Quasi-invariants and free multi-arrangements
- Date: Friday 28 September 2018, 16:00 – 17:00
- Location: Mathematics Level 8, MALL 1, School of Mathematics
- Type: Seminars, Applied Mathematics, Integrable Systems
- Cost: free
Misha Feigin, University of Glasgow. Part of the integrable systems seminars series.
Quasi-invariants are special polynomials associated with a finite reflection group W. For Coxeter groups W they appeared in the work of Chalykh and Veselov in 1990 on quantum Calogero-Moser systems. Feigin and Veselov conjectured and Etingof and Ginzburg proved in 2002 that quasi-invariants form a free module over W-invariant polynomials. A proof of this fact via representation theory of Cherednik algebras was given by Berest, Etingof and Ginzburg in 2003. These results were generalized to complex reflection groups W by Berest and Chalykh in 2011, and they also introduced quasi-invariants with values in W-modules.
I am going to review these results and discuss their implications for theory of free arrangements of hyperplanes initiated by Saito. Thus it leads to new free multi-arrangements associated with finite complex reflection groups. This generalises recent work of Hoge, Mano, Roehrle and Stump who established freeness of such arrangements with extra restrictions. Freeness of multi-arrangements for real Coxeter groups was established by Abe, Terao and Wakamiko in 2011, generalising earlier work of Terao.
The talk is based on joint work with T.Abe, N.Enomoto and M.Yoshinaga.