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

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## Abstract

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 *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 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.

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### References

- 1.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.Google Scholar - 2.E. Allen, “A conjecture of Procesi and the straightening algorithm of G.C. Rota,”
*Proc. Nat. Acad. Sci.***89**(1992), 3980–3984.Google Scholar - 3.A. Garsia, “Combinatorial methods in the theory of Cohen-Macaulay rings,”
*Adv. Math.***38**(1980), 229–266.Google Scholar - 4.A. Garsia, “Unpublished classroom notes,” Winter 1991, University of California, San Diego, CA.Google Scholar
- 5.A. Garsia and I. Gessel, “Permutation statistics and partitions,”
*Adv. Math.***31**(1979), 288–305.Google Scholar - 6.B. Gordon, “Two theorems on multipartite partitions,”
*J. London Math. Soc.***38**(1963), 459–464.Google Scholar - 7.P.A. MacMahon,
*Combinatory Analysis I–II*, Cambridge University Press, London/New York, 1916; Chelsea, New York, 1960.Google Scholar - 8.V. Reiner, “Quotients of Coxeter complexes and
*P*-partitions,”*Mem. Amer. Math. Soc.***95**(460), 1992.Google Scholar - 9.
- 10.