The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials
E.E. Allen
DOI: 10.1023/A:1022481303750
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 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.
Pages: 5–16
Keywords: descent monomial; diagonally symmetric polynomials; polynomial quotient ring
Full Text: PDF
References
1. E. Allen, "A conjecture of Procesi and a new basis for the graded left regular representation of Sn," Ph.D. Thesis, University of California, San Diego, CA, 1991.
2. E. Allen, "A conjecture of Procesi and the straightening algorithm of G.C. Rota," Proc. Nat. Acad. Sci. 89 (1992), 3980-3984.
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5. A. Garsia and I. Gessel, "Permutation statistics and partitions," Adv. Math. 31 (1979), 288-305.
6. B. Gordon, "Two theorems on multipartite partitions," J. London Math. Soc. 38 (1963), 459-464.
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9. R. Stanley, "Ordered structures and partitions," Mem. Amer. Math. Soc. 119, 1972.
10. R. Steinberg, "On a theorem of Pittie," Topology 14 (1975), 173-177.
2. E. Allen, "A conjecture of Procesi and the straightening algorithm of G.C. Rota," Proc. Nat. Acad. Sci. 89 (1992), 3980-3984.
3. A. Garsia, "Combinatorial methods in the theory of Cohen-Macaulay rings," Adv. Math. 38 (1980), 229-266.
4. A. Garsia, "Unpublished classroom notes," Winter 1991, University of California, San Diego, CA.
5. A. Garsia and I. Gessel, "Permutation statistics and partitions," Adv. Math. 31 (1979), 288-305.
6. B. Gordon, "Two theorems on multipartite partitions," J. London Math. Soc. 38 (1963), 459-464.
7. P.A. MacMahon, Combinatory Analysis /-//, Cambridge University Press, London/New York, 1916; Chelsea, New York, 1960.
8. V Reiner, "Quotients of Coxeter complexes and P-partitions," Mem. Amer. Math. Soc. 95 (460), 1992.
9. R. Stanley, "Ordered structures and partitions," Mem. Amer. Math. Soc. 119, 1972.
10. R. Steinberg, "On a theorem of Pittie," Topology 14 (1975), 173-177.