Diagonal Alternants
François Bergeron (UQÀM)
Abstract:
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.