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31 
Combinatorics and the KP HierarchyCarrell, Sean January 2009 (has links)
The study of the infinite (countable) family of partial differential equations
known as the Kadomtzev  Petviashvili (KP) hierarchy has received much interest in
the mathematical and theoretical physics community for over forty years. Recently
there has been a renewed interest in its application to enumerative combinatorics
inspired by Witten's conjecture (now Kontsevich's theorem).
In this thesis we provide a detailed development of the KP hierarchy and some of
its applications with an emphasis on the combinatorics involved. Up until now, most
of the material pertaining to the KP hierarchy has been fragmented throughout the
physics literature and any complete accounts have been for purposes much diff erent
than ours.
We begin by describing a family of related Lie algebras along with a module
on which they act. We then construct a realization of this module in terms of
polynomials and determine the corresponding Lie algebra actions. By doing this
we are able to describe one of the Lie group orbits as a family of polynomials and the
equations that de fine them as a family of partial diff erential equations. This then
becomes the KP hierarchy and its solutions. We then interpret the KP hierarchy
as a pair of operators on the ring of symmetric functions and describe their action
combinatorially. We then conclude the thesis with some combinatorial applications.

32 
Combinatorics and the KP HierarchyCarrell, Sean January 2009 (has links)
The study of the infinite (countable) family of partial differential equations
known as the Kadomtzev  Petviashvili (KP) hierarchy has received much interest in
the mathematical and theoretical physics community for over forty years. Recently
there has been a renewed interest in its application to enumerative combinatorics
inspired by Witten's conjecture (now Kontsevich's theorem).
In this thesis we provide a detailed development of the KP hierarchy and some of
its applications with an emphasis on the combinatorics involved. Up until now, most
of the material pertaining to the KP hierarchy has been fragmented throughout the
physics literature and any complete accounts have been for purposes much diff erent
than ours.
We begin by describing a family of related Lie algebras along with a module
on which they act. We then construct a realization of this module in terms of
polynomials and determine the corresponding Lie algebra actions. By doing this
we are able to describe one of the Lie group orbits as a family of polynomials and the
equations that de fine them as a family of partial diff erential equations. This then
becomes the KP hierarchy and its solutions. We then interpret the KP hierarchy
as a pair of operators on the ring of symmetric functions and describe their action
combinatorially. We then conclude the thesis with some combinatorial applications.

33 
On the Representation Theory of Semisimple Lie GroupsAlFaisal, Faisal January 2010 (has links)
This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of BorelWeilBott, CasselmanOsborne and Kostant. The first of these realizes all the irreducible holomorphic representations of a complex semisimple Lie group G in the cohomology of certain sheaves of equivariant line bundles over the flag variety of G. The latter two theorems describe the Lie algebra cohomology of a maximal nilpotent subalgebra of Lie(G) with coefficients in an irreducible Lie(G)module. Applications to geometry and representation theory are given. Also included is a brief overview of Schmid's farreaching generalization of the BorelWeilBott theorem to the setting of unitary representations of real semisimple Lie groups on (possibly infinitedimensional) Hilbert spaces.

34 
Invariant Lie polynomials in two and three variables.Hu, Jiaxiong 21 August 2009 (has links)
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 3dimensional natural representation of an 8dimensional 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 openended problem.

35 
Products of representations of the symmetric group and noncommutative versionsMoreira Rodriguez, Rivera Walter 10 October 2008 (has links)
We construct a new operation among representations of the symmetric group that
interpolates between the classical internal and external products, which are defined in
terms of tensor product and induction of representations. Following Malvenuto and
Reutenauer, we pass from symmetric functions to noncommutative symmetric functions
and from there to the algebra of permutations in order to relate the internal and
external products to the composition and convolution of linear endomorphisms of the
tensor algebra. The new product we construct corresponds to the Heisenberg product
of endomorphisms of the tensor algebra. For symmetric functions, the Heisenberg
product is given by a construction which combines induction and restriction of representations.
For noncommutative symmetric functions, the structure constants of
the Heisenberg product are given by an explicit combinatorial rule which extends a
wellknown result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra.
We describe the dual operation among quasisymmetric functions in terms of
alphabets.

36 
On the Representation Theory of Semisimple Lie GroupsAlFaisal, Faisal January 2010 (has links)
This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of BorelWeilBott, CasselmanOsborne and Kostant. The first of these realizes all the irreducible holomorphic representations of a complex semisimple Lie group G in the cohomology of certain sheaves of equivariant line bundles over the flag variety of G. The latter two theorems describe the Lie algebra cohomology of a maximal nilpotent subalgebra of Lie(G) with coefficients in an irreducible Lie(G)module. Applications to geometry and representation theory are given. Also included is a brief overview of Schmid's farreaching generalization of the BorelWeilBott theorem to the setting of unitary representations of real semisimple Lie groups on (possibly infinitedimensional) Hilbert spaces.

37 
Matrix Problems and their Relation to the Representation Theory of Quivers and PosetsCicala, Daniel January 2014 (has links)
Techniques from the theory of matrix problems have proven to be helpful for studying problems within representation theory. In particular, matrix problems are well suited to use in problems related to classifying indecomposable representations of quivers and of posets. However, throughout the literature, there are many different types of matrix problems and little clarification of the relationships between them. In this thesis, we choose six types of matrix problems, place them all within a common framework and find correspondences between them. Moreover, we show that their use in the classification of finitedimensional representations of quivers and posets are, in general, wellfounded. Additionally, we investigate a direct relationship between the problem of classifying quiver representations and the problem of classifying poset representations.

38 
On the Construction of Supercuspidal Representations: New Examples from Shallow CharactersGastineau, Stella Sue January 2022 (has links)
Thesis advisor: Mark Reeder / This thesis contributes to the construction of supercuspidal representations in small residual characteristics. Let G be a connected, quasisplit, semisimple reductive algebraic group defined and quasisplit over a nonarchimedean local field k and splitting over a tamely, totally ramified extension of k. To each parahoric subgroup of G(k), Moy and Prasad have attached a natural filtration by compact open subgroups, the first of which is called the prounipotent radical of the parahoric subgroup. The first main result of this thesis is to characterize shallow characters of a prounipotent radical, those being complex characters that vanish on the smallest MoyPrasad subgroup containing all commutators of linearlydependent affine kroot groups. Through lowrank examples, we illustrate how this characterization can be used to explicitly construct all shallow characters. Next, we provide a natural sufficient condition under which a shallow character compactly induces as a direct sum of supercuspidal representations of G(k). Through examples, however, we show that this sufficient condition need not be necessary, all while constructing new supercuspidal representations of Sp_4(k) when p = 2 and the split form of G_2 over k when p = 3. This work extends the construction of the simple supercuspidal representations given by Gross and Reeder and the epipelagic supercuspidal representations given by Reeder and Yu. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

39 
On supersingular representations of GL(2, D) with a division algebra D over a padic fieldWijerathne, Wijerathne Mudiyanselage Menake 01 August 2022 (has links) (PDF)
Let D be a division algebra over a padic field of characteristic 0. We investigate the modp supersingular representations of GL(2, D) by computing a basis for the space of invariants of a certain quotient under the prop Iwahori subgroup. This generalizes the previous works of Hendel and Schein.

40 
Representations and actions of Hopf algebrasYammine, Ramy January 2021 (has links)
The larger area of my thesis is Algebra; more specifically, my work belongs to the following two major branches of Algebra:
\emph{representation theory} and \emph{invariant theory}.
In brief, the objective of representation theory is to investigate algebraic objects through
their actions on vector spaces;
this allows the welldeveloped toolkit of linear algebra to be brought to bear on
complex algebraic problems.
The theory has played a crucial role in nearly every subdiscipline of pure mathematics.
Outside of pure mathematics,
representation theory has been successfully used, for instance,
in the study of symmetries of physical systems
and in describing molecular structures in physical chemistry.
Invariant theory is another classical algebraic theme permeating virtually all areas
of pure mathematics and some areas of applied mathematics as well, notably coding theory.
The theory studies actions of algebraic objects, traditionally groups and Lie algebras,
on algebras, that is, vector spaces that are equipped with a multiplication.
\bigskip
The representation theory of (associative) algebras provides a useful setting in which to studymany aspects of the two most classical flavors of representation theory under a common umbrella:
representations of groups and of Lie algebras. However,
it turns out that general algebras fail to capture certain features of group representations
and the same can be said for representations of Lie algebras as well.
The additional structure that is needed in order to access these features is
naturally provided by the important class of \emph{Hopf algebras}.
Besides unifying the representation theories of groups and of Lie algebras, Hopf algebras serve a similar
purpose in invariant theory,
allowing for a simultaneous treatment of group actions (by automorphisms)
and Lie algebras (by derivations) on algebras.
More importantly, actions of Hopf algebras have the potential of capturing additional aspects
of the structure of algebras they act on, uncovering features that cannot be
accessed by ordinary
groups or Lie algebras.
\bigskip
Presently, the theory of Hopf algebras is still nowhere near thelevel that has been achieved for groups and for Lie algebras over the course of the past century
and earlier.
This thesis aims to make a contribution to the representation and invariant theories of Hopf algebras,
focusing for the most part on Hopf algebras that are not necessarily
finite dimensional.
Specifically, the contributions presented here can be grouped under two headings:
\smallskip
\noindent\qquad(i) \textbf{ Invariant Theory:} Hopf algebra actions and prime spectra, and\smallskip
\noindent\qquad(ii)\textbf{ Representation Theory:} the adjoint representation of a Hopf algebra.
\smallskip
In the work done under the heading (i), we were able to use the action of cocommutative Hopf algebras on other algebras to "stratify" the prime spectrum of the algebra being acted upon, and then express each stratum in terms of the spectrum of a commutative domain. Additionally, we studied the transfer of properties between an ideal in the algebra being acted upon, and the largest subideal of that ideal, stable under the action. We were able to achieve results for various families of acting Hopf algebras, namely \emph{cocommutative} and \emph{connected} Hopf algebras.\\The main results concerning heading (ii) concerned the subalgebra of locally finite elements of a Hopf algebra, often called the finite part of the Hopf algebra. This is a subalgebra containing the center that was used successfully to study the ring theoretic properties of group algebras, Lie algebras, and other classical structures.
We prove that the finite is not only a subalgebra, but a coideal subalgebra in general, and in the case of (almost) cocommuative Hopf algebra, it is indeed a Hopf subalgebra.
The results in this thesis generalize earlier theorems that were proved for the prototypical special classes of Hopf algebras: group algebras and enveloping algebras of Lie algebras. / Mathematics

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