Diagonal Alternants
François Bergeron (UQÀM)


We consider, on the space of polynomials in 2n variables X = x1, x2...,xn and Y=y1, y2, ... ,yn, the usual action of the group Sn x Sn. Using a classical result of Steinberg, this space Q[X,Y] can be viewed as a n!2 dimensional module over the invariants of the group. This is to say that polynomials in X and Y can be uniquely decomposed as linear expressions in covariants, with coefficients that are invariants.  We uses theses results, together with restriction to Sn (considered as a diagonal subgroup), to decompose diagonal alternants. In particular, we give an explicit basis for diagonal alternants, modulo the ideal generated by products of symmetric polynomials in X and Y. The construction of this basis involves a very nice classification of configurations on n points in R2.