We studied the subjectivity of word maps on SU(n) and the length of the shortest elements in the central series of free group of rank 2 with some applications to almost laws in compact groups.

When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation. The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).

We study character generating functions (character generators) of simple Lie algebras. The expression due to Patera and Sharp, derived from the Weyl character formula, is ¯rst re- viewed. A new general formula is then found. It makes clear the distinct roles of \outside" and \inside" elements of the integrity basis, and helps determine their quadratic incompati- bilities. We review, analyze and extend the results obtained by Gaskell using the Demazure character formulas. We ¯nd that the fundamental generalized-poset graphs underlying the character generators can be deduced from such calculations. These graphs, introduced by Baclawski and Towber, can be simpli¯ed for the purposes of constructing the character generator. The generating functions can be written easily using the simpli¯ed versions, and associated Demazure expressions. The rank-two algebras are treated in detail, but we believe our results are indicative of those for general simple Lie algebras. / vii, 92 leaves ; 29 cm.

The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned. / Thesis (M.Sc. (School of Mathematical Sciences)) - University of KwaZulu-Natal, Pietermaritzburg, 2008.

When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation. The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).

Thesis--Leyden. / In Periodical Room.

Thesis (doctoral)--Technische Universiteit Eindhoven, 1989. / Includes indexes. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (p. 117-120).

Universiẗat, Diss., 2004--Saarbrücken.

Η θεωρία των συμμετρικών χώρων αποτελεί μια σπουδαία κλάση των ομογενών χώρων, με εφαρμογές σε πολλούς κλάδους των μαθηματικών όπως στην αλγεβρική και την διαφορική γεωμετρία. Σε αυτήν την εργασία θα δώσουμμε τον ορισμό των συμμετρικών χώρων, τα βασικά τους χαρακτηριστικά και την ταξινόμησή τους. Θα περιγράψουμε τους χώρους αυτούς κυρίως αλγεβρικά, οπότε δεν θεωρείται απαραίτητο από τον αναγνώστη να γνωρίζει εκτενώς την θεωρία της διαφορικής γεωμετρίας για να κατανοήσει πλήρως την εργασία. / The theory of symmetric spaces is an important class of homogeneous spaces, with applications in many branches of mathematics such as algebraic and differential geometry. In this work we will define the symmetric spaces, their key features and sort them. We will describe these spaces mainly algebraic, so it is not considered necessary by the reader to know in detail the theory of differential geometry to understand the work.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-graduação em Física / Made available in DSpace on 2012-10-24T18:37:40Z (GMT). No. of bitstreams: 1 263132.pdf: 526017 bytes, checksum: 5840fcf1dc0e49a55cf108092da716f2 (MD5) / Monopolos magnéticos têm sido objetos de grande interesse nos últimos anos, principalmente por serem previstos em algumas teorias de grande unificação e por, possivelmente, serem relevantes no fenômeno do confinamento em QCD. Consideramos uma teoria de Yang-Mills-Higgs com simetria de gauge SU(n) quebrada espontaneamente em SO(n) que apresenta condições topológicas necessárias para a existência de monopolos Z2. Construímos as formas assintóticas desses monopolos, considerando duas quebras distintas do SU(n) em SO(n), e verificamos que os monopolos fundamentais estão associados aos pesos da representação definidora da álgebra so(n)v.