Spelling suggestions: "subject:"depresentation theory"" "subject:"prepresentation theory""
11 
The centre of a blockSchwabrow, Inga January 2016 (has links)
Let G be a finite group and F a field. The group algebra FG decomposes as a direct sum of twosided ideals, called the blocks of FG. In this thesis the structure of the centre of a block for various groups is investigated. Studying these subalgebras yields information about the relationship between two block algebras and, in certain cases, forms a vital tool in establishing the nonexistence of an important equivalence in the context of modular representation theory. In particular, the focus lies on determining the Loewy structure for the centre of a block, which so far has not been studied in detail but is fundamental in gaining a better understanding of the block itself. For finite groups G with nonabelian, trivial intersection Sylow psubgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of G and its Sylow normaliser. This was already known for the Suzuki groups; the techniques will be generalised to extend the result to cover the Ree groups of type ^2G_2(q).In addition, the three sporadic simple groups with the trivial intersection property, M_11, McL and J_4, together with some small projective special unitary groups are studied with respect to showing the nonexistence of an isomorphism between the centre of the principal block and the centre of its Brauer correspondent. Finally, the Loewy structure of centres of various principal blocks are calculated. In particular, some small sporadic simple groups and groups with normal, elementary abelian Sylow psubgroups are considered. For the latter, some specific formulae for the Loewy length are derived, which generalises recent results on groups with cyclic Sylow psubgroups.

12 
The Convex Hull of the Highest Weight Orbit and the Carathéodory OrbitopeRedding, Nigel January 2017 (has links)
In this thesis, we study the polynomial equations that describe the highest weight
orbit of an irreducible finite dimensional highest weight module under a semisimple
Lie group. We also study the connection of the convex hull of this orbit and the
Carathéodory orbitope.

13 
Some representation theory of the group Sl*(2,A) where A=M(2,O/p^2) and * equals transposeWright, Carmen 01 December 2012 (has links)
Let A be a ring with involution *. The group Sl*(2,A), defined by Pantoja and SotoAndrade, is a noncommutative version of Sl(2,F) where F is a field. In the case of A being artinian, they determined when Sl*(2,A) admitted a Bruhat presentation, and with Gutiérrez, constructed a representation for Sl*(2,A) from its generators. In particular, if A=Mn(F) and * is transposition, then Sl*(2,A) = Sp(2n,F). In this paper, we are interested in the representation theory of G=Sp4(O/p2) where A=M2(O/p2) and O is a local ring with prime ideal p. It has a normal, abelian subgroup K, and by Clifford's theorem we can find distinct irreducible representations of G starting with onedimensional representations of K. The outline of our strategy will be demonstrated in the example of finding irreducible representations of SL2,(O/p2).

14 
Versal deformation rings of modules over Brauer tree algebrasWackwitz, Daniel Joseph 01 July 2015 (has links)
This thesis applies methods from the representation theory of finite dimensional algebras, specifically Brauer tree algebras, to the study of versal deformation rings of modules for these algebras. The main motivation for studying Brauer tree algebras is that they generalize pmodular blocks of group rings with cyclic defect groups.
We consider the case when Λ is a Brauer tree algebra over an algebraically closed field K and determine the structure of the versal deformation ring R(Λ,V) of every indecomposable Λmodule V when the Brauer tree is a star whose exceptional vertex is at the center. The ring R(Λ,V) is a complete local commutative Noetherian Kalgebra with residue field K, which can be expressed as a quotient ring of a power series algebra over K in finitely many commuting variables. The defining property of R(Λ,V) is that the isomorphism class of every lift of V over a complete local commutative Noetherian Kalgebra R with residue field K arises from a local ring homomorphism α : R(Λ, V )→R and that α is unique if R is the ring of dual numbers k[t]/(t2). In the case where Λ is a star Brauer tree algebra and V is an indecomposable Λmodule such that the Kdimension of Ext1Λ(V,V) is equal to R, we explicitly determine generators of an ideal J of k[[t1,....,tr]] such that R(Λ,V)≅k[[t1,....,tr]]/J.

15 
Tilting and Relative Theories in SubcategoriesMohammed, Soud January 2008 (has links)
<p>We show that, over an artin algebra, the tilting functor preserves (co)tilting modules in the subcategories associated to the functor. We also give a sufficient condition for the category of modules of finite projective dimension over an artin algebra to be contravariantly finite in the category of all finitely generated modules over the artin algebra. This is a sufficient condition for the finitistic dimension of the artin algebra to be finite [3].</p><p>We also develop relative theory and in certain subcategories of the module category over an artin algebra in the sense of [10,11]. We use the theory to generalize the main result of [26]</p>

16 
Tilting and Relative Theories in SubcategoriesMohammed, Soud January 2008 (has links)
We show that, over an artin algebra, the tilting functor preserves (co)tilting modules in the subcategories associated to the functor. We also give a sufficient condition for the category of modules of finite projective dimension over an artin algebra to be contravariantly finite in the category of all finitely generated modules over the artin algebra. This is a sufficient condition for the finitistic dimension of the artin algebra to be finite [3]. We also develop relative theory and in certain subcategories of the module category over an artin algebra in the sense of [10,11]. We use the theory to generalize the main result of [26]

17 
Lie Algebras of Differential Operators and DmodulesDonin, Dmitry 20 January 2009 (has links)
In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two
problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic
structure of differential and pseudodifferential operators. We define the KricheverNovikov type Lie algebras of
differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and
central extensions. We show that the corresponding algebras of meromorphic differential operators and
pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic
vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols
holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the
similarities and differences with the case of smooth symbols on the circle.
The second problem is related to the geometry of differential operators and its connection with representations of
semisimple Lie algebras. We show that the semiregular module, naturally associated with a graded semisimple
complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of
the diagonal in the square of the flag variety. Namely, we consider the BeilinsonBernstein localization
of the semiregular module and show that it is isomorphic to the Dmodule obtained by applying the
EmertonNadlerVilonen geometric Jacquet functor to the Dmodule of distributions on the square of the flag variety
with support on the diagonal.

18 
Transfer Relations in Essentially Tame Local Langlands CorrespondenceTam, KamFai 07 January 2013 (has links)
Let $F$ be a nonArchimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this formula to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results.
(i) To show that the automorphic induction character identity is equal to the spectral transfer character identity when both are normalized by the same Whittaker data.
(ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by LanglandsShelstad $\chi$data and to relate BushnellHenniart's rectifiers to certain transfer factors.

19 
Lie Algebras of Differential Operators and DmodulesDonin, Dmitry 20 January 2009 (has links)
In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two
problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic
structure of differential and pseudodifferential operators. We define the KricheverNovikov type Lie algebras of
differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and
central extensions. We show that the corresponding algebras of meromorphic differential operators and
pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic
vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols
holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the
similarities and differences with the case of smooth symbols on the circle.
The second problem is related to the geometry of differential operators and its connection with representations of
semisimple Lie algebras. We show that the semiregular module, naturally associated with a graded semisimple
complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of
the diagonal in the square of the flag variety. Namely, we consider the BeilinsonBernstein localization
of the semiregular module and show that it is isomorphic to the Dmodule obtained by applying the
EmertonNadlerVilonen geometric Jacquet functor to the Dmodule of distributions on the square of the flag variety
with support on the diagonal.

20 
Transfer Relations in Essentially Tame Local Langlands CorrespondenceTam, KamFai 07 January 2013 (has links)
Let $F$ be a nonArchimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this formula to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results.
(i) To show that the automorphic induction character identity is equal to the spectral transfer character identity when both are normalized by the same Whittaker data.
(ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by LanglandsShelstad $\chi$data and to relate BushnellHenniart's rectifiers to certain transfer factors.

Page generated in 0.13 seconds