Normal Supercharacter Theories
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Classification of irreducible characters of some families of groups, for example, the family of the groups of unipotent upper-triangular matrices, is a "wild" problem. To have a tame and tractable theory for the groups of unipotent-upper triangular matrices Andr and Yan introduced the notion of supercharacter theory. Diaconis and Issacs axiomatized the concept of supercharacter theory for any group. In this thesis, for an arbitrary group G, by using sublattices of the lattice of normal subgroups containing the trivial subgroup and G, we build a family of integral supercharacter theories, called normal supercharacter theories (abbreviated NSCT). We present a recursive formula for supercharacters in an NSCT. The finest NSCT is constructed from the whole lattice of normal subgroups of G, and is a mechanism to study the behavior of conjugacy classes by the lattice of normal subgroups. We will uncover a relation between the finest NSCT, faithful irreducible characters, and primitive central idempotents. We argue that NSCT cannot be obtained by previous known supercharacter theory constructions, but it is related to *-products of some certain supercharacter theories. We also construct an NSCT for the family of groups of unipotent upper-triangular matrices. These groups are crucial to the supercharacter theory. The supercharacters of the resulting NSCT are indexed by Dyck paths, which are combinatorial objects that are central to several areas of algebraic combinatorics. Finally, we show that this supercharacter construction is identical to Scott Andrews' construction after gluing the superclasses and the supercharacters by the action of the torus group.