Upper bounds in affine Weyl groups under the weak order.

*(English)*Zbl 0952.20031Let \((W,S)\) 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.

Reviewer: Chen Chengdong (Shanghai)

##### MSC:

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

06A07 | Combinatorics of partially ordered sets |