## Diagonal Alternants.

### Francois Bergeron (UQAM)

Abstract: We consider, on the space of polynomials in
$2 n$ variables

$X=x_1,\ldots,x_n$ and $Y=y_1,\ldots,y_n$, the usual action of the group

$S_n\times S_n$. 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

$R^2$.