Summary: Let $R( X) = Q[ x _{1}, x _{2}, \dots , x _{n}]$ be the ring of polynomials in the variables $X = { x _{1}, x _{2}, \dots , x _{n}}$ and $R*( X)$ denote the quotient of $R( X)$ by the ideal generated by the elementary symmetric functions. Given a $_{ s} ( X)$ = Õ $_{ s $_ i$ \succ s $_ i + 1 $ ( x _{ s $_1 $ x _{ s $_2 frac14 $x _{ s $_ i )$ \_\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 $X = { x _{1}, x _{2}, \dots , x _{n}}$ and $Y = { y _{1}, y _{2}, \dots , y _{n}}$. The diagonal action of s $P( X, Y) = P( x _{ s $_1 $ , x _{ s $_2 , frac14 , $x _{ s $_ n $ , y _{ s $_1 $ , y _{ s $_2 , frac14 , $y _{ s $_ n $ ) \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 ^ rgr( X, Y) be the subring of R( X, Y) which is invariant under the diagonal action. Let R ^ rgr*( X, Y) denote the quotient of R ^ rgr( 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 ^ rgr*( 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 ^ rgr$*( X, Y)$ in terms of their respective bases.
