Let Sn be the symmetric group acting on C[x1, . . . , xn]. The classical symmetric coinvariant
algebra C[x1, . . . , xn]Sn is the quotient of C[x1, . . . , xn] by the ideal generated by symmetric
polynomials vanishing at (0, . . . , 0). According to a classical result, it is isomorphic to C[Sn] as
Sn-module. In [Comm. Math. Phys. 251 (2004), no. 3, 427–445; MR2102326 (2005m:17005)],
B. L. Fe˘ıgin and S. A. Loktev defined the symmetric coinvariant algebra A
n
Sn
, where A is the
coordinate ring of an affine variety M over C. In the paper under review the author deals with
A = C[x, y]/(xy), the coordinate ring ofM = {(x, y) 2 C2: xy = 0}. In this case the symmetric
coinvariant algebra is Rn = A
n/Jn, where Jn is the ideal of A
n generated by the elementary
symmetric polynomials ei = ei(x1, . . . , xn), fi = fi(x1, . . . , xn), 1 i n.
He introduces a generalization of Rn, Rn
i,j = A
n/In
i,j , for 1 i, j n, and gives its Snmodule
structure when i+j n+1. This description is then used to show that, as Sn-module,
Rn
= C[Sn] (n−1)IndSn
S2L1,1, where IndSn
S2L(1,1) = C[Sn]
C[S2] L(1,1) and L(1,1) is the sign
representation of S2. This result, combined with a theorem contained in the paper by Fe˘ıgin and
Loktev quoted above, gives another description of the slr+1-module structure of the local Weyl
module at the double point 0 ofM for slr+1
A