[Geom.,Alg.&Phys.] Can we describe canonical bases as the solutions of optimization problems?
ABSTRACT
Canonical bases (e.g. of representations of quantum groups, of quantum groups themselves, of Hecke algebras, and of their representations) are mysterious and fundamental objects. Their definition typically involves the quantum parameter q in a critical way, via some form of "bar invariance" or "self-duality". Typically, their definition is elementary and combinatorial, and yet they have many deep properties — e.g. positivity and connections to representation theory or geometry. In the story I would like to tell one inverts the picture: one asks for the “simplest” or “smallest” basis which has positive structure constants. Remarkably, in many small examples, there is a unique solution and one recovers the canonical basis (often after specialising q->1) in an elementary way. This is typically not the case in large examples though, and the failure is tied to subtle geometry / representation theoretic behaviour. I would also like to advertise this problem as a place where Reinforcement Learning methods might find an interesting application (this was the original motivation for this project, but has not yet been pursued). This is joint work with Tom Goertzen (https://arxiv.org/abs/2604.18894). (Zoom Meeting ID: 850 9457 8742, Passcode: 324001, link: https://kias-re-kr.zoom.us/j/85094578742?pwd=8edh0bmChHB3jKphDdiJNRxKZOLIDq.1)(https://sites.google.com/view/gapkias)
