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Home > Vol 3, No 1 (1994)

E.E. Allen

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 _s ( X) = Õ _{s i \succ s i + 1} ( x _{s 1} x _{s 2} \frac{1}{4}  x _{s i} ) _σ(X) = \prod\nolimits_{σ_i \succ σ_{i + 1} } {(x_{σ_1 } x_{σ_2 } \ldots x_{σ_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 s P( X, Y) = P( x _{s 1} , x _{s 2} , \frac{1}{4} , x _{s n} , y _{s 1} , y _{s 2} , \frac{1}{4} , y _{s n} ) σP(X,Y) = P(x_{σ_1 } ,x_{σ_2 } , \ldots ,x_{σ_n } ,y_{σ_1 } ,y_{σ_2 } , \ldots ,y_{σ_n } ) 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.

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