Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
17,998
result(s) for
"Representations of groups."
Sort by:
The Representation Theory of the Increasing Monoid
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation
category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish
properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous
connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle
algebras.
eBook
On Fusion Systems of Component Type
This memoir begins a program to classify a large subclass of the class of simple saturated 2-fusion systems of component type. Such a
classification would be of great interest in its own right, but in addition it should lead to a significant simplification of the proof
of the theorem classifying the finite simple groups.
Why should such a simplification be possible? Part of the answer lies in the
fact that there are advantages to be gained by working with fusion systems rather than groups. In particular one can hope to avoid a
proof of the B-Conjecture, a important but difficult result in finite group theory, established only with great effort.
But in
addition, the program involves a reorganization of the treatment of “groups of component type”, or perhaps more accurately, of “fusion
systems of component type”. The groups of component type should be viewed as “odd” groups, in that most examples are groups of Lie type
over fields of odd order. The remaining simple groups should be viewed as “even” groups, since most of the examples in this class are of
Lie type over fields of even order. There are corresponding notions of “odd” and “even” 2-fusion systems.
In our program the
class of odd groups, and/or fusion systems, is contracted in a carefully chosen manner, so as to avoid difficulties associated to
certain “standard form problems”. This has the effect of greatly simplifying the treatment of the odd 2-fusion systems, and then also
the treatment of the odd simple groups. Of course the flip side of such a reorganization is to enlarge the class of even objects, so
that the approach may make it more difficult to treat that class. But it is our sense that the trade off should lead to a net
simplification.
This change in the partition of simple groups into odd and even groups is not dissimilar to the one in the
program of Gorenstein, Lyons, and Solomon (hereafter referred to as GLS) to rewrite the proof of the classification.
In the
introduction, we expand upon these themes, making them a bit more precise, supplying some background, and eventually stating some of our
major theorems. Then in the body of the paper, we fill in details and begin the actual program.
eBook
The Irreducible Subgroups of Exceptional Algebraic Groups
This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields
of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group
A result of Liebeck and Testerman shows that each irreducible connected subgroup
eBook
Infinite-Dimensional Representations of 2-Groups
A ‘2-group’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations
on vector spaces, 2-groups have representations on ‘2-vector spaces’, which are categories analogous to vector spaces. Unfortunately,
Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this
reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called ‘measurable categories’ (since they
are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie
2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of
the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct
sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and
sub-intertwiners—features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and
intertwiners. We also study ‘irretractable’ representations—another feature not seen in ordinary group representation theory. Finally,
we argue that measurable categories equipped with some extra structure deserve to be considered ‘separable 2-Hilbert spaces’, and
compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.
eBook
To an effective local Langlands Correspondence
Let F be a non-Archimedean local field. Let \\mathcal{W}_{F} be the Weil group of F and \\mathcal{P}_{F} the wild inertia subgroup of \\mathcal{W}_{F}. Let \\widehat {\\mathcal{W}}_{F} be the set of equivalence classes of irreducible smooth representations of \\mathcal{W}_{F}. Let \\mathcal{A}^{0}_{n}(F) denote the set of equivalence classes of irreducible cuspidal representations of \\mathrm{GL}_{n}(F) and set \\widehat {\\mathrm{GL}}_{F} = \\bigcup _{n\\ge 1} \\mathcal{A}^{0}_{n}(F). If \\sigma \\in \\widehat {\\mathcal{W}}_{F}, let ^{L}{\\sigma }\\in \\widehat {\\mathrm{GL}}_{F} be the cuspidal representation matched with \\sigma by the Langlands Correspondence. If \\sigma is totally wildly ramified, in that its restriction to \\mathcal{P}_{F} is irreducible, the authors treat ^{L}{\\sigma} as known. From that starting point, the authors construct an explicit bijection \\mathbb{N}:\\widehat {\\mathcal{W}}_{F} \\to \\widehat {\\mathrm{GL}}_{F}, sending \\sigma to ^{N}{\\sigma}. The authors compare this \"naïve correspondence\" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of \"internal twisting\" of a suitable representation \\pi (of \\mathcal{W}_{F} or \\mathrm{GL}_{n}(F)) by tame characters of a tamely ramified field extension of F, canonically associated to \\pi . The authors show this operation is preserved by the Langlands correspondence.
eBook
Deformation Theory and Local-Global Compatibility of Langlands Correspondences
The deformation theory of automorphic representations is used to study local properties of Galois representations associated to
automorphic representations of general linear groups and symplectic groups. In some cases this allows to identify the local Galois
representations with representations predicted by a local Langlands correspondence.
eBook