Combinatorics of affine Springer fibers and combinatorial wall-crossing

Abstract

This thesis deals with several combinatorial problems in representation theory. The first part of the thesis studies the combinatorics of affine Springer fibers of type A. In particular, we give an explicit description of irreducible components of Fl[subscript tS] and calculate the relative positions between two components. We also study the lowest two-sided Kazhdan-Lusztig cell and establish a connection with the affine Springer fibers, which is compatible with the affine matrix ball construction algorithm. The results also prove a special case of Lusztig's conjecture. The work in this part include joint work with Pablo Boixeda. In the second part, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition. This result gives a special situation where column regularization, can be used to understand the complicated Mullineux map, and also proves a special case of Bezrukavnikov's conjecture. Furthermore, we prove a condition under which the two maps are exactly the same, generalizing the work of Bessenrodt, Olsson and Xu. The combinatorial constructions is related to the Iwahori-Hecke algebra and the global crystal basis of the basic [ ... ]-module and we provide several conjectures regarding the q-decomposition numbers and generalizations of results due to Fayers. This part is a joint work with Panagiotis Dimakis and Allen Wang.