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The total character of a finite group G is the sum of the irreducible characters of G. When the total character of a finite group can be written as a monic polynomial with integer coefficients in an irreducible character of G, we say that G is a total character group. In this thesis we examine the total character of the dicyclic group of order 4n, the non-abelian groups of order p^3, and the symmetric group on n elements for all n ≥ 1. The dicyclic group of order 4n is a total character group precisely when n is congruent to 2 or 3 mod 4, and the associated polynomial is a sum of Chebyshev polynomials of the second kind. The irreducible characters paired with these polynomials are exactly the faithful characters of the dicyclic group. In contrast, the non-abelian groups of order p^3 and the symmetric group on n elements with n ≥ 4 are not total character groups. Finally, we examine the special case when G is a total character group and the polynomial is of degree 2. In this case, we say that G is a quadratic total character group. We classify groups which are both quadratic total character groups and p-groups.

In the first part of the thesis we define an automorphism φn for each star graph Stn of degree n-1, which yields permutations of labels for the edges of Stn taken from the set of integers {1,..., [n/2c]}. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known two-labelled Hamilton cycle H1 2(n) in Stn. The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies in computer science. All our results improve on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs.

The character theory of the symmetric group is a powerful method of studying enu- merative questions about factorizations of permutations, which arise in areas including topology, geometry, and mathematical physics. This method relies on having an encoding of the enumerative problem in the centre Z(n) of the algebra C[S_n] spanned by the symmetric group S_n. This thesis develops methods to deal with permutation factorization problems which cannot be encoded in Z(n). The (p,q,n)-dipole problem, which arises in the study of connections between string theory and Yang-Mills theory, is the chief problem motivating this research. This thesis introduces a refinement of the (p,q,n)-dipole problem, namely, the (a,b,c,d)- dipole problem. A Join-Cut analysis of the (a,b,c,d)-dipole problem leads to two partial differential equations which determine the generating series for the problem. The first equation determines the series for (a,b,0,0)-dipoles, which is the initial condition for the second equation, which gives the series for (a,b,c,d)-dipoles. An analysis of these equa- tions leads to a process, recursive in genus, for solving the (a,b,c,d)-dipole problem for a surface of genus g. These solutions are expressed in terms of a natural family of functions which are well-understood as sums indexed by compositions of a binary string. The combinatorial analysis of the (a,b,0,0)-dipole problem reveals an unexpected fact about a special case of the (p,q,n)-dipole problem. When q=n−1, the problem may be encoded in the centralizer Z_1(n) of C[S_n] with respect to the subgroup S_{n−1}. The algebra Z_1(n) has many combinatorially important similarities to Z(n) which may be used to find an explicit expression for the genus polynomials for the (p,n−1,n)-dipole problem for all values of p and n, giving a solution to this case for all orientable surfaces. Moreover, the algebraic techniques developed to solve this problem provide an alge- braic approach to solving a class of non-central problems which includes problems such as the non-transitive star factorization problem and the problem of enumerating Z_1- decompositions of a full cycle, and raise intriguing questions about the combinatorial significance of centralizers with respect to subgroups other than S_{n−1}.

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter.

The character theory of the symmetric group is a powerful method of studying enu- merative questions about factorizations of permutations, which arise in areas including topology, geometry, and mathematical physics. This method relies on having an encoding of the enumerative problem in the centre Z(n) of the algebra C[S_n] spanned by the symmetric group S_n. This thesis develops methods to deal with permutation factorization problems which cannot be encoded in Z(n). The (p,q,n)-dipole problem, which arises in the study of connections between string theory and Yang-Mills theory, is the chief problem motivating this research. This thesis introduces a refinement of the (p,q,n)-dipole problem, namely, the (a,b,c,d)- dipole problem. A Join-Cut analysis of the (a,b,c,d)-dipole problem leads to two partial differential equations which determine the generating series for the problem. The first equation determines the series for (a,b,0,0)-dipoles, which is the initial condition for the second equation, which gives the series for (a,b,c,d)-dipoles. An analysis of these equa- tions leads to a process, recursive in genus, for solving the (a,b,c,d)-dipole problem for a surface of genus g. These solutions are expressed in terms of a natural family of functions which are well-understood as sums indexed by compositions of a binary string. The combinatorial analysis of the (a,b,0,0)-dipole problem reveals an unexpected fact about a special case of the (p,q,n)-dipole problem. When q=n−1, the problem may be encoded in the centralizer Z_1(n) of C[S_n] with respect to the subgroup S_{n−1}. The algebra Z_1(n) has many combinatorially important similarities to Z(n) which may be used to find an explicit expression for the genus polynomials for the (p,n−1,n)-dipole problem for all values of p and n, giving a solution to this case for all orientable surfaces. Moreover, the algebraic techniques developed to solve this problem provide an alge- braic approach to solving a class of non-central problems which includes problems such as the non-transitive star factorization problem and the problem of enumerating Z_1- decompositions of a full cycle, and raise intriguing questions about the combinatorial significance of centralizers with respect to subgroups other than S_{n−1}.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-09-20T13:36:29Z No. of bitstreams: 1 sarahribeirodejesusferreira.pdf: 854513 bytes, checksum: bdb519074051d0889c62002f16fe1a8e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-10-01T19:17:08Z (GMT) No. of bitstreams: 1 sarahribeirodejesusferreira.pdf: 854513 bytes, checksum: bdb519074051d0889c62002f16fe1a8e (MD5) / Made available in DSpace on 2018-10-01T19:17:08Z (GMT). No. of bitstreams: 1 sarahribeirodejesusferreira.pdf: 854513 bytes, checksum: bdb519074051d0889c62002f16fe1a8e (MD5) Previous issue date: 2018-08-13 / Nesse trabalho, apresentamos a teoria de representação básica do grupo simétrico e seus aspectos combinatórios. O objetivo principal desse trabalho é construir um conjunto completo de representações irredutíveis e não equivalentes do grupo simétrico, em termos da sua partição e conceitos combinatórios relacionados com o tableau de Young. Veremos que esse objeto combinatório nos fornecerá duas maneiras de descrever as representações irredutíveis do grupo simétrico, uma via politablóides e uma alternativa via idempotentes da álgebra de grupo, e que, na verdade, essas duas abordagens são isomorfas. Iremos abordar alguns resultados interessantes, como a regra de Young, a regra da ramificação e o algoritmo combinatório da correspondência de Robinson-Schensted. / In this work, we present the basic representation theory of the symmetric group and its combinatorial aspects. The main objective of this work is to construct a complete set of irreducible and inequivalent representations of the symmetric group, in terms of its partition and combinatorial concepts related to Young’s tableau. We will see that this combinatorial object will provide us two ways of describing the irreducible representations of the symmetric group, a politabloid pathway, and an alternative via idempotent group algebra, and that, in fact, these two approaches are isomorphic. We will cover some interesting results, such as the Young’s rule, the branching rule, and the Robinson-Schensted’s combinatorial matching algorithm.

We have found an operation between permutations in the symmetric group Sn upon which we have experimentally derived results that can be linked to desirable properties in cryptography, mainly in the domain of one-way functions. From it, we have implemented a beta version of an algorithm for a hashing function by exploiting the operation’s low computational cost for speed and its properties for security. Its design makes it resistant to length extension attacks and the encoding of blocks into permutations suggests that any differential cryptanalysis technique that is based on bit conditions should be useless against it. More precisely, when measuring the evolution of differences in the compression function, bit-based distances such as the exclusive-or distance should be replaced by another type of distance, still to be determined in future research. In this work we will present the algorithm and introduce a new framework of cryptanalysis for collision and preimage attacks in order to somehow measure its security. Once this is done, we will run comparison tests against MD5 and SHA256 in order to externally evaluate our algorithm in terms of speed, weaknesses and strength.

Graded Hecke algebras are deformations of skew group algebras which arise from a group acting on a polynomial ring. Over fields of characteristic zero, these deformations have been studied in depth and include both symplectic reflection algebras and rational Cherednik algebras as examples. In Lusztig's graded affine Hecke algebras, the action of the group is deformed, but not the commutativity of the vectors. In Drinfeld's Hecke algebras, the commutativity of the vectors is deformed, but not the action of the group. Lusztig's algebras are all isomorphic to Drinfeld's algebras in the nonmodular setting. We find new deformations in the modular setting, i.e., when the characteristic of the underlying field divides the order of the group. We use Poincare-Birkhoff-Witt conditions to classify these deformations arising from the symmetric group acting on a polynomial ring in arbitrary characteristic, including the modular case.

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