Diagonal Alternants

François Bergeron (UQÀM)

**Abstract:**

We consider, on the space of polynomials in *2n* variables *X = x*_{1},
x_{2}...,x_{n} and *Y=y*_{1},
y_{2}, ... ,y_{n}, the
usual action of the group S_{n} x S_{n}.
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 S_{n} (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 *R*^{2}.