Browsing by Author "Herman, Allen"

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Open Access
Cameron-Liebler Sets for 2-Transitive Groups
(Faculty of Graduate Studies and Research, University of Regina, 2020-11) Palmarin, Daniel Michael; Fallat, Shaun; Meagher, Karen; Herman, Allen; Butz, Cortney
This research was conducted on 2-transitive groups whose minimal normal subgroup is abelian. Suppose G is such a group and ΓG is its derangement graph. Any maximum coclique S of ΓG has a characteristic vector xS. Each xS is a boolean vector contained in a particular module, which is called the permutation module MP. This module has a dimension of 1 + (n − 1)2, where n = deg(G), and it is spanned by {xij | i,j∈{1,...,n}}, where each xij is the characteristic vector of Sij, the set of permutations that map i to j. Apart from the xij, which correspond to the stabilizers of G and their cosets, this research set out to find any other boolean vectors that are contained in Mp using linear programming. Henceforth, such boolean vectors are defined to be Cameron-Liebler sets for 2-transitive groups. In addition to finding Cameron-Liebler sets, analyses were performed on each group to determine: (1) whether the strict EKR property holds; (2) the number of maximum cocliques that are subgroups, cosets, or neither; (3) isomorphism classes and conjugacy classes of the maximum cocliques that are subgroups; (4) the dimension of C′, the maximum cliques that are subgroups (along with their right cosets), and C, all maximum cliques; and (5) the spectrum of ΓG and whether the ratio bound is satisfied with equality.
Open Access
Classification of Subcategories in Abelian Categories and Triangulated Categories
(Faculty of Graduate Studies and Research, University of Regina, 2016-09) Liu, Yong; Stanley, Donald; Szechtman, Fernando; Herman, Allen; Yao, Yiyu; Krause, Henning
Two approaches for classifying subcategories of a category are given. We examine the class of Serre subcategories in an abelian category as our first target, using the concepts of monoform objects and the associated atom spectrum [13]. Then we generalize this idea to give a classification of nullity classes in an abelian category, using premonoform objects instead to form a new spectrum so that there is a bijection between the collection of nullity classes and that of closed and extension closed subsets of the spectrum. Additionally, we impose a natural sheaf structure induced by the center of a category on the atom spectrum, over which the sheaves of modules over the structure sheaf are also discussed. The second approach is enlightened by the lattice structure implicitly shown in the statements of classification of the subcategories in an abelian category. We introduce a new concept of classifying space of subcategories, those subcategories satifying finitely many closure operations, in an either abelian or triangulated category. We show that a class of subcategories is classified by a topological space if these subcategories are primely generated. Many well-known results fit into our framework, such as Neeman’s classification [19] of localizing subcategories of the derived category D(R) of a commutative Noetherian ring R, etc.
Open Access
Corrigendum to “The recognition problem for table algebras and reality-based algebras” [J. Algebra 479 (2017) 173–191]
(2019) Herman, Allen; Muzychuk, Mikhail; Xu, Bangteng
This note reports and corrects an error in the above article.
Open Access
Cubical Blakers-Massey Theorem for CDGA
(Faculty of Graduate Studies and Research, University of Regina, 2017-06) Hu, Yang; Stanley, Donald; Herman, Allen; Songhafouo Tsopmene, Paul Arnaud; Gilligan, Bruce
We state and prove an algebraic version of the Blakers-Massey Theorem. The Blakers-Massey Theorem is a classical result in homotopy theory that measures the obstruction to homotopy excision. It also measures how far a homotopy pushout square of topological spaces is from being a homotopy pullback. This theorem can be generalized to higher-dimensional cubical diagrams of topological spaces, where it measures how far a cubical homotopy colimit is from being a homotopy limit. This work is inspired by Quillen's rational homotopy theory, where commutative di erential graded algebras (or CDGAs for short) over the eld Q of rational numbers are algebraic models. We construct the Blakers-Massey Theorem for n-cubes of (simply connected) CDGAs, measuring how far an n-cube of CDGAs which is a homotopy limit is from being a homotopy colimit.
Open Access
Decomposition of Certain Representations Into A Direct Sum of Indecomposable Representations
(Faculty of Graduate Studies and Research, University of Regina, 2019-04) Zhang, Yihui; Stanley, Donald; Herman, Allen; Frankland, Martin; Rayan, Steven
Representations of quivers and posets are produced by persistent homology. It is possible to decompose such representations into direct sums of indecomposable representations. The indecomposable representations of quivers and posets that arise from one dimensional persistent homology are well understood. However, the same is not true for multidimensional persistent homology. This thesis finds such a list of indecomposable representations for a very simple case of a poset that can arise from multidimensional persistent homology, and proves that it is possible to decompose a representation of such a poset into a direct sum of these indecomposables.
Open Access
Eigenvalues of K-Uniform Hypergraphs
(Faculty of Graduate Studies and Research, University of Regina, 2017-08) Gorr, Adam Vernon; Meagher, Karen; Fallat, Shaun; Herman, Allen; Gosselin, Shonda; Butz, Cory
We de ne two separate attempts to generalize the de nition of eigenvalues to hypergraphs and show several results related to each. The rst approach is rooted in 2-dimensional matrices and allows for the generalization of many results from graph theory. The second approach covered is more sophisticated and may only be applied to k-uniform hypergraphs. We include the development of a sound algorithm using the resultant of polynomials that can be used for any k-uniform hypergraph. Speci c examples are provided to demonstrate the power of the algorithm. Further, we show that certain results hold for the eigenvalues and associated eigenvectors of k-uniform hypergraphs and those hypergraphs obtained from combinatorial designs such as Steiner triple systems.
Open Access
Enriched model categories and the Dold-Kan correspondence
(Faculty of Graduate Studies and Research, University of Regina, 2024-10) Ngopnang Ngompe, Arnaud; Frankland, Martin; Stanley, Donald; Fallat, Shaun; Herman, Allen; Zilles, Sandra; Ponto, Kate
The work we present in this thesis is an application of the monoidal properties of the Dold–Kan correspondence and is constituted of two main parts. In the first one, we observe that by a theorem of Christensen and Hovey, the category of nonnegatively graded chain complexes of left R-modules has a model structure, called the Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. Hence, the Dold–Kan correspondence induces a model structure on the category of simplicial left R-modules and some properties, notably it is monoidal. In the second part, we observe that changing the enrichment of an enriched, tensored and cotensored category along the Dold–Kan correspondence does not preserve the tensoring nor the cotensoring. Thus, we generalize this observation to any weak monoidal Quillen adjunction and we give an insight of which properties are preserved and which are weakened after changing the enrichment of an enriched model category along a right weak monoidal Quillen adjoint.
Open Access
Equivariant LS-Category and Equivariant Topological Complexity
(Faculty of Graduate Studies and Research, University of Regina, 2016-05) Bayeh, Marzieh; Stanley, Donald; Meagher, Karen; Gilligan, Bruce; Herman, Allen; Yao, Yiyu; Oprea, John
In this thesis we consider topological spaces endowed with an action of a topological group, and we develop a new concept to study these spaces. This concept is called orbit class and is often a good replacement for the well-known concept or- bit type. Using the concept of orbit class, we de ne a partial ordering on the set of all orbit classes. This partial order not only gives a partition on the topological space based on the orbits, but it also gives a discrete combinatorial translation of the topological space. We also use the properties of the orbit class to study equivariant LS-category and equivariant topological complexity. Equivariant LS-category was introduced by Marzantowicz in 1989, as a generalization of LS-category. Since then, equivariant LS-category has been studied by mathematicians and many results with di erent conditions have been developed. Equivariant topological complexity was introduced by Colman and Grant in 2012, as a generalization of topological complexity. In 2015, Lubawski and Marzantowicz introduced the invariant topological complexity as another generalization of the topological complexity and they claimed that their proposed invariant is more e cient than the equivariant topological complexity. In this thesis we study the equivariant LS-category and give some new results found by applying the properties of orbit class. We also study both the equivariant topological complexity and the invariant topological complexity. By using results from orbit class we show that in most cases the invariant topological complexity is in nite. In particular, if a topological space has more than one minimal orbit class then the invariant topological complexity is in nite. Finally, we study some particular cases of locally standard torus manifolds, and calculate their LS-category, topological complexity, equivariant LS-category, and invariant topological complexity. We also give counterexamples to two theorems from a published paper by Colman and Grant [10], and prove a modi ed version of one of those theorems.
Open Access
The Erdős-Ko-Rado Theorem for Transitive Permutation Groups
(Faculty of Graduate Studies and Research, University of Regina, 2022-03) Razafimahatratra, Andriaherimanana Sarobidy; Meagher, Karen; Fallat, Shaun; Herman, Allen; Yang, Boting; Mojallal, Seyed Ahmad; Bamberg, John
Given a transitive permutation group $G \le Sym(\Omega)$, a subset $F$ of $G$ is $\textit {intersecting}$ if any two elements of $F$ agree on some elements of $\Omega$. We are interested in the problem of finding the structure of the largest intersecting families of $G$. This problem is the analogue of the $\textit {Erdős-Ko-Rado (EKR) Theorem}$ for transitive permutation groups. We say that a transitive group $G \le Sym(\Omega )$ has the $\textit{EKR property}$ if any intersecting set of $G$ has size at most the order of a stabilizer of a point of $G$. Moreover, $G$ has the $\textit {strict-EKR property}$ if the largest intersecting sets in $G$ are cosets of a stabilizer of a point of $G$. In this thesis, we use various algebraic techniques to prove EKR-type results for finite transitive groups. In particular, we prove that the action of the symmetric group on the 2-tuples with distinct entries and 2-subsets of $[n]$ have the EKR property, and construct families of transitive groups that are as far away as possible from having the EKR property. Then, we show that any transitive subgroup of $GL_2(q)$ acting on the non-zero vectors of $\mathbb F^2_q$ has the EKR property. We also prove that for any odd primes $p$, the size of the largest intersecting set in a transitive group of degree $2p$ is at most twice the order of a point stabilizer. In addition, we show that if $G$ is transitive of degree a product of two odd primes, then $G$ has the EKR property whenever the socle of $G$ admits an imprimitive subgroup.
Open Access
Extensions of the Erdős-Ko-Rado Theorem to Perfect Matchings
(Faculty of Graduate Studies and Research, University of Regina, 2022-03-31) Nasrollahi Shirazi, Mahsa; Meagher, Karen; Fallat, Shaun; Herman, Allen; Zilles, Sandra; Nasserasr, Shahla; Guo, Krystal
One of the important results in extremal set theory is the Erdős-Ko-Rado (EKR) theorem which gives a tight upper bound on the size of intersecting sets. The focus of this thesis is on extensions of the EKR theorem to perfect matchings and uniform set partitions. Two perfect matchings are said to be t-intersecting if they have at least t edges in common. In 2017, Godsil and Meagher algebraically proved the EKR theorem for intersecting perfect matchings on the complete graph with 2k vertices. In 2017, Lindzey presented an asymptotic refinement of the EKR theorem on perfect matchings. In this thesis, we extend their results to 2-intersecting and also to set-wise 2-intersecting perfect matchings. These results are not asymptotic. A perfect matching is in fact a special case of a uniform set partition. Another focus of this thesis is on partially 2-intersecting uniform set partitions. We find the largest set of 2-intersecting uniform set partitions, when the number of parts is sufficiently large. The result on uniform set partitions is part of a joint research project with Karen Meagher and Brett Stevens.
Open Access
The Fundamental Modules of the Classical Lie Algebras
(Faculty of Graduate Studies and Research, University of Regina, 2012-02) Krimker Fernandez, Gustavo Sergio; Szechtman, Fernando; Herman, Allen; Volodin, Andrei; Gilligan, Bruce; Zhao, Kaiming
The main objective of this Thesis is the construction of the fundamental modules of the classical Lie algebras. Weyl’s Theorem shows that if L is a semisimple Lie algebra, then any finite dimensional L−module is a direct sum of irreducible L−modules. Since the classical algebras are semisimple, we just need the irreducible modules in order to obtain the others. On the other hand, the fundamental modules give us every irreducible L− module and, therefore, every finite dimensional L−module.
Open Access
Maximum Intersecting Families of Permutations
(Faculty of Graduate Studies and Research, University of Regina, 2013-07) Ahmadi, Bahman; Meagher, Karen; Fallat, Shaun; Herman, Allen; Zilles, Sandra; Dukes, Peter
In extremal set theory, the Erd}os-Ko-Rado (EKR) theorem gives an upper bound on the size of intersecting k-subsets of the set {1; : : : ;n}. Furthemore, it classi es the maximum-sized families of intersecting k-subsets. It has been shown that similar theorems can be proved for other mathematical objects with a suitable notion of \intersection". Let G B Sym(n) be a permutation group with its permutation action on the set {1; : : : ;n}. The intersection for the elements of G is de ned as follows: two permutations ; > G are intersecting if (i) = (i) for some i > {1; : : : ;n}. A subset S of G is, then, intersecting if any pair of its elements is intersecting. We say G has the EKR property if the size of any intersecting subset of G is bounded above by the size of a point stabilizer in G. If, in addition, the only maximum-sized intersecting subsets are the cosets of the point-stabilizers in G, then G is said to have the strict EKR property. It was rst shown by Cameron and Ku [10] that the group G = Sym(n) has the strict EKR property. Then Godsil and Meagher presented an entirely di erent proof of this fact using some algebraic properties of the symmetric group. A similar method was employed to prove that the projective general linear group PGL(2; q), with its natural action on the projective line Pq, has the strict EKR property. The main objective in this thesis is to formally introduce this method, which we call the module method, and show that this provides a standard way to prove Erd}os-Ko-Rado theorems for other permutation groups. We then, along with proving Erd}os-Ko-Rado theorems for various groups, use this method to prove some permutation groups have the strict EKR property. We will also show that this method can be useful in characterizing the maximum independent sets of some Cayley graphs. To explain the module method, we need some facts from representation theory of groups, in particular, the symmetric group. We will provide the reader with a su cient level of background from representation theory as well as graph theory and linear algebraic facts about graphs.
Open Access
Musings on matchings, matrices, and multiplicities
(Faculty of Graduate Studies and Research, University of Regina, 2024-07) Parenteau, Johnna Michele; Fallat, Shaun; Herman, Allen; Meagher, Karen
The Parter-Wiener Theorem is a celebrated contribution to the inverse eigenvalue problem for trees due to its determination of vertices whose removal affects multiplicities of eigenvalues in a non-intuitive manner. For a more general graph, G, that contains cycles, the construction of the weighted matching polynomial and its many properties are derived. These properties are shown to determine a relationship between the multiplicities of the roots of the weighted matching polynomial and the graph operation of vertex deletion in G, which is the operation at the core of the Parter-Wiener Theorem. Solutions for locating vertices whose removal increases the multiplicity of a root are presented, which gives rise to a new classification of graphs, called SRSI graphs. These graphs, along with graphs that have Hamilton paths, are determined to have a trivial variation of the Parter-Wiener Theorem. In an effort to determine the location of Parter vertices, vertices are categorized into classes based on their effects of root multiplicities, and, in the case of zero roots, the location of Parter vertices are explicitly noted. Moreover, computational results regarding the process of categorizing vertices into these classes are outlined, and the Vandermonde eigenvector test is established with the assistance of companion matrices. A myriad of results throughout the thesis are then used to determine a partially-generalized Parter-Wiener Theorem for this weighted matching polynomial.
Open Access
On Koszul duality between polynomial and exterior algebras
(Faculty of Graduate Studies and Research, University of Regina, 2023-01) Heenwalle Gedara, Fatima Ahmed Aboalkasem; Frankland, Martin; Herman, Allen; Stanley, Donald; Peschke, George
In this thesis we will study Ext-algebras over a polynomial and exterior algebra. We prove the classical fact that the Ext-algebra over a polynomial algebra is exterior and the Ext-algebra over an exterior algebra is polynomial, using the tautological Koszul complex. We also give a proof using Koszul duality for algebras.
Open Access
Orders of Torsion Units of integral reality-based algebras with rational multiplicities
(2018) Herman, Allen; Singh, Gurmail
A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over C whose distinguished basis B contains 1 and is closed under pseudo-inverse. An integral RBA is one whose structure constants in its distinguished basis are integers. If the algebra has a one-dimensional representation taking positive values on B, then we say that the RBA has a positive degree map. These RBAs have a standard feasible trace, and the multiplicities of the irreducible characters in the standard feasible trace are the multiplicities of the RBA. In this paper, we show that for integral RBAs with positive degree map whose multiplicities are rational, any finite subgroup of torsion units whose elements are all of degree 1 and have algebraic integer coefficients must have order dividing a certain positive integer determined by the degree map and the multiplicities. The paper concludes with a thorough investigation of the properties of RBAs that force multiplicities to be rational.
Open Access
Representations of McLain Groups
(Faculty of Graduate Studies and Research, University of Regina, 2016-06) Izadi, Mohammadali; Szechtman, Fernando; Herman, Allen; Velez, Maria; Gilligan, Bruce; Garcia, Gaston
Basic modules of McLain groups are defined and investigated. These are a (possibly infinite dimensional) generalization of Andre’s basic modules of the multiplicative group of upper triangular square matrices over a finite field with 1’s on the main diagonal. The ring R need not be finite or commutative and modules of a McLain group are allowed to be infinite dimensional over an arbitrary field F. The set , totally ordered by , is allowed to be infinite. We show that distinct basic modules are disjoint, determine the dimension of the endomorphism algebra of a basic module, find when a basic module is irreducible, and we study the problem of finding a decomposition of a basic module as direct sum of irreducible submodules.
Open Access
Summarizing Conditional Preference Networks
(Faculty of Graduate Studies and Research, University of Regina, 2019-04) Ali, Abu Mohammad Hammad; Zilles, Sandra; Mouhoub, Malek; Hamilton, Howard; Herman, Allen
Preference modeling has been studied extensively in the literature, and has applications in recommender systems and automated decision-making. The eventual objective of working with preference models is to be able to reason about preferences over objects, often referred to as outcomes. In most of the literature, each outcome is described as an assignment of values to a set of attributes. Representing and reasoning about preferences over outcomes calls for efficient preference models. In this thesis, we focus on one such model, Con- ditional Preference Networks (CP-nets). A CP-net is a graphical model that captures the preferences of an individual using a directed graph, with vertices representing attributes and edges representing dependency relations between attributes. Information about the preferential dependence/independence be- tween attributes can be leveraged to efficiently order outcomes without exhaus- tively comparing all attributes in a pair of outcomes. In most existing studies, it is assumed that each individual user has their unique CP-net representing their preferences. In this thesis, we propose an approach to aggregate the preferences of multiple users via a single CP-net, while minimizing disagree- ment with individual users. We assume that each user has their preferences represented via a separable CP-net, i.e., a CP-net without any edges between attributes. Our goal is to represent the preferences of a group of users using a single CP-net, referred to as a summary CP-net. We present two algorithms that assume all the input CP-nets are separable, with results on correctness and complexity for each algorithm. We also present a discussion on some important properties of CP-nets and the impact these have on our algorithms.
Open Access
A survey of semisimple algebras in algebraic combinatorics
(Springer Nature, 2021-10-21) Herman, Allen
This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent(aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1’s matrix, and contain a subset $\Delta$ that sums to the identity matrix. The special case when $\Delta$ is a singleton is the important case of an adjacency algebra of a ﬁnite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.
Open Access
The Involutive Double Coset Property for String C-goups of Affine Type
(University of Calgary, 2024) Herman, Allen; Shalabi, Roqayia
In this article we complete the classification of infinite affine Coxeter group types with the property that every double coset relative to the first parabolic subgroup is represented by an involution. This involutive double coset property was established earlier for the Coxeter groups of type $\tilde{C}_2$ and $\tilde{G}_2$, we complete the classification by showing it also holds for type $\tilde{F}_4$ and the types $\tilde{C}_n$ for all $n$. As this property is inherited by all string $C$-groups of these types, it follows that the corresponding abstract regular polytopes will have polyhedral realization cones.
Open Access
The Macdonald group
(Faculty of Graduate Studies and Research, University of Regina, 2024-02) Montoya Ocampo, Alexander; Szechtman, Fernando; Herman, Allen; Gilligan, Bruce
Given α ∈ Z, the Macdonald group G(α) is defined by G(α) = ⟨ A,B | A[A,B] = Aα, B[B,A] = Bα ⟩. It is known that G(α) is finite if and only if α ̸= 1, in which case the prime factors of |G(α)| are those of α − 1. It is also known that G(α) is nilpotent in certain cases. We show that G(α) is always nilpotent, so that for α ̸= 1, G(α) is the direct product of its Sylow subgroups. In the first third of the thesis, we determine the order, upper and lower central series, nilpotency class, and exponent of each of these Sylow subgroups. For the remaining two thirds of the thesis we concentrate on the Sylow 2-subgroup J = J(α) of G(α), so we assume that α = 1 + 2mℓ, where m ≥ 1 and ℓ is odd. We show that J has presentation J = ⟨ x, y | x[x,y] = x1+2mℓ, y[y,x] = y1+2mℓ, x23m−1 = 1 = y23m−1⟩, order 27m−3, and nilpotency class 5 if m > 1 and 3 if m = 1. In the middle third of the thesis, we determine the automorphism groups of the 2-groups J, H = J/Z(J) and K = H/Z(H), where |H| = 26m−3 and |K| = 25m−3. Explicit multiplication, power, and commutator formulas for J, H, and K are given, and used in the calculation of Aut(J), Aut(H), and Aut(K). In the final third of the thesis, we consider the infinite family of finite 2-groups {J(α)}α̸=1 and settle the following isomorphism problem: given α ̸= 1 ̸= α′ ∈ Z, when are J(α) and J(α′) isomorphic?