36 research outputs found
Unitary equivalence to a complex symmetric matrix: an algorithm
We present a necessary and sufficient condition for a 3 by 3 matrix to be
unitarily equivalent to a symmetric matrix with complex entries, and an
algorithm whereby an arbitrary 3 by 3 matrix can be tested. This test
generalizes to a necessary and sufficient condition that applies to almost
every n by n matrix. The test is constructive in that it explicitly exhibits
the unitary equivalence to a complex symmetric matrix.Comment: 10 page
Representation theory in chiral conformal field theory: from fields to observables
This article develops new techniques for understanding the relationship
between the three different mathematical formulations of two-dimensional chiral
conformal field theory: conformal nets (axiomatizing local observables), vertex
operator algebras (axiomatizing fields), and Segal CFTs. It builds upon
previous work which introduced a geometric interpolation procedure for
constructing conformal nets from VOAs via Segal CFT, simultaneously relating
all three frameworks. In this article, we extend this construction to study the
relationship between the representation theory of conformal nets and the
representation theory of vertex operator algebras. We define a correspondence
between representations in the two contexts, and show how to construct
representations of conformal nets from VOAs. We also show that this
correspondence is rich enough to relate the respective 'fusion product'
theories for conformal nets and VOAs, by constructing local intertwiners (in
the sense of conformal nets) from intertwining operators (in the sense of
VOAs). We use these techniques to show that all WZW conformal nets can be
constructed using our geometric interpolation procedure.Comment: 79 pages. v2: minor revisions and update
Categorified trace for module tensor categories over braided tensor categories
Given a braided pivotal category and a pivotal module tensor
category , we define a functor , called the associated categorified trace. By a result of
Bezrukavnikov, Finkelberg and Ostrik, the functor
comes equipped with natural isomorphisms , which we call the
traciators. This situation lends itself to a diagramatic calculus of `strings
on cylinders', where the traciator corresponds to wrapping a string around the
back of a cylinder. We show that in fact has a much
richer graphical calculus in which the tubes are allowed to branch and braid.
Given algebra objects and , we prove that
and are again algebra objects.
Moreover, provided certain mild assumptions are satisfied,
and are
semisimple whenever and are semisimple.Comment: 49 pages, many figure
Projections and idempotents with fixed diagonal and the homotopy problem for unit tight frames
We investigate the topological and metric structure of the set of idempotent
operators and projections which have prescribed diagonal entries with respect
to a fixed orthonormal basis of a Hilbert space. As an application, we settle
some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of
the set of unit-norm tight frames.Comment: New title and introductio
Unitary Equivalence to a Complex Symmetric Matrix: Low Dimensions
A matrix T∈Mn(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above