54 research outputs found
Two-Rowed Hecke Algebra Representations at Roots of Unity
In this paper, we initiate a study into the explicit construction of
irreducible representations of the Hecke algebra of type in
the non-generic case where is a root of unity. The approach is via the
Specht modules of which are irreducible in the generic case, and
possess a natural basis indexed by Young tableaux. The general framework in
which the irreducible non-generic -modules are to be constructed is set
up and, in particular, the full set of modules corresponding to two-part
partitions is described. Plentiful examples are given.Comment: LaTeX, 9 pages. Submitted for the Proceedings of the 4th
International Colloquium ``Quantum Groups and Integrable Systems,'' Prague,
22-24 June 199
Generalised -manifolds
We define new Riemannian structures on 7-manifolds by a differential form of
mixed degree which is the critical point of a (possibly constrained)
variational problem over a fixed cohomology class. The unconstrained critical
points generalise the notion of a manifold of holonomy , while the
constrained ones give rise to a new geometry without a classical counterpart.
We characterise these structures by the means of spinors and show the
integrability conditions to be equivalent to the supersymmetry equations on
spinors in supergravity theory of type IIA/B with bosonic background fields. In
particular, this geometry can be described by two linear metric connections
with skew torsion. Finally, we construct explicit examples by using the device
of T-duality.Comment: 27 pages. v2: references added. v3: wrong argument (Theorem 3.3) and
example (Section 4.1) removed, further examples added, notation simplified,
all comments appreciated. v4:computation of Ricci tensor corrected, various
minor changes, final version of the paper to appear in Comm. Math. Phy
On the Representation Theory of an Algebra of Braids and Ties
We consider the algebra introduced by F. Aicardi and J.
Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor
space representation for and show that this is faithful. We use
it to give a basis for and to classify its irreducible
representations.Comment: 24 pages. Final version. To appear in Journal of Algebraic
Combinatorics
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
We present a systematic approach to studying the geometric aspects of Vinberg
theta-representations. The main idea is to use the Borel-Weil construction for
representations of reductive groups as sections of homogeneous bundles on
homogeneous spaces, and then to study degeneracy loci of these vector bundles.
Our main technical tool is to use free resolutions as an "enhanced" version of
degeneracy loci formulas. We illustrate our approach on several examples and
show how they are connected to moduli spaces of Abelian varieties. To make the
article accessible to both algebraists and geometers, we also include
background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the
occasion of his 65th birthday; v2: fixed some typos and added reference
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
Ferromagnetic Ordering of Energy Levels for Symmetric Spin Chains
We consider the class of quantum spin chains with arbitrary
-invariant nearest neighbor interactions, sometimes
called for the quantum deformation of , for
. We derive sufficient conditions for the Hamiltonian to satisfy the
property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the
property that the ground state energy restricted to a fixed total spin subspace
is a decreasing function of the total spin. Using the Perron-Frobenius theorem,
we show sufficient conditions are positivity of all interactions in the dual
canonical basis of Lusztig. We characterize the cone of positive interactions,
showing that it is a simplicial cone consisting of all non-positive linear
combinations of "cascade operators," a special new basis of
intertwiners we define. We also state applications to
interacting particle processes.Comment: 23 page
Parabolically induced representations of graded Hecke algebras
We study the representation theory of graded Hecke algebras, starting from
scratch and focusing on representations that are obtained with induction from a
discrete series representation of a parabolic subalgebra. We determine all
intertwining operators between such parabolically induced representations, and
use them to parametrize the irreducible representations.Comment: In the second version several new results have been added to prove
some claims from the last page of the first version. In the third version the
introduction has been extended and we determine the global dimension of a
graded Hecke algebr
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