The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials
 E.E. Allen
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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.
 E. Allen, “A conjecture of Procesi and a new basis for the graded left regular representation of S _{ n },” Ph.D. Thesis, University of California, San Diego, CA, 1991.
 E. Allen, “A conjecture of Procesi and the straightening algorithm of G.C. Rota,” Proc. Nat. Acad. Sci. 89 (1992), 3980–3984.
 A. Garsia, “Combinatorial methods in the theory of CohenMacaulay rings,” Adv. Math. 38 (1980), 229–266.
 A. Garsia, “Unpublished classroom notes,” Winter 1991, University of California, San Diego, CA.
 A. Garsia and I. Gessel, “Permutation statistics and partitions,” Adv. Math. 31 (1979), 288–305.
 B. Gordon, “Two theorems on multipartite partitions,” J. London Math. Soc. 38 (1963), 459–464.
 P.A. MacMahon, Combinatory Analysis I–II, Cambridge University Press, London/New York, 1916; Chelsea, New York, 1960.
 V. Reiner, “Quotients of Coxeter complexes and Ppartitions,” Mem. Amer. Math. Soc. 95 (460), 1992.
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 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
 19940101
 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