This thesis is concerned with three distinct, but closely related, research topics focusing on the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin by obtaining a new canonical form for nilpotent matrices, up to similarity, which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi;j) -> (xn+1-j;n+1-i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group, we thus obtain a unified approach to computing representatives for nilpotent orbits for all classical groups G. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487]. We give a case-free proof of the conjectures of Lusztig from that paper. This presents a uniform picture of the unipotent elements of G, which can be viewed as an extension of the Dynkin-Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra g and the coadjoint action of G on g*. We also obtain several general results about the Hesselink stratification and Fq-rational structures on G-modules. Our third topic is concerned with generalised Gelfand-Graev representations of finite groups of Lie type. Let u be a unipotent element in such a group GF and let Γu be the associated generalised Gelfand-Graev representation of GF . Under the assumption that G has a connected centre, we show that the dimension of the endomorphism algebra of Γu is a polynomial in q (the order of the associated finite field), with degree given by dimCG(u). When the centre of G is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of q, unless one adopts a convention of considering separately various congruence classes of q. Subject to such a convention, we extend our result. We also present computational data related to the main theoretical results. In particular, tables of our canonical forms are given in the appendices, as well as tables of dimension polynomials for endomorphism algebras of generalised Gelfand-Graev representations, together with the relevant GAP source code.

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In this paper, we use the techniques of plethystic substitution to reformulate the difference raising operators presented by Di Francesco and Kedem. A connection between these operators and Shimozono and Zabrocki's parabolic Jing operators is presented. In particular, we find that these operators are a renormalization of a particular case of the parabolic Jing operators. / Master of Science / In response to an open problem in Physics, an idea is presented by Di Francesco and Kedem in [1]. A connection between this idea and a Math idea presented by Shimozono and Zabrocki in [9] is presented. It is common that unknown overlap exists when authors from different fields work on similar problems. This connection is seen once the techniques used by Di Francesco and Kedem are interpreted in the language used by Shimozono and Zabrocki. In particular, we find that the idea in [1] is a specialization of that in [9].

By formulating the six dimensional (0,2) superconformal field theory X[j] on a Riemann surface decorated with certain codimension two defects, a multitude of four dimensional N=2 supersymmetric field theories can be constructed. In this dissertation, various aspects of this construction are investigated in detail for j=A,D,E. This includes, in particular, an exposition of the various partial descriptions of the codimension two defects that become available under dimensional reductions and the relationships between them. Also investigated is a particular observable of this class of four dimensional theories, namely the partition function on the four sphere and its relationship to correlation functions in a class of two dimensional non-rational conformal field theories called Toda theories. It is pointed out that the scale factor that captures the Euler anomaly of the four dimensional theory has an interpretation in the two dimensional language, thereby adding one of the basic observables of the 4d theory to the 4d/2d dictionary. / text

Student Number : 9206749J - PhD thesis - School of Mathematics - Faculty of Science / For right near-rings the left representation has always been considered the natural one. A study of right representation for right distributively generated (d.g.) near-rings was initiated by Rahbari and this work is extended here to introduce radical-like objects in the near-ring R using right R-groups. The right radicals rJ0(R), rJ1/2(R) and rJ2(R) are defined as counterparts of the left radicals J0(R), J1/2(R) and J2(R) respectively, and their properties are discussed. Of particular interest are the relationships between the left and right radicals. It is shown for example that for all finite d.g. near-rings R with identity, J2(R) = rJ0(R) = rJ1/2(R) = rJ2(R). A right anti-radical, rSoi(R), is defined for d.g. near-rings with identity, using a construction that is analogous to that of the (left) socle-ideal, Soi(R). In particular, it is shown that for finite d.g. near-rings with identity, an ideal A is contained in rSoi(R) if and only if A \ J2(R) = (0). The relationship between the left and right socle-ideals is investigated, and it is established that rSoi(R) #18; Soi(R) for d.g. near-rings with identity and satisfying the descending chain condition for left R-subgroups.

The Hessenberg representation is a representation of the symmetric group afforded on the cohomology ring of a regular semisimple Hessenberg variety. We study this representation via a combinatorial presentation called GKM Theory. This presentation allows for the study of the representation entirely from a graph. The thesis derives a combinatorial construction of a basis of the equivariant cohomology as a free module over a polynomial ring. This generalizes classical constructions of Schubert classes and divided difference operators for the equivariant cohomology of the flag variety.

Voting theory as been explored mathematically since the 1780’s. Many people have tackled parts of it using various tools, and now we shall look at it through the eyes of a representation theorist. Each vote can be thought of as a permutation of the symmetric group, Sn, and a poll is similar to a linear combination of these elements. Specifically, we will focus on translating and generalizing the works of Donald Saari into more algebraic terms to discover not just one space, but a whole isotypic component essential to positional voting.

In voting theory, simple questions can lead to convoluted and sometimes paradoxical results. Recently, mathematician Donald Saari used geometric insights to study various voting methods. He argued that a particular positional voting method (namely that proposed by Borda) minimizes the frequency of paradoxes. We present an approach to similar ideas which draw from group theory and algebra. In particular, we employ tools from representation theory on the symmetric group to elicit some of the natural behaviors of voting profiles. We also make generalizations to similar results for partially ranked data.

Irreducible representations of a finite group over a field are important because all representations of a group are direct sums of irreducible representations. Maschke tells us that if φ is a representation of the finite group G of order n on the m-dimensional space V over the field K of complex numbers and if U is an invariant subspace of φ, then U has a complementary reducing subspace W . The objective of this thesis is to find all irreducible representations of the dihedral group D2n. The reason we will work with the dihedral group is because it is one of the first and most intuitive non-abelian group we encounter in abstract algebra. I will compute the representations and characters of D2n and my thesis will be an explanation of these computations. When n = 2k + 1 we will show that there are k + 2 irreducible representations of D2n, but when n = 2k we will see that D2n has k + 3 irreducible rep- resentations. To achieve this we will first give some background in group, ring, module, and vector space theory that is used in representation theory. We will then explain what general representation theory is. Finally we will show how we arrived at our conclusion.

In 1949, Wever observed that the degree d of an invariant Lie polynomial must be a multiple of the number q of generators of the free Lie algebra. He also found that there are no invariant Lie polynomials in the following cases: q = 2, d = 4; q = 3, d = 6; d = q ≥ 3. Wever gave a formula for the number of invariants for q = 2 in the natural representation of sl(2). In 1958, Burrow extended Wevers formula to q > 1 and d = mq where m > 1. In the present thesis, we concentrate on finding invariant Lie polynomials (simply called Lie invariants) in the natural representations of sl(2) and sl(3), and in the adjoint representation of sl(2). We first review the method to construct the Hall basis of the free Lie algebra and the way to transform arbitrary Lie words into linear combinations of Hall words. To find the Lie invariants, we need to find the nullspace of an integer matrix, and for this we use the Hermite normal form. After that, we review the generalized Witt dimension formula which can be used to compute the number of primitive Lie invariants of a given degree. Secondly, we recall the result of Bremner on Lie invariants of degree ≤ 10 in the natural representation of sl(2). We extend these results to compute the Lie invariants of degree 12 and 14. This is the first original contribution in the present thesis. Thirdly, we compute the Lie invariants in the adjoint representation of sl(2) up to degree 8. This is the second original contribution in the present thesis. Fourthly, we consider the natural representation of sl(3). This is a 3-dimensional natural representation of an 8-dimensional Lie algebra. Due to the huge number of Hall words in each degree and the limitation of computer hardware, we compute the Lie invariants only up to degree 12. Finally, we discuss possible directions for extending the results. Because there are infinitely many different simple finite dimensional Lie algebras and each of them has infinitely many distinct irreducible representations, it is an open-ended problem.