  We study the polynomial algebra (over a ring containing the rationals) in an
n by m matrix of variables, and subject to the relation that says that the
product of any two variables in the same column is zero. We show that the
sub-algebra of polynomials, which are invariant under n! permutations of the
columns, is a quotient of the polynomial algebra in m variables; the quotient
map sends the ith variable to the sum of the entries in the ith row of the
matrix. - An application in synthetic differential geometry is sketched

Commutative Algebra