# The quasiinvariants of the symmetric group

Abstract : For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-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 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 $[n-1,1]$.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [5 references]

https://hal.inria.fr/hal-01185154
Contributor : Coordination Episciences Iam <>
Submitted on : Wednesday, August 19, 2015 - 11:42:12 AM
Last modification on : Wednesday, August 7, 2019 - 12:19:20 PM
Long-term archiving on: : Friday, November 20, 2015 - 10:32:10 AM

### File

dmAJ0152.pdf
Publisher files allowed on an open archive

### Identifiers

• HAL Id : hal-01185154, version 1

### Citation

Jason Bandlow, Gregg Musiker. The quasiinvariants of the symmetric group. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.599-610. ⟨hal-01185154⟩

Record views