Journal of Algebraic Combinatorics

, Volume 3, Issue 1, pp 5–16

# The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

• E.E. Allen
Article

DOI: 10.1023/A:1022481303750

Allen, E. Journal of Algebraic Combinatorics (1994) 3: 5. doi:10.1023/A:1022481303750

## 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 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 = {x1, x2, ..., xn} and Y = {y1, y2, ..., yn}. The diagonal action of σ ∈ Sn 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 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.

descent monomial diagonally symmetric polynomials polynomial quotient ring