1. CMB Online first
|The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames|
In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.
Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearlet
Categories:43A65, 42C40, 42C15
2. CMB 2011 (vol 56 pp. 13)
|Ordering the Representations of $S_n$ Using the Interchange Process|
Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec$ on the irreducible representations of $S_n$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the ``Aldous order'' completely is a generalized question. We show a few additional entries for this order.
Keywords:Aldous' conjecture, interchange process, symmetric group, representations
Categories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35