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Articles 1 - 9 of 9
Full-Text Articles in Algebra
Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula
Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula
All NMU Master's Theses
The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these "even index-sequences" are particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the …
Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier
Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier
Theses and Dissertations--Mathematics
In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of …
Representations Of Finite Groups And Diagrammatic Algebras, Hudson Yeend
Representations Of Finite Groups And Diagrammatic Algebras, Hudson Yeend
CMC Senior Theses
Representation theory allows mathematicians to study abstract mathematical objects using the powerful and concrete tools of linear algebra. This thesis aims to present some foundational concepts in representation theory and apply these concepts to specific groups and algebras. We begin by examining representations of finite groups, culminating with a proof of Maschke's theorem. We then use the correspondence between a group and its group algebra to segue into a study of representations of diagrammatic algebras, where we introduce analogous notions of decomposition. We end with a study of quiver representations, noting that Gabriel's theorem and the kQ-modular structure transcend …
Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson
Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson
LSU Doctoral Dissertations
We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an …
Permutations, Representations, And Partition Algebras: A Random Walk Through Algebraic Statistics, Ian Shors
Permutations, Representations, And Partition Algebras: A Random Walk Through Algebraic Statistics, Ian Shors
HMC Senior Theses
My thesis examines a class of functions on the symmetric group called permutation statistics using tools from representation theory. In 2014, Axel Hultman gave formulas for computing expected values of permutation statistics sampled via random walks. I present analogous formulas for computing variances of these statistics involving Kronecker coefficients – certain numbers that arise in the representation theory of the symmetric group. I also explore deep connections between the study of moments of permutation statistics and the representation theory of the partition algebras, a family of algebras introduced by Paul Martin in 1991. By harnessing these partition algebras, I derive …
The Decomposition Of The Space Of Algebraic Curvature Tensors, Katelyn Sage Risinger
The Decomposition Of The Space Of Algebraic Curvature Tensors, Katelyn Sage Risinger
Electronic Theses, Projects, and Dissertations
We decompose the space of algebraic curvature tensors (ACTs) on a finite dimensional, real inner product space under the action of the orthogonal group into three inequivalent and irreducible subspaces: the real numbers, the space of trace-free symmetric bilinear forms, and the space of Weyl tensors. First, we decompose the space of ACTs using two short exact sequences and a key result, Lemma 3.5, which allows us to express one vector space as the direct sum of the others. This gives us a decomposition of the space of ACTs as the direct sum of three subspaces, which at this point …
Noncommutative Tensor Triangular Geometry And Its Applications To Representation Theory, Kent Barton Vashaw
Noncommutative Tensor Triangular Geometry And Its Applications To Representation Theory, Kent Barton Vashaw
LSU Doctoral Dissertations
One of the cornerstones of the representation theory of Hopf algebras and finite tensor categories is the theory of support varieties. Balmer introduced tensor triangular geometry for symmetric monoidal triangulated categories, which united various support variety theories coming from disparate areas such as homotopy theory, algebraic geometry, and representation theory. In this thesis a noncommutative version will be introduced and developed. We show that this noncommutative analogue of Balmer's theory can be determined in many concrete situations via the theory of abstract support data, and can be used to classify thick tensor ideals. We prove an analogue of prime ideal …
Clifford Algebra Decompositions Of Conformal Orthogonal Group Elements, G. Stacey Staples, David Wylie
Clifford Algebra Decompositions Of Conformal Orthogonal Group Elements, G. Stacey Staples, David Wylie
SIUE Faculty Research, Scholarship, and Creative Activity
Beginning with a finite-dimensional vector space V equipped with a nondegenerate quadratic form Q, we consider the decompositions of elements of the conformal orthogonal group COQ(V), defined as the direct product of the orthogonal group OQ(V) with dilations. Utilizing the correspondence between conformal orthogonal group elements and ``decomposable'' elements of the associated Clifford algebra, ClQ(V), a decomposition algorithm is developed. Preliminary results on complexity reductions that can be realized passing from additive to multiplicative representations of invertible elements are also presented with examples. The approach here is …
On Representations Of Semigroups Having Hypercube-Like Cayley Graphs, Cody Cassiday, G. Stacey Staples
On Representations Of Semigroups Having Hypercube-Like Cayley Graphs, Cody Cassiday, G. Stacey Staples
SIUE Faculty Research, Scholarship, and Creative Activity
The $n-dimensional hypercube, or n-cube, is the Cayley graph of the Abelian group Z2n. A number of combinatorially-interesting groups and semigroups arise from modified hypercubes. The inherent combinatorial properties of these groups and semigroups make them useful in a number of contexts, including coding theory, graph theory, stochastic processes, and even quantum mechanics. In this paper, particular groups and semigroups whose Cayley graphs are generalizations of hypercubes are described, and their irreducible representations are characterized. Constructions of faithful representations are also presented for each semigroup. The associated semigroup algebras are realized within the context …