Björner and Wachs proved that under the weak order every quotient of a Coxeter group is a meet semi-lattice, and in the finite case is a lattice. In this paper, we examine the case of an affine Weyl group W with corresponding finite Weyl group W0. In particular, we show that the quotient of W by W0 is a lattice and that up to isomorphism this is the only quotient of W which is a lattice. We also determine that the question of which pairs of elements of W have upper bounds can be reduced to the analogous question within a particular finite subposet.
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- 1.Björner, A., Wachs, M. (1988) Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc. 308, 1–37.Google Scholar
- 2.Eriksson, K. (1996) Reduced words in affine Coxeter groups, Discrete Math. 157, 127–146.Google Scholar
- 3.Humphreys, J. (1990) Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge.Google Scholar
- 4.Proctor, R. A. (1984) Bruhat Lattices, Plane partition generating functions, and minuscule representations, European J. Combin. 5, 331–350.Google Scholar
- 5.Shi, J.Y. (1987) Sign types corresponding to an affine Weyl group, J. London Math. Soc. 35, 56–74.Google Scholar
- 6.Stanley, R. P. (1986) Enumerative Combinatorics, Wadsworth & Brooks/Cole, Monterey, CA.Google Scholar
- 7.Stembridge, J. R. (1996) On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5, 353–385.Google Scholar
- 8.Waugh, D. J. (1997) Quotients groups under the weak order, PhD Thesis, University of Michigan, Ann Arbor.Google Scholar