Codes and Expansions (CodEx) Seminar
Clay Shonkwiler (Colorado State University)
A Lie Algebraic Perspective on Frame Theory
The space of Hermitian matrices with fixed spectrum is a natural object from the perspective of Lie theory; precisely, it has a natural interpretation as a coadjoint orbit of the unitary group. In turn, coadjoint orbits are also natural from the perspective of symplectic geometry, and this interpretation leads to a remarkably simple proof of the frame homotopy conjecture for complex frames, which says that the space of unit-norm tight frames is path-connected.
For a long time, I found it frustrating that I didn't know of a similar interpretation of the space of symmetric matrices with fixed spectrum. Inevitably this turned out to be due to my own ignorance, as I realized when I learned about the Cartan decomposition of semisimple Lie algebras, which is a sort of generalization of the polar decomposition.
The Cartan decomposition (and related ideas like the Kostant convexity theorem and Terng's isoparametric submanifolds) provides a unifying perspective on real, complex, and quaternionic frames which I will describe in reasonably concrete terms. As a particular application, I will discuss joint work with Tom Needham (Florida State University) in which we prove the analog of the frame homotopy conjecture for quaternionic frames.