110 research outputs found
The Weyl group of the fine grading of associated with tensor product of generalized Pauli matrices
We consider the fine grading of sl(n,\mb C) induced by tensor product of
generalized Pauli matrices in the paper. Based on the classification of maximal
diagonalizable subgroups of PGL(n,\mb C) by Havlicek, Patera and Pelantova,
we prove that any finite maximal diagonalizable subgroup of PGL(n,\mb C)
is a symplectic abelian group and its Weyl group, which describes the symmetry
of the fine grading induced by the action of , is just the isometry group of
the symplectic abelian group . For a finite symplectic abelian group, it is
also proved that its isometry group is always generated by the transvections
contained in it
A note on the Schur multiplier of a nilpotent Lie algebra
For a nilpotent Lie algebra of dimension and dim, we find
the upper bound dim, where denotes the
Schur multiplier of . In case the equality holds if and only if
, where is an abelian Lie algebra of dimension
and H(1) is the Heisenberg algebra of dimension 3.Comment: Paper in press in Comm. Algebra with small revision
Detection of Symmetry Protected Topological Phases in 1D
A topological phase is a phase of matter which cannot be characterized by a
local order parameter. It has been shown that gapped phases in 1D systems can
be completely characterized using tools related to projective representations
of the symmetry groups. We show how to determine the matrices of these
representations in a simple way in order to distinguish between different
phases directly. From these matrices we also point out how to derive several
different types of non-local order parameters for time reversal, inversion
symmetry and symmetry, as well as some more general cases
(some of which have been obtained before by other methods). Using these
concepts, the ordinary string order for the Haldane phase can be related to a
selection rule that changes at the critical point. We furthermore point out an
example of a more complicated internal symmetry for which the ordinary string
order cannot be applied.Comment: 12 pages, 9 Figure
The Drinfel'd Double and Twisting in Stringy Orbifold Theory
This paper exposes the fundamental role that the Drinfel'd double \dkg of
the group ring of a finite group and its twists \dbkg, \beta \in
Z^3(G,\uk) as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold
theories and their twistings.
The results pertain to three different aspects of the theory. First, we show
that --Frobenius algebras arising in global orbifold cohomology or K-theory
are most naturally defined as elements in the braided category of
\dkg--modules. Secondly, we obtain a geometric realization of the Drinfel'd
double as the global orbifold --theory of global quotient given by the
inertia variety of a point with a action on the one hand and more
stunningly a geometric realization of its representation ring in the braided
category sense as the full --theory of the stack . Finally, we show
how one can use the co-cycles above to twist a) the global orbifold
--theory of the inertia of a global quotient and more importantly b) the
stacky --theory of a global quotient . This corresponds to twistings
with a special type of 2--gerbe.Comment: 35 pages, no figure
Profinite completion of Grigorchuk's group is not finitely presented
In this paper we prove that the profinite completion of
the Grigorchuk group is not finitely presented as a profinite
group. We obtain this result by showing that H^2(\mathcal{\hat
G},\field{F}_2) is infinite dimensional. Also several results are proven about
the finite quotients including minimal
presentations and Schur Multipliers
D-branes and Discrete Torsion II
We derive D-brane gauge theories for C^3/Z_n x Z_n orbifolds with discrete
torsion and study the moduli space of a D-brane at a point. We show that, as
suggested in previous work, closed string moduli do not fully resolve the
singularity, but the resulting space -- containing n-1 conifold singularities
-- is somewhat surprising. Fractional branes also have unusual properties.
We also define an index which is the CFT analog of the intersection form in
geometric compactification, and use this to show that the elementary D6-brane
wrapped about T^6/Z_n x Z_n must have U(n) world-volume gauge symmetry.Comment: harvmac, 25 p
Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields
Let be a finite Galois extension of fields with abelian Galois group
. A self-dual normal basis for is a normal basis with the
additional property that for .
Bayer-Fluckiger and Lenstra have shown that when , then
admits a self-dual normal basis if and only if is odd. If is an
extension of finite fields and , then admits a self-dual normal
basis if and only if the exponent of is not divisible by . In this
paper we construct self-dual normal basis generators for finite extensions of
finite fields whenever they exist.
Now let be a finite extension of \Q_p, let be a finite abelian
Galois extension of odd degree and let \bo_L be the valuation ring of . We
define to be the unique fractional \bo_L-ideal with square equal to
the inverse different of . It is known that a self-dual integral normal
basis exists for if and only if is weakly ramified. Assuming
, we construct such bases whenever they exist
gauge symmetry and Tri-bimaximal mixing
We study an effective gauge theory whose gauge group is a semidirect product
with and being a
connected Lie group and a finite group, respectively. The semidirect product is
defined through a projective homomorphism (i.e., homomorphism up to
the center of ) from into . The (linear)
representation of is made from and a projective representation of
over . To be specific, we take as
and as . It is noticed that
the irreducible projective representations of are
three-dimensional in spite of its Abelian nature. We give a toy model on the
lepton mixing which illustrates the peculiar feature of such gauge symmetry. It
is shown that under a particular vacuum alignment the tri-bimaximal mixing
matrix is reproduced.Comment: 10 page
D-branes on Singularities: New Quivers from Old
In this paper we present simplifying techniques which allow one to compute
the quiver diagrams for various D-branes at (non-Abelian) orbifold
singularities with and without discrete torsion. The main idea behind the
construction is to take the orbifold of an orbifold. Many interesting discrete
groups fit into an exact sequence . As such, the orbifold
is easier to compute as and we present graphical rules which
allow fast computation given the quiver.Comment: 25 pages, 13 figures, LaTe
Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence
Consider the generalized iterated wreath product of symmetric groups. We give a complete description of the traversal
for the generalized iterated wreath product. We also prove an existence of a
bijection between the equivalence classes of ordinary irreducible
representations of the generalized iterated wreath product and orbits of labels
on certain rooted trees. We find a recursion for the number of these labels and
the degrees of irreducible representations of the generalized iterated wreath
product. Finally, we give rough upper bound estimates for fast Fourier
transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv
admin note: text overlap with arXiv:1409.060
- …