The quasiinvariants of the symmetric group

The quasiinvariants of the symmetric group

Blog Article

For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $mathbf{QI_m}(G)$ the ring of iphone 13 dallas $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov.These form a nested series of rings, with $mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $mathbf{QI}_{infty}(G)$ the usual ring of invariants.Remarkably, the ring $mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$.

However, even in the case of the symmetric group, no basis for $mathbf{QI_m}(G)$ is known.We provide a new read more description of $mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$.