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Browsing Mathematics & Statistics by Author "Bergeron, Nantel"
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Item Open Access Normal Supercharacter Theories(2018-03-01) Aliniaeifard, Farid; Bergeron, NantelClassification 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.Item Open Access Polarization Operators in Superspace(2024-07-18) Chan, Kelvin Tian Yi; Bergeron, NantelThe classical coinvariant rings and its variants are quotient rings with rich connections to combinatorics, symmetric function theory and geometry. Studies of a generalization of the classical coinvariant rings known as the diagonal harmonics have fruitfully produced many interesting discoveries in combinatorics including the q, t-Catalan numbers and the Shuffle Theorem. The super coinvariant rings are a direct generalization of the classical coinvariant rings to one set of commuting variables and one set of anticommuting variables. N. Bergeron, Li, Machacek, Sulzgruber, and Zabrocki conjectured in 2018 that the super coinvariant rings are representation theoretic models for the Delta Conjecture at t = 0. In this dissertation, we explore the super coinvariant rings using algebraic and combina- torial methods. In particular, we study the alternating component of the super harmonics and discover a novel basis using polarization operators. We use polarization equivalence to establish a triangularity relation between the new basis and a known basis due to two groups of researchers Bergeron, Li, Machacek, Sulzgruber, and Zabrocki and Swanson and Wallach. Furthermore, we prove a folklore result on the cocharge statistics of standard Young tableaux and propose a basis for every irreducible representation appearing in the super harmonics.Item Open Access Second-order finite free probability(2024-03-16) McConnell, Curran; Bergeron, NantelFinite free probability is a new field lying at the intersection of random matrix theory and non-commutative probability. It is called “finite” because unlike traditional free probability, which takes the perspective of operators on infinite-dimensional vector spaces, finite free probability focuses on the study of d × d matrices. Both fields study the behaviour of the eigenvalues of random linear transformations under addition. Finite free probability seeks in particular to characterize random matrices in terms of their (random) characteristic polynomials. I studied the covariance between the coefficients of these polynomials, in order to deepen our knowledge of how random characteristic polynomials fluctuate about their expected values. Focusing on a special case related to random unitary matrices, I applied the representation theory of the unitary group to derive a combinatorial summation expression for the covariance.Item Open Access Theta Maps for Combinatorial Hopf Algebras(2018-11-21) Li, Shu Xiao; Bergeron, NantelThis 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.Item Open Access Topics in Fomin-Kirillov Algebra(2022-03-03) Homayouni, Sirous; Bergeron, NantelWe introduce the notion of being 'z-star' for homogeneous polynomials. By proving a theorem plus developing a conjecture we state that, with a graded lexicographic monomial ordering, the reduced Grobner basis, for the ideal I generated by the relations of Fomin-Kirilov algebra FK(n), consists of 'z-star' polynomials. For general n, we find the character of the Fomin-Kirillov a lgebra, for some finite usual degrees with general set partition degree. We find the decomposition of this character in irreducible characters where this decomposition stabilizes at some enough big n. We develop a quotient of FK(n), denoted by FKCn (n), by making the quotient of the free algebra generated by the edges of an n-cycle, compared to the associated complete graph, where the ideal is generated by the relations of FK(n) except for letting the missing edges equal to zero and keeping only the edges of the polygon. We find the character map of algebra FKCn (n) and prove that the dimension of it equals the Lucas Number Ln and its Hilbert series is q-Lucas polynomial. We consider the commutative quotient of Fomin-Kirilov algebra, denoted by FK(n) and find its Grobner basis.