90,856 research outputs found
On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two
We study the irreducible complex representations of general linear groups
over principal ideal local rings of length two with a fixed finite residue
field. We construct a canonical correspondence between the irreducible
representations of all such groups which preserves dimensions. For general
linear groups of order three and four over these rings, we construct all the
irreducible representations. We show that the the problem of constructing all
the irreducible representations of all general linear groups over these rings
is not easier than the problem of constructing all the irreducible
representations of the general linear groups over principal ideal local rings
of arbitrary length in the function field case.Comment: 21 page
Thin coverings of modules
Thin coverings are a method of constructing graded-simple modules from simple
(ungraded) modules. After a general discussion, we classify the thin coverings
of (quasifinite) simple modules over associative algebras graded by finite
abelian groups. The classification uses the representation theory of cyclotomic
quantum tori. We close with an application to representations of multiloop Lie
algebras
Fast Fourier Transforms for Finite Inverse Semigroups
We extend the theory of fast Fourier transforms on finite groups to finite
inverse semigroups. We use a general method for constructing the irreducible
representations of a finite inverse semigroup to reduce the problem of
computing its Fourier transform to the problems of computing Fourier transforms
on its maximal subgroups and a fast zeta transform on its poset structure. We
then exhibit explicit fast algorithms for particular inverse semigroups of
interest--specifically, for the rook monoid and its wreath products by
arbitrary finite groups.Comment: ver 3: Added improved upper and lower bounds for the memory required
by the fast zeta transform on the rook monoid. ver 2: Corrected typos and
(naive) bounds on memory requirements. 30 pages, 0 figure
Homological algebra of twisted quiver bundles
Several important cases of vector bundles with extra structure (such as Higgs
bundles and triples) may be regarded as examples of twisted representations of
a finite quiver in the category of sheaves of modules on a
variety/manifold/ringed space. We show that the category of such
representations is an abelian category with enough injectives by constructing
an explicit injective resolution. Using this explicit resolution, we find a
long exact sequence that computes the Ext groups in this new category in terms
of the Ext groups in the old category. The quiver formulation is directly
reflected in the form of the long exact sequence. We also show that under
suitable circumstances, the Ext groups are isomorphic to certain
hypercohomology groups.Comment: 20 pages; v2: substantially revised version; v3: minor clarifications
and correction
Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations
We present a necessary and sufficient condition for a finite-dimensional
highest weight representation of the loop algebra to be irreducible. In
particular, for a highest weight representation with degenerate parameters of
the highest weight, we can explicitly determine whether it is irreducible or
not. We also present an algorithm for constructing finite-dimensional highest
weight representations with a given highest weight. We give a conjecture that
all the highest weight representations with the same highest weight can be
constructed by the algorithm. For some examples we show the conjecture
explicitly. The result should be useful in analyzing the spectra of integrable
lattice models related to roots of unity representations of quantum groups, in
particular, the spectral degeneracy of the XXZ spin chain at roots of unity
associated with the loop algebra.Comment: 32 pages with no figure; with corrections on the published versio
Weyl approach to representation theory of reflection equation algebra
The present paper deals with the representation theory of the reflection
equation algebra, connected with a Hecke type R-matrix. Up to some reasonable
additional conditions the R-matrix is arbitrary (not necessary originated from
quantum groups). We suggest a universal method of constructing finite
dimensional irreducible non-commutative representations in the framework of the
Weyl approach well known in the representation theory of classical Lie groups
and algebras. With this method a series of irreducible modules is constructed
which are parametrized by Young diagrams. The spectrum of central elements
s(k)=Tr_q(L^k) is calculated in the single-row and single-column
representations. A rule for the decomposition of the tensor product of modules
into the direct sum of irreducible components is also suggested.Comment: LaTeX2e file, 27 pages, no figure
McKay Graphs and Modular Representation Theory
Ordinary representation theory has been widely researched to the extent that there is a well-understood method for constructing the ordinary irreducible characters of a finite group. In parallel, John McKay showed how to associate to a finite group a graph constructed from the group\u27s irreducible representations. In this project, we prove a structure theorem for the McKay graphs of products of groups as well as develop formulas for the graphs of two infinite families of groups. We then study the modular representations of these families and give conjectures for a modular version of the McKay graphs
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