116 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
Pulmonary tuberculosis in a South African regional emergency centre: Can infection control be improved to lower the risk of nosocomial transmission?
Background. George Regional Hospital (GRH) is a 272-bed regional referral hospital for the Eden and Central Karoo districts, Western Cape Province, South Africa. The perception among emergency centre (EC) staff is that a high burden of tuberculosis (TB) is being diagnosed and that infection control procedures are currently lacking, leading to a high risk of nosocomial transmission.Objectives. To establish the burden of pulmonary TB (PTB) presenting to GRH via the EC and audit current infection prevention and control practices, to quantify the risk of transmission of PTB in the EC and to establish whether infection control measures are inadequate.Methods. An audit of infection control based on the Centers for Disease Control audit tool for TB, analysis of results, and implementation of new infection control measures including a new standard operating procedure based on a set of triage criteria.Results. Implementation of new triage criteria and a standard operating procedure led to the longest length of stay of a patient with suspected TB in the EC being reduced by 40% (from 142 hours to 84 hours). The average time between seeing a doctor and leaving the EC for patients with suspected TB was reduced by 20% (from 20 hours 40 minutes to 16 hours 45 minutes).Conclusion. Simple measures implemented in the EC led to a significant reduction in the time patients with suspected or confirmed TB spent in the EC. This should lead to a reduced risk of nosocomial transmission of TB to both staff and patients
Cellular structure of -Brauer algebras
In this paper we consider the -Brauer algebra over a commutative
noetherian domain. We first construct a new basis for -Brauer algebras, and
we then prove that it is a cell basis, and thus these algebras are cellular in
the sense of Graham and Lehrer. In particular, they are shown to be an iterated
inflation of Hecke algebras of type Moreover, when is a field of
arbitrary characteristic, we determine for which parameters the -Brauer
algebras are quasi-heredity. So the general theory of cellular algebras and
quasi-hereditary algebras applies to -Brauer algebras. As a consequence, we
can determine all irreducible representations of -Brauer algebras by linear
algebra methods
Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or
B--M--W) algebra by lifting bases for cell modules of
is given. By iterating this procedure, we produce cellular bases for B--M--W
algebras on which a large abelian subalgebra, generated by elements which
generalise the Jucys--Murphy elements from the representation theory of the
Iwahori--Hecke algebra of the symmetric group, acts triangularly. The
triangular action of this abelian subalgebra is used to provide explicit
criteria, in terms of the defining parameters and , for B--M--W algebras
to be semisimple. The aforementioned constructions provide generalisations, to
the algebras under consideration here, of certain results from the Specht
module theory of the Iwahori--Hecke algebra of the symmetric group
Enlarged symmetry algebras of spin chains, loop models, and S-matrices
The symmetry algebras of certain families of quantum spin chains are
considered in detail. The simplest examples possess m states per site (m\geq2),
with nearest-neighbor interactions with U(m) symmetry, under which the sites
transform alternately along the chain in the fundamental m and its conjugate
representation \bar{m}. We find that these spin chains, even with {\em
arbitrary} coefficients of these interactions, have a symmetry algebra A_m much
larger than U(m), which implies that the energy eigenstates fall into sectors
that for open chains (i.e., free boundary conditions) can be labeled by j=0, 1,
>..., L, for the 2L-site chain, such that the degeneracies of all eigenvalues
in the jth sector are generically the same and increase rapidly with j. For
large j, these degeneracies are much larger than those that would be expected
from the U(m) symmetry alone. The enlarged symmetry algebra A_m(2L) consists of
operators that commute in this space of states with the Temperley-Lieb algebra
that is generated by the set of nearest-neighbor interaction terms; A_m(2L) is
not a Yangian. There are similar results for supersymmetric chains with
gl(m+n|n) symmetry of nearest-neighbor interactions, and a richer
representation structure for closed chains (i.e., periodic boundary
conditions). The symmetries also apply to the loop models that can be obtained
from the spin chains in a spacetime or transfer matrix picture. In the loop
language, the symmetries arise because the loops cannot cross. We further
define tensor products of representations (for the open chains) by joining
chains end to end. The fusion rules for decomposing the tensor product of
representations labeled j_1 and j_2 take the same form as the Clebsch-Gordan
series for SU(2). This and other structures turn the symmetry algebra \cA_m
into a ribbon Hopf algebra, and we show that this is ``Morita equivalent'' to
the quantum group U_q(sl_2) for m=q+q^{-1}. The open-chain results are extended
to the cases |m|< 2 for which the algebras are no longer semisimple; these
possess continuum limits that are critical (conformal) field theories, or
massive perturbations thereof. Such models, for open and closed boundary
conditions, arise in connection with disordered fermions, percolation, and
polymers (self-avoiding walks), and certain non-linear sigma models, all in two
dimensions. A product operation is defined in a related way for the
Temperley-Lieb representations also, and the fusion rules for this are related
to those for A_m or U_q(sl_2) representations; this is useful for the continuum
limits also, as we discuss in a companion paper
Carter-Payne homomorphisms and Jantzen filtrations
We prove a q-analogue of the Carter-Payne theorem in the case where the
differences between the parts of the partitions are sufficiently large. We
identify a layer of the Jantzen filtration which contains the image of these
Carter-Payne homomorphisms and we show how these homomorphisms compose.Comment: 30 page
Representation-theoretic derivation of the Temperley-Lieb-Martin algebras
Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the
quotients of the Hecke algebra that admit only representations corresponding to
Young diagrams with a given maximum number of columns (or rows), are obtained,
making explicit use of the Hecke algebra representation theory. Similar
techniques are used to construct the algebras whose representations do not
contain rectangular subdiagrams of a given size.Comment: 12 pages, LaTeX, to appear in J. Phys.
On the Two-Point Correlation Function for the Invariant Spin One-Half Heisenberg Chain at Roots of Unity
Using tensor calculus we compute the two-point scalar operators
(TPSO), their averages on the ground-state give the two-point correlation
functions. The TPSOs are identified as elements of the Temperley-Lieb algebra
and a recurrence relation is given for them. We have not tempted to derive the
analytic expressions for the correlation functions in the general case but got
some partial results. For , all correlation functions are
(trivially) zero, for , they are related in the continuum to the
correlation functions of left-handed and right-handed Majorana fields in the
half plane coupled by the boundary condition. In the case , one
gets the correlation functions of Mittag's and Stephen's parafermions for the
three-state Potts model. A diagrammatic approach to compute correlation
functions is also presented.Comment: 19 pages, LaTeX, BONN-HE-93-3
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