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IntroSurvey of Representation Theory

Abstract

There could be thousands of Introductions/Surveys of representation theory, given that it is an enormous field. This is just one of them, quite personal and informal. It has an increasing level of difficulty; the first part is intended for final year undergrads. We explain some basics of representation theory, notably Schur–Weyl duality and representations of the symmetric group. We then do the quantum version, introduce Kazhdan–Lusztig theory, quantum groups and their categorical versions. We then proceed to a survey of some recent advances in modular representation theory. We finish with 20 open problems and a song of despair.

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Notes

  1. “Cahier I”, page I.100.

  2. Discussing mathematics curriculum reform at Princeton University (1910), as quoted in Abraham P. Hillman, Gerald L. Alexanderson, “Abstract Algebra: A First Undergraduate Course” (1994).

  3. “The World of Mathematics” (1956) p.1534.

  4. Quoted in “Out of the Mouths of Mathematicians” (1993) by R. Schmalz.

  5. George Whitelaw Mackey “Group Theory and its Significance”, Proceedings American Philosophical Society (1973), 117, No. 5, 380.

  6. Joseph A. Gallian “Contemporary Abstract Algebra” (1994) p. 55.

  7. This fits well with the fact that in classical mechanics there is a set describing the states of a system, while in quantum physics the states are vector lines in a Hilbert space H, thus in the projective Hilbert space P(H).

  8. This “ignoring” business bothers me. I would like some slight twist in the analogy for the numbers to really fit. I find intriguing that \((q-1)^r\) is the number of points in a maximal torus T, so one could replace \(G({\mathbb {F}}_q)\) by \(G({\mathbb {F}}_q)/T({\mathbb {F}}_q)\) or even by \(G({\mathbb {F}}_q)/B({\mathbb {F}}_q)\) (with B a Borel) for the numbers to have the correct limit, but this only give homogeneous spaces and not groups, and part of the magic of this analogy is that it also extends to the level of representations (see Sect. 4.3). One may counterargue this by saying that each homogeneous space has a corresponding groupoid (à la Connes), and maybe the groupoid representation theory of \(G({\mathbb {F}}_q)/B({\mathbb {F}}_q)\) is similar to the group representation theory of the whole group. Another option, staying in the realm of groups, is that maybe there is something like an \({\mathbb {F}}_q\)-version (instead of the usual \({\mathbb {Z}}\) version) of the affine Weyl group of cardinality \((q-1)^r\times {\mathrm {card}}(W), \) that would make the numbers agree.

  9. This is a particular example (for \({\mathrm{GL}}_n\) and \(S_n\)) of a general construction that associates to a connected reductive algebraic group over k its Weyl group \(W:=N_G(T)/T\), where T is a maximal torus.

  10. For a heuristic argument, it is quite obvious that diagonal matrices admit an \(r{\mathrm {th}}\)-root, so the same is valid for every diagonalizable matrix, a set of matrices that is dense in \({\mathrm{GL}}_m({\mathbb {C}})\).

  11. If instead of looking these objects as groups, one looks them as topological spaces, \({\mathrm{GL}}_n({\mathbb {C}})\) is connected and \(S_n\) is discrete, so any continuous homomorphism from \({\mathrm{GL}}_n({\mathbb {C}})\) to \(S_n\) is also constant. Boring\({}^2\).

  12. In categorical terms, this theorem says that the category of representations of \(S_n\) over \({\mathbb {C}}\) is semisimple.

  13. I learnt this notation from Wolfgang Soergel, and I love it, although it is impossible not to ask oneself, what are the names of the elements between b and c?.

  14. This is non-standard notation. I call them like that in analogy with flag varieties in \({\mathrm{GL}}_n({\mathbb {F}}_q)\) in the sense of Sect. 3.1.

  15. In the literature, an element of \(\mathcal {FL}(\lambda )\) is called a \(\lambda \)-tabloid.

  16. In the literature, an element of \(\mathcal {OFL}(\lambda )\) is called a \(\lambda \)-tableau.

  17. In the book32 that I cited there is a mistake in the formula, the \(\prod _i (\lambda _i)!\) is missing (thanks Geordie Williamson for noticing the mistake and Valentin Feray for explaining me the correct version of the theorem).

  18. Technically \(\lambda '\) is characterized uniquely by the condition \(\lambda _i\ge j\) if and only if \(\lambda _j'\ge i\).

  19. Usually in the presentation of this theory one associate to each \(\lambda \) partition of n a Young diagram and the transpose partition is literally the transpose of the Young diagram, thus explaining the name “transpose”. I will not explain Young diagrams.

  20. Game of Thrones reference.

  21. Fast question: do open problems give the set of all mathematical problems the structure of a topological space?.

  22. See105, p. 230] where an amusing (and false) theory of angles is developed by Averroes to explain why Aristotle was right.

  23. Two polyhedra are called dual if the vertices of one correspond to the faces of the other, and the edges of one correspond to the edges of the other, in an incidence-preserving way.

  24. By a basic result of the theory, all representations are direct limits of finite-dimensional representations. That is why I assume our representations to be finite-dimensional.

  25. It is an abelian category, Noetherian, Artinian, it is the heart of a t-structure, the simple objects are very easy to compute, etc.

  26. Let us ignore this word for now.

  27. Something very important about this equivalence and that is being ignored here, is that the general equivalence is between perverse sheaves on the affine Grassmannian associated to a group G and algebraic representations of \({}^LG,\) the Langlands dual group. We do not see this phenomenon here, because \({\mathrm{GL}}_n\) is self-dual.

  28. This is a humble reference to the famous GAGA paper (géométrie algebrique et géométrie analytique)107 by Serre.

  29. If the reader is tempted to believe that the sets \(\le \theta (w)\) with \(w\in W_a\) are easy, she can check the paper10 by Björner and Ekedahl that appeared in Annals of Mathematics in 2009, where the main theorem (proved with fancy mathematics) is that if \(f_i\) is the number of elements in \(\le \theta (w)\) of length i then \(f_i\le f_j\) if \(0\le i<j\le l(w)-i.\)

  30. This argument is like killing a fly with a bazooka, but I do not know an easier argument.

  31. Although with this procedure the family might not be “flat”, i.e., writing down an arbitrary definition would produce something that has smaller dimension in general. It is a bit of a miracle that this does not happen for Coxeter groups.

  32. There is an old controversy in the mathematical community on whether André Weil and Andrew Wiles are the same person or not. Some argue it is just the pronunciation of the same name in French and English. Some go even further and suggest that André Weil’s sister, famous philosopher Simone Weil is the same person as Andrew Wiles’s sister, the most decorated gymnast of all time: Simone Wiles. On such a delicate matter I prefer not to pronounce myself (just to be clear, this footnote is a joke).

  33. I have not explained what a formal character is because I do not want to introduce Lie algebras, but they are analogues of characters for simple groups as explained in 4.2.6 and are probably the most important piece of data one can extract from a representation.

  34. I was searching in the dictionary for a synonym of “monstrous” and this beautiful, self-explanatory word appeared.

  35. Beware that the tensor product is over \({\mathbb {C}}\). If it was over A one could see an A-module as a coherent sheaf and take the standard tensor product of coherent sheaves.

  36. The naive idea of defining \(h\cdot (m\otimes n):=(h\cdot m)\otimes (h\cdot n)\) is not well defined as \(((h+h')\cdot m)\otimes ((h+h')\cdot n)\) is usually different from \((h\cdot m)\otimes (h\cdot n)+(h'\cdot m)\otimes (h'\cdot n)\).

  37. In Crane’s paper one can find the very surprising fact that even Albert Einstein was at the end of his career thinking these kind of ideas. In a letter to Paul Langevin, Einstein said “The other possibility leads in my opinion to a renunciation of the space–time continuum, and to a purely algebraic physics.”

  38. The only Lie algebra for which a classification of the irreducible representations is known is \(\mathfrak {sl}_2({\mathbb {C}})\), see16.

  39. Recall the business of needing a square root of q in the Hecke algebra.

  40. While writing this paper, a new preprint68 was posted on the arXiv with the quite surprising comment that light leaves might be relevant in cryptography.

  41. One could think that going up one step in the categorical ladder gives rise naturally to one more rank involved, but this is not always the case as for “singular Soergel bimodules”, the relations have no bound in the number of simple reflections involved.

  42. Technically, to “behave well” means that the map ch descends to an isomorphism of \({\mathbb {Z}}[v,v^{-1}]\)-algebras \(\mathrm {ch}:[\mathcal {H}^k(W)]\cong H(W)\), where \([\cdot ]\) means the split Grothendieck group of an additive category.

  43. If one uses Schanuel’s improved version of the Euler characteristic, one could be tempted to admit that \({\mathbb {F}}_{-1}={\mathbb {R}}.\)

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Acknowledgements

You would not be reading this if it was not for Apoorva Khare, who was the engine for this work and also, with his warmth, kept me going in times of despair (my baby Gael was 1–5 months old when this paper was written). I would like to thank warmly Jorge Soto-Andrade for his many comments that improved an earlier version of this paper. Also to Juan Camilo Arias and Karina Batistelli for carefully correcting earlier versions of this manuscript. I would also like to thank Stephen Griffeth, Giancarlo Lucchini, David Plaza, Gonzalo Jimenez and Felipe Gambardella for helpful comments. Finally, enormous thanks to Geordie Williamson who corrected the paper in super detail, finding some errors in a previous version. This project was funded by ANID project Fondecyt regular 1200061.

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Funding was provided by Fondecyt (Grant no. 1200061).

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Libedinsky, N. IntroSurvey of Representation Theory. J Indian Inst Sci (2022). https://doi.org/10.1007/s41745-022-00301-4

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