The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

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Abstract

Let R(X) = Q[x1, x2, …, xn] be the ring of polynomials in the variables X = {x1, x2, …, xn} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a σ ∈ Sn, we let gσ(X) = ∏σ1>σi + 1 (xσ1xσ2 … xσi). 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, …, xn} and Y = {y1, y2, …, yn}. The diagonal action of σ ∈ Sn on polynomial P(X, Y) is defined as σ P(X, Y) = P(xσ1, xσ2, …, xσn, yσ1, yσ2, …, yσn). Let Rρ(X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let Rρ*(X, Y) denote the quotient of Rρ(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 Rρ*(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 Rρ*(X, Y) in terms of their respective bases.

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