On Using Graphical Calculi: Centers, Zeroth Hochschild Homology and Possible Compositions of Induction and Restriction Functors in Various Diagrammatical Algebras

• Main Author: Brichard, Joelle
• Language: English
• Published: 2011
• Subjects: Mathematics
• Online Access: https://doi.org/10.7916/D87H1RKG

This thesis is divided into three chapters, each using certain graphical calculus in a slightly different way. In the first chapter, we compute the dimension of the center of the 0-Hecke algebra Hn and of the Nilcoxeter algebra NCn using a calculus of diagrams on the Moebius band. In the case of the Nilcoxeter algebra, this calculus is shown to produce a basis for Z(NCn) and the table of multiplication in this basis is shown to be trivial. We conjecture that a basis for Z(Hn) can also be obtained in a specic way from this topological calculus. In the second chapter, we also use a calculus of diagrams on the annulus and the Moebius band to determine the zeroth Hochschild Homology of Kuperberg's webs for rank two Lie algebras. We use results from Sikora and Westbury to prove the linear independence of these webs on these surfaces. In the third chapter, we use other diagrams to attempt to find explicitely the possible compositions of the induction and restriction functors in the cyclotomic quotients of the NilHecke algebra. We use a computer program to obtain partial results.