Michael Strayer (UNC-CH), GMA Seminar

Title: $d$-complete Posets and Representations of Lie Algebras

Abstract: In the first part of this talk, I will give the definition of $d$-complete poset as well as some of the basic properties.  This will be in the context of general, “uncolored” $d$-complete posets.  One particularly interesting result generalizes the famous “hook length formula” for counting the dimension of an irreducible representation of the symmetric group corresponding to a certain Young diagram.  This part of the talk will be aimed toward first-year students, since representations of finite groups is a topic that comes up in the spring semester algebra class.  The second part of the talk will be aimed more toward second-year students, as a number of them are taking Lie Algebras right now.  In this part of the talk, I will introduce the notion of “coloring” a $d$-complete poset, and discuss how these (now colored) posets can completely describe certain “minuscule” representations of simple Lie algebras.  I will provide all necessary definitions.  If time remains, I will talk a bit about my research with Dr. Proctor.