invariants.

Applications to the, currently very active, area of exceptional orthogonal polynomials, the classical problem of describing the polynomials with given multiple roots with prescribed multiplicities, and to the theory of the fractional quantum Hall effect in two-dimensional electron systems will also be studied.

### Funded by

- Swedish Research Council (VR) (Public, Sweden)