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
$R^2$.