Qualifying Exam

Speaker:  Evan Henning, University of Kentucky

Title:  Permutation Hopf algebras

Abstract:

Hopf algebras are a very natural algebraic structure in the study of combinatorics. This is due to the fact that many combinatorial objects admit canonical operations of combination and decomposition. 

One of the most foundational combinatorial objects of mathematics is that of permutations. There are many Hopf algebras which can be put on the linear span of permutations, the most famous of which is the Malvenuto and Reutenauer Hopf algebra of permutations. This is a shuffle-like Hopf algebra that is self-dual and surjects onto the Hopf algebra of quasisymmetric functions. In this talk, we focus on the work of Mingze Zhao and Huilan Li and consider two other Hopf algebras that can be put on permutations. 

These two new Hopf algebras give permutations a tensor and shuffle algebra structure. We will prove the duality between these Hopf algebras and will consider a generalization of these Hopf algebras, which can be extended to juggling.