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York University

1. Li, Shu Xiao. Theta Maps for Combinatorial Hopf Algebras.

Degree: PhD, Mathematics & Statistics, 2018, York University

URL: http://hdl.handle.net/10315/35599

This thesis introduces a way to generalize of peak algebra. There are several equivalent denitions for the peak algebra. Stembridge describes it via enriched P-partitions to generalize marked shifted tableaux and Schur's Q functions. Nyman shows that it is a the sum of permutations with the same peak set. Aguiar, Bergeron and Sottile show that the peak algebra is the odd Hopf sub-algebra of quasi symmetric functions using their theory of combinatorial Hopf algebras.
In all these cases, there is a very natural and well-behaved Hopf algebra morphism from quasi-symmetric functions or non-commutative symmetric functions to their respective peak algebra, which we call the theta map. This thesis focuses on generalizing the peak algebra by constructing generalized theta maps for an arbitrary combinatorial Hopf algebra.
The motivating example of this thesis is the Malvenuto-Reutenauer Hopf algebra of permutations. Our main result is a combinatorial description of all of the theta maps of this Hopf algebra whose images are generalizations of the peak algebra. We also give a criterion to check whether a map is a theta map, and we nd theta maps for Hopf sub-algebras of quasi-symmetric functions. We also show the existence of theta maps for any commutative and cocommutative Hopf algebras. From there, we study the diagonally symmetric functions and diagonally quasi-symmetric functions. Lastly, we describe theta maps for a Hopf algebra V on permutations.
*Advisors/Committee Members: Bergeron, Nantel (advisor).*

Subjects/Keywords: Mathematics; Combinatorics; Hopf algebra; Symmetric functions

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Li, S. X. (2018). Theta Maps for Combinatorial Hopf Algebras. (Doctoral Dissertation). York University. Retrieved from http://hdl.handle.net/10315/35599

Chicago Manual of Style (16^{th} Edition):

Li, Shu Xiao. “Theta Maps for Combinatorial Hopf Algebras.” 2018. Doctoral Dissertation, York University. Accessed November 26, 2020. http://hdl.handle.net/10315/35599.

MLA Handbook (7^{th} Edition):

Li, Shu Xiao. “Theta Maps for Combinatorial Hopf Algebras.” 2018. Web. 26 Nov 2020.

Vancouver:

Li SX. Theta Maps for Combinatorial Hopf Algebras. [Internet] [Doctoral dissertation]. York University; 2018. [cited 2020 Nov 26]. Available from: http://hdl.handle.net/10315/35599.

Council of Science Editors:

Li SX. Theta Maps for Combinatorial Hopf Algebras. [Doctoral Dissertation]. York University; 2018. Available from: http://hdl.handle.net/10315/35599

York University

2. Aliniaeifard, Farid. Normal Supercharacter Theories.

Degree: PhD, Mathematics & Statistics, 2018, York University

URL: http://hdl.handle.net/10315/34307

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.
*Advisors/Committee Members: Bergeron, Nantel (advisor).*

Subjects/Keywords: Mathematics; Character Theory; Supercharacter Theory; Lattice Theory; Normal Subgroups; Group Theory; Lattice of Normal Subgroups; Primitive Central Idempotents

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Aliniaeifard, F. (2018). Normal Supercharacter Theories. (Doctoral Dissertation). York University. Retrieved from http://hdl.handle.net/10315/34307

Chicago Manual of Style (16^{th} Edition):

Aliniaeifard, Farid. “Normal Supercharacter Theories.” 2018. Doctoral Dissertation, York University. Accessed November 26, 2020. http://hdl.handle.net/10315/34307.

MLA Handbook (7^{th} Edition):

Aliniaeifard, Farid. “Normal Supercharacter Theories.” 2018. Web. 26 Nov 2020.

Vancouver:

Aliniaeifard F. Normal Supercharacter Theories. [Internet] [Doctoral dissertation]. York University; 2018. [cited 2020 Nov 26]. Available from: http://hdl.handle.net/10315/34307.

Council of Science Editors:

Aliniaeifard F. Normal Supercharacter Theories. [Doctoral Dissertation]. York University; 2018. Available from: http://hdl.handle.net/10315/34307