Swapneel Mahajan

Abstract: Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Björner. In the first chapter, we give an axiomatic treatment of either concept and show their equivalence. We also axiomatize duality in this setting. As applications of these ideas, we prove a duality theorem on buildings and give a geometric interpretation of the flag h vector. The former may be regarded as a q-analogue of the Dehn-Sommerville equations. We also briefly discuss the connection with the random walks introduced by Bidigare, Hanlon and Rockmore.

The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group S_n, i.e., the Coxeter group of type A_{n–1}. In the second chapter, we give analogous shuffles for the Coxeter groups of type B_n and D_n. These can be interpreted as shuffles on a "signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also give new joker shuffles of type A_{n–1} and briefly discuss the generalization to buildings which leads to q-analogues.