Abstracts
for the Seminar

Fall 2022

**Speaker: **Grant Barkley, Harvard University

**Title: **Extending the weak Bruhat order

**Time:** 2:30 PM, Monday, November 28, 2022

**Place:** Malott 206

**Abstract:** The weak order is a partial order on the elements of a Coxeter group W. Its Hasse diagram describes the 1-skeleton of the permutahedron. For finite Coxeter groups, the weak order has joins and meets, making it a lattice. But in infinite Coxeter groups, we can fail to have common upper bounds for two elements. Matthew Dyer introduced a poset called extended weak order, motivated by the geometry of the weak order, which contains the usual weak order as a subposet. It is conjectured that the extended weak order is always a lattice, even for infinite Coxeter groups. Recently with David Speyer we have proved this conjecture for the case of affine Coxeter groups. We'll discuss these ideas, focusing on the infinite symmetric group and affine symmetric group.

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