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### Crossover morita equivalences for blocks of the covering groups of the symmetric and alternating groups

### Goldstone boson counting in linear sigma models with chemical potential

We analyze the effects of finite chemical potential on spontaneous breaking
of internal symmetries within the class of relativistic field theories
described by the linear sigma model. Special attention is paid to the emergence
of ``abnormal'' Goldstone bosons with quadratic dispersion relation. We show
that their presence is tightly connected to nonzero density of the Noether
charges, and formulate a general counting rule. The general results are
demonstrated on an SU(3)xU(1) invariant model with an SU(3)-sextet scalar
field, which describes one of the color-superconducting phases of QCD.Comment: 10 pages, REVTeX4, 4 eps figures, v2: general discussion in Sec. IV
expanded and improved, references added, other minor corrections throughout
the tex

### Cliffordons

At higher energies the present complex quantum theory with its unitary group
might expand into a real quantum theory with an orthogonal group, broken by an
approximate $i$ operator at lower energies. Implementing this possibility
requires a real quantum double-valued statistics. A Clifford statistics,
representing a swap (12) by a difference $\gamma_1-\gamma_2$ of Clifford units,
is uniquely appropriate. Unlike the Maxwell-Boltzmann, Fermi-Dirac,
Bose-Einstein, and para- statistics, which are tensorial and single-valued, and
unlike anyons, which are confined to two dimensions, Clifford statistics are
multivalued and work for any dimensionality. Nayak and Wilczek proposed a
Clifford statistics for the fractional quantum Hall effect. We apply them to
toy quanta here. A complex-Clifford example has the energy spectrum of a system
of spin-1/2 particles in an external magnetic field. This supports the proposal
that the double-valued rotations --- spin --- seen at current energies might
arise from double-valued permutations --- swap --- to be seen at higher
energies. Another toy with real Clifford statistics illustrates how an
effective imaginary unit $i$ can arise naturally within a real quantum theory.Comment: 15 pages, no figures; original title ("Clifford statistics") changed;
to appear in J. Math. Phys., 42, 2001. Key words: Clifford statistics,
cliffordons, double-valued representations of permutation groups, spin, swap,
imaginary unit $i$, applications to quantum space-time and the Standard
Model. Some of these results were presented at the American Physical Society
Centennial Meeting, Atlanta, March 25, 199

### Randomized Rounding for the Largest Simplex Problem

The maximum volume $j$-simplex problem asks to compute the $j$-dimensional
simplex of maximum volume inside the convex hull of a given set of $n$ points
in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this
problem which achieves an approximation ratio of $e^{j/2 + o(j)}$. The problem
is known to be $\mathrm{NP}$-hard to approximate within a factor of $c^{j}$ for
some constant $c > 1$. Our algorithm also gives a factor $e^{j + o(j)}$
approximation for the problem of finding the principal $j\times j$ submatrix of
a rank $d$ positive semidefinite matrix with the largest determinant. We
achieve our approximation by rounding solutions to a generalization of the
$D$-optimal design problem, or, equivalently, the dual of an appropriate
smallest enclosing ellipsoid problem. Our arguments give a short and simple
proof of a restricted invertibility principle for determinants

### Optimizing local protocols implementing nonlocal quantum gates

We present a method of optimizing recently designed protocols for
implementing an arbitrary nonlocal unitary gate acting on a bipartite system.
These protocols use only local operations and classical communication with the
assistance of entanglement, and are deterministic while also being "one-shot",
in that they use only one copy of an entangled resource state. The optimization
is in the sense of minimizing the amount of entanglement used, and it is often
the case that less entanglement is needed than with an alternative protocol
using two-way teleportation.Comment: 11 pages, 1 figure. This is a companion paper to arXiv:1001.546

### Decomposition of fractional quantum Hall states: New symmetries and approximations

We provide a detailed description of a new symmetry structure of the monomial
(Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall
states first obtained in Ref. 1, which we now extend to spin-singlet states. We
show that the Haldane-Rezayi spin-singlet state can be obtained without exact
diagonalization through a differential equation method that we conjecture to be
generic to other FQH model states. The symmetry rules in Ref. 1 as well as the
ones we obtain for the spin singlet states allow us to build approximations of
FQH states that exhibit increasing overlap with the exact state (as a function
of system size). We show that these overlaps reach unity in the thermodynamic
limit even though our approximation omits more than half of the Hilbert space.
We show that the product rule is valid for any FQH state which can be written
as an expectation value of parafermionic operators.Comment: 22 pages, 8 figure

### $q$-Trinomial identities

We obtain connection coefficients between $q$-binomial and $q$-trinomial
coefficients. Using these, one can transform $q$-binomial identities into a
$q$-trinomial identities and back again. To demonstrate the usefulness of this
procedure we rederive some known trinomial identities related to partition
theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which
have recently arisen in their study of the $\phi_{2,1}$ and $\phi_{1,5}$
perturbations of minimal conformal field theory.Comment: 21 pages, AMSLate

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