Symmetric coinvariant algebras and local Weyl modules at a double point

The symmetric coinvariant algebra is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular representation of Sn. Replacing with , we introduce another symmetric coinvariant algebra and determine its Sn-module structure. As an application, we determine the -module structure of the local Weyl module at a double point for .