22,529 research outputs found
Lowest-energy states in parity-transformation eigenspaces of SO(N) spin chain
We expand the symmetry of the open finite-size SO(N) symmetric spin chain to
O(N). We partition its space of states into the eigenspaces of the parity
transformations in the flavor space, generating the subgroup
. It is proven that the lowest-energy states in these
eigenspaces are nondegenerate and assemble in antisymmetric tensors or
pseudotensors. At the valence-bond solid point, they constitute the
-fold degenerate ground state with fully broken parity-transformation
symmetry.Comment: 11 pages, final versio
Dependence of Supertropical Eigenspaces
We study the pathology that causes tropical eigenspaces of distinct
supertropical eigenvalues of a nonsingular matrix , to be dependent. We show
that in lower dimensions the eigenvectors of distinct eigenvalues are
independent, as desired. The index set that differentiates between subsequent
essential monomials of the characteristic polynomial, yields an eigenvalue
, and corresponds to the columns of the eigenmatrix from
which the eigenvectors are taken. We ascertain the cause for failure in higher
dimensions, and prove that independence of the eigenvectors is recovered in
case a certain "difference criterion" holds, defined in terms of disjoint
differences between index sets of subsequent coefficients. We conclude by
considering the eigenvectors of the matrix A^\nabla : = \det(A)^{-1}\adj(A)
and the connection of the independence question to generalized eigenvectors.Comment: The first author is sported by the French Chateaubriand grant and
INRIA postdoctoral fellowshi
Hua operators, Poisson transform and relative discrete series on line bundle over bounded symmetric domains
Let be a bounded symmetric domain and its Shilov
boundary. We consider the action of on sections of a homogeneous line
bundle over and the corresponding eigenspaces of -invariant
differential operators. The Poisson transform maps hyperfunctions on the to
the eigenspaces. We characterize the image in terms of twisted Hua operators.
For some special parameters the Poisson transform is of Szeg\"o type mapping
into the relative discrete series; we compute the corresponding elements in the
discrete series
Instanton Floer homology and the Alexander polynomial
The instanton Floer homology of a knot in the three-sphere is a vector space
with a canonical mod 2 grading. It carries a distinguished endomorphism of even
degree,arising from the 2-dimensional homology class represented by a Seifert
surface. The Floer homology decomposes as a direct sum of the generalized
eigenspaces of this endomorphism. We show that the Euler characteristics of
these generalized eigenspaces are the coefficients of the Alexander polynomial
of the knot. Among other applications, we deduce that instanton homology
detects fibered knots.Comment: 25 pages, 6 figures. Revised version, correcting errors concerning
mod 2 gradings in the skein sequenc
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