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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 knowledge, 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:

http://en.wikipedia.org/wiki/Unsolved_problems_in_mathematics

http://garden.irmacs.sfu.ca/

http://maven.smith.edu/~orourke/TOPP/

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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Alexander Chervov
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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 07 '12 at 13:18
  • @Joel 1) arXiv has math.RT, so let me say I mean the same. 2) thank you for the comment, however it is not clear for me why narrowing would be helpful. The other MO questions cited above seems to be of the same "narrowness", and as for me it is quite appropriate. – Alexander Chervov May 09 '12 at 17:45
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    One of them will be "Artin's Holomorphy Conjecture". Here is a report on current progress. http://www.math.tifr.res.in/~dprasad/artin.pdf – Sungjin Kim May 15 '12 at 08:16
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    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 09:53
  • http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=5001752 Some questions about p-groups A Mann – Alexander Chervov Sep 09 '12 at 16:40
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    http://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 06:32
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    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). http://mathoverflow.net/questions/21071/can-we-bound-degrees-of-complex-irreps-in-terms-of-the-average-conjugacy-class-si/21091#21091 http://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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    Brauer height-zero conjecture http://www.encyclopediaofmath.org/index.php/Brauer_height-zero_conjecture http://projecteuclid.org/euclid.pja/1330958574 – Alexander Chervov Oct 10 '12 at 10:36
  • B is an HZ-block if and only if D is abelian – Alexander Chervov Oct 10 '12 at 10:39
  • http://www.dmi.unisa.it/ischia2010/talks/Navarro.pdf Recent advancs on BHZ http://personalpages.manchester.ac.uk/staff/Charles.Eaton/emfinal.pdf Extending Brauer's Height Zero Conjecture to blocks with nonabelian defect groups Charles W. Eatonand Alexander Moreto We propose a generalization of Brauer's Height Zero Conjecture that considers positive heights. We give strong evidence supporting one half of the generalization and obtain some partial results regarding the other half. Mentions Robinson conjectures – Alexander Chervov Oct 10 '12 at 11:04
  • http://www.uv.es/amoquin/30.pdf A proof of Huppert’s - conjecture for nonsolvable groups Huppert’s - conjectures say the following: (i) There is a real-valued function f such that |(G)| <= f((G)) for every finite group G. (ii) If G is solvable, then |(G)| <= 2(G). Until – Alexander Chervov Oct 10 '12 at 11:06
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    Main Problems in the Representation Theory of Finite Groups http://www.ehu.es/emsweekend/ficheros/GNAVARRO.pdf – Alexander Chervov Oct 10 '12 at 11:58
  • http://www.math.ku.dk/english/research/conferences/group.actions2011/problem.session.seattle96-maybe.pdf/ Problems in rep. theory and H^* – Alexander Chervov Oct 10 '12 at 12:39
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    Computing the dimensions of the irreducible representations in positive characteristic (or, more generally, their decomposition numbers or Brauer characters). Similarly, determining the characters of the irreducible representations of the finite groups of Lie type in positive characteristic. Until recently the James and Lusztig conjectures, respectively, gave answers to these questions in some cases, but in 2013 Williamson gave counterexamples to both of these conjectures. – Andrew Dec 15 '13 at 14:50
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    I am an outsider to these topics, but I tend to view several open problems in algebraic combinatorics as also being questions of representation theory, and this includes all sorts of "positivity" conjectures... – Suvrit Apr 10 '15 at 13:15
  • @AlexanderChervov Re Jo"el's comment, for instance just in the Langlands program, which deals with automorphic and Galois representations (and connections to other things) there are huge swaths of open problems. One difference is that this post asks for big & small questions, not just big questions. Even so, the other questions seem more focused (e.g., reread the algebraic geometry one). – Kimball May 05 '15 at 02:10
  • McKay conjecture: http://arizona.openrepository.com/arizona/bitstream/10150/621714/1/nt19.pdf IRREDUCIBLE REPRESENTATIONS OF ODD DEGREE GABRIEL NAVARRO AND PHAM HUU TIEP Abstract. McKay’s original observation on characters of odd degrees of finite groups is reduced to almost simple groups.
    1. Introduction

    In 1971 John McKay made an observation that changed the course of the Representation Theory of Finite Groups: “In this note we observe that the number m2(G) of inequivalent irreducible complex representations of odd degree of a finite group G is a power of two for many groups G” [M].

     – Alexander Chervov Jun 22 '17 at 14:41
  • What McKay was noticing was a particular (but fundamental) case of what later has become known as the McKay Conjecture: if G is a finite group, p is any prime and P ∈ Sylp (G), then mp(G) = mp(NG(P)), where now mp(G) is the number of inequivalent irreducible complex representations of degree not divisible by p of a finite group G. If true, this is an astonishing fact, since global information of a finite group G is going to be encoded in a local small subgroup of G, the Sylow normalizer NG(P). – Alexander Chervov Jun 22 '17 at 14:41
  • The Major Problems in Group Representation Theory David A. Craven 18th November 2009 http://web.mat.bham.ac.uk/D.A.Craven/docs/seminars/181109transcript.pdf In group representation theory, there are many unsolved conjectures, most of which try to understand the involved relationship between a finite group G and the normalizers of p-subgroups NG(Q), where Q is often the Sylow p-subgroup, but will frequently be smaller than the Sylow. – Alexander Chervov Jun 22 '17 at 14:45
  • Alperin’s fusion theorem proves that the conjugacy of elements in a given Sylow psubgroup is governed by the normalizers of non-trivial p-subgroups. In the abelian case, an old theorem of Burnside proves that NG(P) controls fusion in P, where P ∈ Sylp (G), so we should expect the structure there to be fairly transparent compared to other groups – Alexander Chervov Jun 22 '17 at 14:45
  • In this lecture we will see a variety of conjectures linking the representation theories of finite groups and normalizers of p-subgroups. The first of these were numerical, linking the (complex) character degrees of G with NG(P), and then became more structural. The most structural of them all – Brou´e’s conjecture – details the precise nature of the control of the representation theory of G be NG(P) in the case where P is abelian. A common generalization of Alperin’s weight conjecture and Brou´e’s conjecture would be a very interesting development. – Alexander Chervov Jun 22 '17 at 14:45
  • Some open problems here: MODULAR REPRESENTATIONS, OLD AND NEW BHAMA SRINIVASAN To the memory of Harish-Chandra http://web.mat.bham.ac.uk/D.A.Craven/docs/seminars/181109transcript.pdf – Alexander Chervov Jun 22 '17 at 14:50
  • Problem session: http://www.math.ku.dk/english/research/conferences/2011/group.actions2011/problem.session.seattle96-maybe.pdf problem.session.seattle96-maybe.pdf – Alexander Chervov Jun 22 '17 at 14:54
  • Open problems on Cherednik algebras, symplectic reflection algebras, and related topics. by P. Etingof and V. Ginzburg. http://icms.org.uk/downloads/cherpdf/openpro.pdf – Alexander Chervov Jun 22 '17 at 14:59
  • http://www.math.rwth-aachen.de/~Gerhard.Hiss/Presentations/Bristol08.pdf Gerhard.Hiss PROBLEMS IN THE REPRESENTATION THEORY OF FINITE GROUPS OF LIE TYPE Gerhard Hiss Lehrstuhl D für Mathematik RWTH Aachen University Bristol University, Algebra and Geometry Seminar 10. December 2008 – Alexander Chervov Jun 22 '17 at 15:02
  • Geordie Williamson Challenges in the representation theory of finite groups 2016 http://www.maths.usyd.edu.au/u/geordie/Dusseldorf.pdf Geordie Williamson Max Planck Institute, Bonn D¨usseldorf colloquium in mathematics, July 2016. – Alexander Chervov Jun 22 '17 at 15:04
  • OLD : J. Symbolic Computation (1990) 9, 571-582 Some Problems in Computational Representation Theory GERHARD O. MICHLER https://core.ac.uk/download/pdf/82016689.pdf – Alexander Chervov Jun 22 '17 at 15:07
  • Pham Huu Tiep Acta Mathematica Vietnamica March 2014, Volume 39, Issue 1, pp 87–109 Representation of finite groups: conjectures, reductions, and applications https://link.springer.com/article/10.1007/s40306-013-0043-y – Alexander Chervov Jun 22 '17 at 18:18

2 Answers2

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These are some big problems I know about:

  • $e$-positivity of Stanley's chromatic-symmetric functions for incomparability graphs obtained from $3+1$-avoiding posets. Shareshian and Wachs have some recent results related to this that connects these polynomials to representation theory, and they refine this conjecture with a $q$-parameter. Note that Schur-positivity is already known, and that the $e_\lambda$ expands positively into Schurs. It all boils down to finding a partition-valued combinatorial statistic on certain acyclic orientations.

  • Give a combinatorial description of the Kronecker coefficients.

  • Find a combinatorial interpretation of the Littlewood-Richardson coefficients for the Jack polynomials, $J_{\mu} J_{\nu} = \sum_\lambda c^\lambda_{\mu\nu}(\alpha) J_\lambda$. It is conjectured (but not proved) that $c^\lambda_{\mu\nu}(\alpha)$ is a polynomial in $\alpha$ with non-negative integer coefficients. (Here, one needs to be a bit careful with which normalization one chooses).

  • Find a combinatorial description of the multiplicative structure constants for the Schubert polynomials (analogue of the Littlewood-Richardson coefficients in the Schur polynomial case).

  • The different variants of the shuffle conjecture. UPDATE: There is a recent proof on arxiv. There are still some generalizations of this that remains unproved.

  • Prove that LLT polynomials have positive Schur expansion. This is related to the shuffle conjecture and the $qt$-Kostka polynomials, see N. Loehr's notes.

  • Give a combinatorial formula for the non-homogeneous symmetric Jack polynomials (similar to the Knop-Sahi formula for the ordinary Jack polynomials).

  • Give combinatorial descriptions of structure constants that appear in plethystic substitutions, $a^\nu_{\lambda\mu} = \langle s_\lambda[s_\mu], s_\nu \rangle$

  • Find a combinatorial description of the $qt$-Kostka polynomials.

  • Find a combinatorial description of the polynomials $c^\nu_{\lambda\mu}(t)$ in $$s_\lambda(t) \cdot P_\mu(x;t) = \sum_\nu c^\nu_{\lambda\mu}(t) P_\mu(x;t)$$ where $P_\lambda(x;t)$ is the Hall-Littlewood polynomial.

  • Prove (or disprive) that the map $k \mapsto K_{k\lambda,k\mu}$ is a polynomial with non-negative coefficients, where $K_{\lambda\mu}$ is the Kostka coefficient. Or in more general, same question for $k \mapsto c^{k\nu}_{k\lambda,k\mu}$ for the Littlewood-Richardson coefficients. This question goes back to King, Tollu and Toumazet. Note that a proof of the latter would imply the famous saturation conjecture proved by A. Knutson and T. Tao. The conjecture is stated in Rassart's paper, here, where the polynomiality property is proved.
    I have looked on the special case concerning Koskta coefficients, and this lead me to ask this somewhat related question. The negative answer to this question hints that the positivity conjecture might possibly be false, but a minimal counter example lives in a very high dimension, out of reach by exhaustive computer search. This concerns Ehrhart polynomials for non-integral polytopes, and very little is know about this case. One can show that such Ehrhart (quasi)polynomials have properties that Ehrhart polynomials corresponding to integral polytopes lack.

Whenever one seeks a combinatorial description, it is understood that the polynomial or number in question is conjectured to be a polynomial with non-negative integer coefficients.

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Per Alexandersson
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    The second item is Conjecture 8.3 of http://www-math.mit.edu/~rstan/pubs/pubfiles/73.pdf. – Sam Hopkins May 04 '15 at 21:08
  • Could you please expand the last bullet point with some more precise details / links to a formulation of this conjecture. Sounds very interesting! – Suvrit Aug 30 '15 at 02:26
  • @Suvrit Added some more details. Now when you mention it, I might perhaps write down a note on where one might want to seek for a possible counter-example... – Per Alexandersson Aug 30 '15 at 02:59
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There are many open problems on modular representation theory of finite groups. There is a well-known list of problems of R. Brauer which date to around 1960, about ordinary and modular representations of finite groups. There are other major conjectures ( Broue's Abelian defect group conjecture has already been mentioned in comments). There are Alperin's Weight conjecture, Dade's Projective Conjecture, the Isaacs-Navarro conjecture, and several related conjectures. This is only the tip of the iceberg (and only in one corner of representation theory).

Geoff Robinson
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