The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

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 a e Sn, we let g a ( X ) = Yai>ai+1 (xa1,xa2 ... xai). In the late 1970s I. Cessel 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 a e Sn on polynomial P(X, Y) is defined as rP(X, Y) = P(xai, xa2 xan, ya1, ya2 yan). Let R p ( X , Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R p + ( X , Y) denote the quotient of R p ( 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 Rp+(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 Rp *(X, Y) in terms of their respective bases.