The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

Abstract

Let R(X) = Q[x 1, x 2, ..., x n] be the ring of polynomials in the variables X = {x 1, x 2, ..., x n} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a σ ∈ S n, we let g $$_\sigma (X) = \prod\nolimits_{\sigma _i \succ \sigma _{i + 1} } {(x_{\sigma _1 } x_{\sigma _2 } \ldots x_{\sigma _i } } )$$ 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 = {x 1, x 2, ..., x n} and Y = {y 1, y 2, ..., y n}. The diagonal action of σ ∈ S n on polynomial P(X, Y) is defined as $$\sigma P(X,Y) = P(x_{\sigma _1 } ,x_{\sigma _2 } , \ldots ,x_{\sigma _n } ,y_{\sigma _1 } ,y_{\sigma _2 } , \ldots ,y_{\sigma _n } )$$ Let R ρ(X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R ρ*(X, Y) denote the quotient of R ρ(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 ρ*(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 ρ*(X, Y) in terms of their respective bases.