The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

Let R(X) = Q[x₁, x₂, ..., x_n] be the ring of polynomials in the variables X = {x₁, x₂, ..., x_n} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a S_n, we let g In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x₁, x₂, ..., x_n} and Y = {y₁, y₂, ..., y_n}. The diagonal action of S_n on polynomial P(X, Y) is defined as Let R(X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R*(X, Y) denote the quotient of R(X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for R*(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and R*(X, Y) in terms of their respective bases.