188 research outputs found
Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials
The concept of cyclic tridiagonal pairs is introduced, and explicit examples
are given. For a fairly general class of cyclic tridiagonal pairs with
cyclicity N, we associate a pair of `divided polynomials'. The properties of
this pair generalize the ones of tridiagonal pairs of Racah type. The algebra
generated by the pair of divided polynomials is identified as a higher-order
generalization of the Onsager algebra. It can be viewed as a subalgebra of the
q-Onsager algebra for a proper specialization at q the primitive 2Nth root of
unity. Orthogonal polynomials beyond the Leonard duality are revisited in light
of this framework. In particular, certain second-order Dunkl shift operators
provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of
irreducibility is added; v3: version for Linear Algebra and its Applications,
one assumption added in Appendix about eq. (A.2
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Positive maps and trace polynomials from the symmetric group
With techniques borrowed from quantum information theory, we develop a method
to systematically obtain operator inequalities and identities in several matrix
variables. These take the form of trace polynomials: polynomial-like
expressions that involve matrix monomials
and their traces . Our
method rests on translating the action of the symmetric group on tensor product
spaces into that of matrix multiplication. As a result, we extend the polarized
Cayley-Hamilton identity to an operator inequality on the positive cone,
characterize the set of multilinear equivariant positive maps in terms of
Werner state witnesses, and construct permutation polynomials and tensor
polynomial identities on tensor product spaces. We give connections to concepts
in quantum information theory and invariant theory.Comment: 28 pages, 3 figures, 2 tables. Extensively rewritten: asymmetric
maps, proof for Motzkin matrix polynomial, and connections to QIT added.
Comments welcome
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