# Conjectures and open problems in representation theory [closed]

Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?

• Do you want representation theory problems or enumerative geometry problems? – Ben McKay Jan 29 at 12:50
• Dear Ben, one of them or both will be ok. Since I am interesting in both fields and they have some connections to each other. Thank you so much for your advice! – Khanh Nguyen Jan 29 at 12:55
• I guess that for representation theory you want to be more specific since different people mean different things when they say "Representation Theory". For some open problems in the Representation theory of finite dimensional algebras and quivers, see e.g. math.uni-bonn.de/people/schroer/fd-problems.html. – Julian Kuelshammer Jan 29 at 13:10
• The work of J.M. Landsberg on matrix multiplication draws from representation theory and complex algebraic geometry, but maybe not much enumerative geometry, and is driven by conjectures from computer science about the speed with which computers can multiply matrices. – Ben McKay Jan 29 at 14:37
• I'm kind of amazed this question is still open, given how strict math overflow usually is. Regardless of what happens, just a bit of advice Khanh: this question is way too broad. Even if your question was faithful to the title, it would be far too broad. The only real answer is yes, there are many conjectures and open problems in representation theory. The more thought you put into your question, the better answers you will get. – Andy Sanders Jan 29 at 18:29

There are many open, and seemingly deep, conjectures in modular representation theory (or block theory) in connection with enumerating representation-theoretic invariants: a start of a list might be : Brauer's $$k(B)$$-problem, the Alperin-McKay Conjecture, the Alperin Weight Conjecture, Dade's conjectures, Isaacs-Navarro conjecture. Gabriel Navarro has several recent survey papers discussing these and other conjectures.