EPrint Collection - Mathematics
http://hdl.handle.net/1773/2129
2020-03-31T17:05:28ZAffine Partitions and Affine Grassmannians
http://hdl.handle.net/1773/20987
Affine Partitions and Affine Grassmannians
Billey, Sara C.; Mitchell, Stephen C.
We give a bijection between certain colored partitions and the elements in the quotient of an aﬃne Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type An, the aﬃne colored partitions form another family of combinatorial objects in bijection with n + 1 core partitions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the aﬃne Grassmannians in terms of aﬃne partitions and a generalization of Young’s lattice which reﬁnes weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.
Below is the code used to supplement the proofs in "Affine Partitions and Affine Grassmannians" by Sara Billey and Stephen Mitchell. The code includes algorithms for generating elements in Coxeter groups up to some length, the Coxeter matrices for Weyl groups and Affine Weyl groups, algorithms for quotients of Coxeter groups, affine partitions, colored partitions, rank generating functions, Bruhat order, weak order, generalized Young's lattice, etc.
The code supplements the proofs in the paper by proving that the affine partitions in each exceptional type are equinumerous with the minimal length coset representatives for the affine Weyl group mod the Weyl group. The lisp code can be used to identify the generating function for affine partitions. The maple code takes in this generating function, simplifies it, and compares it with Bott's formula.
2008-01-01T00:00:00ZFibrations and Sheaves
http://hdl.handle.net/1773/20977
Fibrations and Sheaves
Warner, Garth
The purpose of this book is to give a systematic treatment of fibration theory and sheaf theory, the emphasis being on the foundational essentials.
2012-12-13T00:00:00ZHomotopical Topos Theory
http://hdl.handle.net/1773/19722
Homotopical Topos Theory
Warner, Garth
The purpose of this book is two-fold: (1) To give a systematic introduction to topos theory from a purely categorical point of view, thus ignoring all logical and algebraic issues. (2) To give an account of the homotopy theory of the simplicial objects in a Grothendieck topos.
2012-05-01T00:00:00ZCategorical Homotopy Theory
http://hdl.handle.net/1773/19589
Categorical Homotopy Theory
Warner, Garth
This book is an account of certain developments in categorical homotopy theory that have taken place since the year 2000. Some aspects have been given the complete treatment (i.e., proofs in all detail), while others are merely surveyed. Therefore a lot of ground is covered in a relatively compact manner, thus giving the reader a feel for the "big picture" without getting bogged down in the "nitty-gritty."
2012-01-01T00:00:00Z