Open problems/questions in representation theory and around?

Question

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.

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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

Answer 1

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.

Answer 2

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).