406 research outputs found
Universality theorems for configuration spaces of planar linkages
We prove realizability theorems for vector-valued polynomial mappings,
real-algebraic sets and compact smooth manifolds by moduli spaces of planar
linkages. We also establish a relation between universality theorems for moduli
spaces of mechanical linkages and projective arrangements.Comment: 45 pages, 15 figures. See also
http://www.math.utah.edu/~kapovich/eprints.htm
On representation varieties of 3-manifold groups
We prove universality theorems ("Murphy's Laws") for representation schemes
of fundamental groups of closed 3-dimensional manifolds. We show that germs of
SL(2,C)-representation schemes of such groups are essentially the same as germs
of schemes of over rational numbers.Comment: 28 page
The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)
In Part 1 of this paper we construct a spectral sequence converging to the
relative Lie algebra cohomology associated to the action of any subgroup of
the symplectic group on the polynomial Fock model of the Weil representation,
see Section 7. These relative Lie algebra cohomology groups are of interest
because they map to the cohomology of suitable arithmetic quotients of the
symmetric space of . We apply this spectral sequence to the case in Sections 8, 9, and 10 to compute the relative Lie
algebra cohomology groups . Here is Minkowski space and
is the subspace of consisting of all products of
polynomials with the Gaussian. In Part 2 of this paper we compute the
cohomology groups using spectral theory and representation theory. In Part 3 of this paper
we compute the maps between the polynomial Fock and cohomology groups
induced by the inclusions .Comment: 64 pages, 5 figure
Saturation and Irredundancy for Spin(8)
We explicitly calculate the triangle inequalities for the group PSO(8).
Therefore we explicitly solve the eigenvalues of sum problem for this group
(equivalently describing the side-lengths of geodesic triangles in the
corresponding symmetric space for the Weyl chamber-valued metric). We then
apply some computer programs to verify two basic questions/conjectures. First,
we verify that the above system of inequalities is irredundant. Then, we verify
the ``saturation conjecture'' for the decomposition of tensor products of
finite-dimensional irreducible representations of Spin(8). Namely, we show that
for any triple of dominant weights a, b, c such that a+b+c is in the root
lattice, and any positive integer N, the tensor product of the irreducible
representations V(a) and V(b) contains V(c) if and only if the tensor product
of V(Na) and V(Nb) contains V(Nc).Comment: 22 pages, 2 figure
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