Abstract
Let W be a Coxeter group. We define an element w ε W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.Billey and M.Haiman, “Schubert polynomials for the classical groups”, J. Amer. Math. Soc. 8 (1995), 443–482.
S.Billey, W.Jockusch, and R.Stanley, “Some combinatorial properties of Schubert polynomials,” J. Alg. Combin. 2 (1993), 345–374.
A.Björner, “Orderings of Coxeter groups,” Contemporary Math. 34 (1984), 175–195.
N.Bourbaki, Groupes et Algebres de Lie, Chaps. IV–VI, Masson, Paris, 1981.
P. Cartier and D. Foata, Problèmes Combinatoires de Commutation et Réarrangements, Lect. Notes in Math. Vol. 85, Springer-Verlag, 1969.
C.K. Fan, A Hecke Algebra Quotient and Properties of Commutative Elements of a Weyl Group, Ph.D. thesis, MIT, 1995.
C.K. Fan, “A Hecke algebra quotient and some combinatorial applications,” J. Alg. Combin., to appear.
S.V. Fomin and A.N. Kirillov, “Combinatorial B a-analogues of Schubert polynomials,” Trans. Amer. Math. Soc., to appear.
J.E.Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge, 1990.
T.K. Lam, B and D Analogues of Stable Schubert Polynomials and Related Insertion Algorithms, Ph.D. thesis, MIT. 1995.
R.A.Proctor, “Bruhat lattices, plane partition generating functions, and minuscule representations,” Europ. J. Combin. 5 (1984), 331–350.
R.P.Stanley, Enumerative Combinatories, Vol. I, Wadsworth & Brooks/Cole, Monterey, 1986.
J.R. Stembridge. “Some combinatorial aspects of reduced words in finite Coxeter groups,” Trans. Amer. Math. Soc., to appear.
G.X. Viennot, “Heaps of pieces I: Basic definitions and combinatorial lemmas,” in Combinatoire Énumérative, G. Labelle and P. Leroux (Eds.), pp. 321–350, Lect. Notes in Math. Vol. 1234, Springer-Verlag, 1985.
Author information
Authors and Affiliations
Additional information
Partially supported by NSF Grants DMS-9057192 and DMS-9401575.
Rights and permissions
About this article
Cite this article
Stembridge, J.R. On the fully commutative elements of Coxeter groups. J Algebr Comb 5, 353–385 (1996). https://doi.org/10.1007/BF00193185
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00193185