Conjectures on the Quotient Ring by Diagonal Invariants
 Mark D. Haiman
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring \(\mathbb{Q}[x_1 , \ldots ,x_n ,y_1 , \ldots ,y_n ]\) in two sets of variables by the ideal generated by all S _{n} invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x _{1}, ..., x _{n}} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.
 G.E. Andrews, “Identities in combinatorics III: A qanalog of the Lagrange inversion theorem,” Proc. AMS 53 (1975), 240–245.
 N. Bergeron and A.M. Garsia, “On certain spaces of harmonic polynomials,” Contemporary Math. 138 (1992), 51–86.
 L. Carlitz and J. Riordan, “Two element lattice permutation numbers and their qgeneralization,” Duke J. Math. 31 (1964), 371–388.
 C. Chevalley, “Invariants of finite groups generated by reflections,” Amer. J. Math. 77 (1955), 778–782.
 J. Fürlinger and J. Hofbauer, “qCatalan numbers,” J. Comb. Theory (A) 40 (1985), 248–264.
 A.M. Garsia, “A qanalogue of the Lagrange inversion formula,” Houston J. Math. 7 (1981), 205–237.
 A.M. Garsia and C. Procesi, “On certain graded S _{ n }modules and the qKostka polynomials,” Adv. Math. 94 (1992), 82–138.
 A.M. Garsia and J. Remmel, “A novel form of qLagrange inversion,” Houston J. Math. 12 503–523.
 M. Gerstenhaber, “On dominance and varieties of commuting matrices,” Ann. Math. 73 (1961), 324–348.
 I. Gessel, “A noncommutative generalization and qanalog of the Lagrange inversion formula,” Trans. AMS 257 (1980), 455–482.
 I. Gessel and D.L. Wang, “Depthfirst search as a combinatorial correspondence,” J. Comb. Theory (A) 26 (1979), 308–313.
 S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984.
 J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, England, 1990.
 A.G. Konheim and B. Weiss, “An occupancy discipline and applications,” SIAM J. Appl. Math. 14 (1966), 1266–1274.
 B. Kostant, “Lie group representations on polynomial rings,” Amer. J. Math. 85 (1963), 327–409.
 G. Kreweras, “Une famille de polynômes ayant plusieurs propriétés énumeratives,” Period. Math. Hungar. 11 (1980), 309–320.
 I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, England, 1979.
 C.L. Mallows and J. Riordan, “The inversion enumerator for labelled trees,” Bull. AMS 74 (1968), 92–94.
 P. Orlik and L. Solomon, “Unitary reflection groups and cohomology,” Invent. Math. 59 (1980), 77–94.
 R.W. Richardson, “Commuting varieties of semisimple Lie algebras and algebraic groups,” Compositio Math. 38 (1979), 311–327.
 G.C. Shephard and J.A. Todd, “Finite unitary reflection groups,” Canad. J. Math. 6 (1954), 274–304.
 J.Y. Shi, “The KazhdanLusztig cells in certain affine Weyl groups,” Lecture Notes in Math 1179, SpringerVerlag, Berlin, 1986.
 R.P. Stanley, Enumerative Combinatorics, Vol. I. Wadsworth & Brooks/Cole, Monterey, CA, 1986.
 R. Steinberg, “Finite reflection groups,” Trans. AMS 91 (1959), 493–504.
 R. Steinberg, “Invariants of finite reflection groups,” Canad. J. Math. 12 (1960), 616–618.
 R. Steinberg, “Differential equations invariant under finite reflection groups,” Trans. AMS 112 (1964), 392–400.
 W.T. Tutte, “A contribution to the theory of chromatic polynomials,” Canad. J. Math. 6 (1953), 80–91.
 N. Wallach, “Invariant differential operators on a reductive Lie algebra and Weyl group representation,” Manuscript, U.C.S.D. (1992).
 H. Weyl, The Classical Groups, Their Invariants and Representations, Second Edition. Princeton University Press, Princeton, NJ, 1949.
 Title
 Conjectures on the Quotient Ring by Diagonal Invariants
 Journal

Journal of Algebraic Combinatorics
Volume 3, Issue 1 , pp 1776
 Cover Date
 19940101
 DOI
 10.1023/A:1022450120589
 Print ISSN
 09259899
 Online ISSN
 15729192
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 diagonal harmonics
 invariant
 Coxeter group
 Authors

 Mark D. Haiman ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 920930112