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What are open problems in representation theory ?

What are the sources (books/papers/sites) discussing this ?

Any kinds of problems/questions are welcome - big/small, vague/concrete. Some estimation of difficulty and importance, as well as, small description, prerequisites and relevant references, ... are welcome.

To the best of my knowlegde there are NO good lists of representation theory problems on the web. E.g. the sites below contain lots of unsolved problem in other areas, but not in representation theory:




MO questions also discuss other fields, but not representation theory:

What are the big problems in probability theory?

What are some open problems in algebraic geometry?

What are some open problems in toric varieties?

More open problems

Open problems with monetary rewards

Open problems in Euclidean geometry?

Open Questions in Riemannian Geometry

What are some of the big open problems in 3-manifold theory?

Open problems in continued fractions theory

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Can you precise what you mean by 'representation theory'? It's a huge, disparate, area of research, and narrowing down a little bit the question would help for better answers. –  Joël May 7 '12 at 13:18
One of them will be "Artin's Holomorphy Conjecture". Here is a report on current progress. math.tifr.res.in/~dprasad/artin.pdf –  i707107 May 15 '12 at 8:16
i707107 Agree. May be you can write it as an answer (hopefully adding some comments). The whole Langlands program is one the main problems in RT. –  Alexander Chervov May 15 '12 at 9:53
arxiv.org/abs/1210.2225 We state Brou´e’s Abelian Defect Group Conjecture[14, Chapter 6.3.3]. Conjecture 1.0.1 (Brou´e). Let G be a finite group and P an abelian p-subgroup. Let b be a block idempotent of OG with defect group P and Brauer correspondent c in NG(P). Then OGb and ONG(P)c are derived equivalent. –  Alexander Chervov Oct 10 '12 at 6:32
If G is solvable, then Gluck's conjecture is that √[G:Fit(G)] ≤ b(G), and this has been verified for solvable G such that G/Φ(G) has an Abelian Sylow 2-subgroup or G such that G″ = 1. (If G is non-abelian simple, then Fit(G)=1, and so the bound cannot hold). mathoverflow.net/questions/21071/… arxiv.org/abs/1009.5434 Gluck’s conjecture has been verified for groups of odd order, solvable groups whose orders are not divisible by 3 (see [15]), and solvable groups with abelian Sylow 2-subgroups –  Alexander Chervov Oct 10 '12 at 10:06

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