# Colloquium: The shortest path poset of Bruhat intervals

**Title:**The shortest path poset of Bruhat intervals

**Speaker:**Saul Blanco Rodriguez, Indiana University Bloomington

https://www.soic.indiana.edu/

**Abstract:** A Coxeter group W is a group generated by reflections; examples are the symmetric group and the hyperoctahedral group. These groups have many interesting combinatorial properties. For instance, one can define a partial order, called the Bruhat order, on the elements of W . If [u,v] is an interval in the Bruhat order, its Bruhat graph, B(u,v) includes the Hasse diagram of the poset [u,v] with edges directed upwards, as well as other edges that I will describe in the talk. While the longest u-v paths in B(u,v) are well-understood (they form a face poset of a regular cell decomposition of a sphere), not much is known about the other u-v paths in B(u,v). In this talk, I will describe what is known of the shortest u-v paths and point out connections to other areas.