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Upper bounds in affine Weyl groups under the weak order. (English) Zbl 0952.20031
Let $$(W,S)$$ be a finitely generated Coxeter group. For $$w=s_1\cdots s_k\in W$$, with $$s_i\in S$$ for all $$i$$ and $$k$$ minimal, we say that $$s_1\cdots s_k$$ is a reduced expression for $$w$$ and $$l(w)=k$$ is the length of $$w$$. For $$x,y\in W$$, we write $$x<_Ly$$ if there is a reduced expression for $$y$$ which ends with a reduced expression for $$x$$. This partial ordering on $$W$$ is called the (left) weak order. In [Trans. Am. Math. Soc. 308, No. 1, 1-37 (1988; Zbl 0659.05007)] A. Björner and M. L. 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, the author examines the case of an affine Weyl group $$W$$ with corresponding finite Weyl group $$W_0$$. In particular, the author shows that the quotient of $$W$$ by $$W_0$$ is a lattice and that up to isomorphism this is the only quotient of $$W$$ which is a lattice. The author also determines 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.

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E15 Combinatorial aspects of groups and algebras (MSC2010) 06A07 Combinatorics of partially ordered sets
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