Journal of Algebraic Combinatorics
, Volume 3, Issue 1, pp 516
First online:
The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials
 E.E. AllenAffiliated withDepartment of Mathematics and Computer Science, Wake Forest University
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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 σ ∈ S _{n}, 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 StanleyReisner 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 σ ∈ S _{n} 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 R ^{ρ}(X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R ^{ρ}*(X, Y) denote the quotient of R ^{ρ}(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 ^{ρ}*(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 ^{ρ}*(X, Y) in terms of their respective bases.
 Title
 The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials
 Journal

Journal of Algebraic Combinatorics
Volume 3, Issue 1 , pp 516
 Cover Date
 199401
 DOI
 10.1023/A:1022481303750
 Print ISSN
 09259899
 Online ISSN
 15729192
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 descent monomial
 diagonally symmetric polynomials
 polynomial quotient ring
 Authors

 E.E. Allen ^{(1)}
 Author Affiliations

 1. Department of Mathematics and Computer Science, Wake Forest University, WinstonSalem, NC, 27109