Patricia Klein (Texas A&M University)

Math & Stats Algebra & Geometry Seminar
Monday, March 24, 2025

Time: 4:30 p.m.  Place: Jeffery Hall, Room 116

Speaker: Patricia Klein (Texas A&M University)

Title: Algebra and geometry of ASM weak order

Abstract: The central objects of study in this talk will be Schubert varieties in the complete flag variety. These Schubert varieties are indexed by permutations, which we will identify with permutation matrices. Schubert varieties can be studied via affine varieties called matrix Schubert varieties, introduced by Fulton. One can associate an arbitrary intersection of matrix Schubert varieties to what is called an alternating sign matrix (ASM). In this talk, we will review the role of matrix Schubert varieties - and their unions and intersections - in Schubert calculus and describe classical uses of permutation matrices and strong Bruhat order in encoding algebro-geometric invariants. We will then describe new work giving relationships between algebro-geometric invariants of unions and intersections of matrix Schubert varieties and ASMs under weak Bruhat order. No prior knowledge of strong or weak Bruhat order will be assumed for this talk. This is joint work with Laura Escobar and Anna Weigandt.