An Analysis of the Multiplicity Spaces in Branching of Symplectic Groups

Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components encode the multiplicities of irreducible representations of $Sp_{2n-2}$ in irreducible representations of $Sp_{2n}$. Our first theorem states that the map taking an element of $Sp_{2n}$ to its principal $n \times (n+1)$ submatrix induces an isomorphism of \B to a different branching algebra \B'. The algebra \B' encodes multiplicities of irreducible representations of $GL_{n-1}$ in certain irreducible representations of $GL_{n+1}$. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of $Sp_{2n}$ to $Sp_{2n-2}$ is canonically an irreducible module for the $n$-fold product of $SL_{2}$. In particular, this induces a canonical decomposition of the multiplicity spaces into one dimensional spaces, thereby resolving the multiplicities.Comment: 32 pages, revised abstract and introduction, and reorganized the structure of the pape

Algorithmic Verification of Continuous and Hybrid Systems

We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661

Bell Violations through Independent Bases Games

In a recent paper, Junge and Palazuelos presented two two-player games exhibiting interesting properties. In their first game, entangled players can perform notably better than classical players. The quantitative gap between the two cases is remarkably large, especially as a function of the number of inputs to the players. In their second game, entangled players can perform notably better than players that are restricted to using a maximally entangled state (of arbitrary dimension). This was the first game exhibiting such a behavior. The analysis of both games is heavily based on non-trivial results from Banach space theory and operator space theory. Here we present two games exhibiting a similar behavior, but with proofs that are arguably simpler, using elementary probabilistic techniques and standard quantum information arguments. Our games also give better quantitative bounds.Comment: Minor update