7,551 research outputs found
On Weyl modules over affine Lie algebras in prime characteristic
We construct a family of homomorphisms between Weyl modules for affine Lie
algebras in characteristic p, which supports our conjecture on the strong
linkage principle in this context. We also exhibit a large class of reducible
Weyl modules beyond level one, for p not necessarily small.Comment: 30 pages, 1 figure, 3 tables; v4: clarifying the statement of
Conjecture 6.1 regarding the strong linkage principl
On Fourier frame of absolutely continuous measures
Let be a compactly supported absolutely continuous probability measure
on , we show that admits Fourier frames if and only if its
Radon-Nikodym derivative is upper and lower bounded almost everywhere on its
support. As a consequence, we prove that if an equal weight absolutely
continuous self-similar measure on admits Fourier frame, then the
measure must be a characteristic function of self-similar tile. In particular,
this shows for almost everywhere , the -Bernoulli
convolutions cannot admit Fourier frames
Some reductions of the spectral set conjecture to integers
The spectral set conjecture, also known as the Fuglede conjecture, asserts
that every bounded spectral set is a tile and vice versa. While this conjecture
remains open on , there are many results in the literature that
discuss the relations among various forms of the Fuglede conjecture on
, and and also the seemingly
stronger universal tiling (spectrum) conjectures on the respective groups. In
this paper, we clarify the equivalences between these statements in dimension
one. In addition, we show that if the Fuglede conjecture on is
true, then every spectral set with rational measure must have a rational
spectrum. We then investigate the Coven-Meyerowitz property for finite sets of
integers, introduced in \cite{CoMe99}, and we show that if the spectral sets
and the tiles in satisfy the Coven-Meyerowitz property, then both
sides of the Fuglede conjecture on are true
- …