Symmetric coinvariant algebras and local Weyl modules at a double point

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The symmetric coinvariant algebra C[x₁, dots, xₙ]ₛₙ 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 C[x] with A = C[x,y]/(xy), we introduce another symmetric coinvariant algebra Aotimes nₛₙ and determine its Sₙ-module structure. As an application, we determine the slᵣ₊₁-module structure of the local Weyl module at a double point for slᵣ₊₁ otimes A.