Skip to main content

IntroSurvey of Representation Theory


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.

This is a preview of subscription content, access via your institution.

Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:
Figure 9:
Figure 10:
Figure 11:
Figure 12:
Figure 13:
Figure 14:


  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 paper14 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 \(\mathfrak {sl}_2({\mathbb {C}})\), see15.

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

  40. While writing this paper, a new preprint67 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 \({\mathbb {F}}_{-1}={\mathbb {R}}.\)


  1. Abe N (2019) On Soergel bimodules. arXiv:1901.02336

  2. Abe N (2021) A homomorphism between Bott–Samelson bimodules. arXiv:2012.09414

  3. Achar PN, Makisumi S, Riche S, Williamson G (2019) Koszul duality for Kac–Moody groups and characters of tilting modules. J Am Math Soc 32(1):261–310

    Article  Google Scholar 

  4. Alexander J (1923) A lemma on systems of knotted curves. Proc Natl Acad 9(3):93–95

    CAS  Article  Google Scholar 

  5. Andersen HH, Jantzen JC, Soergel W (1994) Representations of quantum groups at a \(p\)th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque (220):321

  6. Andruskiewitsch N, Santos WF (2009) The beginnings of the theory of Hopf algebras. Acta Appl Math 108(1):3–17

    Article  Google Scholar 

  7. Andruskiewitsch N, Schneider H-J (2010) On the classification of finite-dimensional pointed Hopf algebras. Ann Math (2) 171(1):375–417

    Article  Google Scholar 

  8. Anno R, Bezrukavnikov R, Mirković I (2015) Stability conditions for Slodowy slices and real variations of stability. Mosc Math J 15(2):187–203 (403)

    Article  Google Scholar 

  9. Belolipetsky M (2004) Cells and representations of right-angled Coxeter groups. Selecta Math (NS) 10(3):325–339

    Article  Google Scholar 

  10. Belolipetsky M, Gunnells P (2015) Kazhdan–Lusztig cells in infinite Coxeter groups. J Gen Lie Theory Appl 9(S1):S1–002, 4

  11. Belolipetsky M, Gunnells P, Scott RA (2014) Kazhdan–Lusztig cells in planar hyperbolic Coxeter groups and automata. Int J Algebra Comput 24(5):757–772

    Article  Google Scholar 

  12. Bernstein J, Lunts V (1994) Equivariant sheaves and functors, vol 1578. Lecture notes in mathematics. Springer, Berlin

    Google Scholar 

  13. Bezrukavnikov R, Riche S (2021) Hecke action on the principal block. arXiv:2009.10587

  14. Björner A, Ekedahl T (2009) On the shape of Bruhat intervals. Ann Math (2) 170(2):799–817

    Article  Google Scholar 

  15. Block RE (1981) The irreducible representations of the Lie algebra \(\mathfrak{sl}_2\) and of the Weyl algebra. Adv Math 39(1):69–110

    Article  Google Scholar 

  16. Blundell C, Buesing L, Davies A, Velickovic P, Williamson G (2021) Towards combinatorial invariance for Kazhdan–Lusztig polynomials. arXiv:2111.15161

  17. Brenti F (1998) Kazhdan–Lusztig and \(R\)-polynomials from a combinatorial point of view. vol 193, pp 93–116. Selected papers in honor of Adriano Garsia (Taormina, 1994)

  18. Bridgeland T (2007) Stability conditions on triangulated categories. Ann Math (2) 166(2):317–345

    Article  Google Scholar 

  19. Bruhat F (1954) Représentations induites des groupes de Lie semi-simples complexes. C R Acad Sci Paris 238:437–439

    Google Scholar 

  20. Burrull G, Libedinsky N, Plaza D (2021) Combinatorial invariance conjecture for \(\widetilde{A}_2\). arXiv:2105.04609

  21. Burrull G, Libedinsky N, Sentinelli P (2019) \(p\)-Jones–Wenzl idempotents. Adv Math 352:246–264

    Article  Google Scholar 

  22. Chari V, Pressley A (1994) A guide to quantum groups. Cambridge University Press, Cambridge

    Google Scholar 

  23. Chen ER, Engel M, Glotzer SC (2010) Dense crystalline dimer packings of regular tetrahedra. Discrete Comput Geom 44(2):253–280

    Article  Google Scholar 

  24. Chevalley C (1955) Sur certains groupes simples. Tohoku Math J 2(7):14–66

    Google Scholar 

  25. Ciappara J (2021) Hecke category actions via Smith–Treumann theory. arXiv:2103.07091

  26. Connes A, Consani C (2011) Characteristic 1, entropy and the absolute point. Noncommutative geometry. Arithmetic, and related topics. Johns Hopkins Univ. Press, Baltimore, pp 75–139

    Google Scholar 

  27. Crane L (1995) Clock and category: is quantum gravity algebraic? J Math Phys 36(11):6180–6193

    Article  Google Scholar 

  28. Crane L, Frenkel IB (1994) Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. vol 35, pp 5136–5154. Topology and physics

  29. Davies A, Velickovic P, Buesing L, Blackwell S, Zheng D, Tomašev N, Tanburn R, Battaglia P, Blundell C, Juhasz A, Lackenby M, Williamson G, Hassabis D, Kohli P (2021) Advancing mathematics by guiding human intuition with AI. Nature 600:70–74

    CAS  Article  Google Scholar 

  30. Deligne P (2007) La catégorie des représentations du groupe symétrique \(S_t\), lorsque \(t\) n’est pas un entier naturel. In: Algebraic groups and homogeneous spaces, volume 19 of Tata Inst. Fund. Res. Stud. Math.. Tata Inst. Fund. Res., Mumbai, pp 209–273

  31. Deninger C (1992) Local \(L\)-factors of motives and regularized determinants. Invent Math 107(1):135–150

    Article  Google Scholar 

  32. Diaconis P (1988) Group representations in probability and statistics, vol 11. Institute of Mathematical Statistics Lecture Notes-Monograph Series. Institute of Mathematical Statistics, Hayward

  33. Dieudonné J (1977) Panorama des mathématiques pures. Gauthier-Villars, Paris. Le choix bourbachique. [The Bourbakian choice]

  34. Dipper R, James G (1991) \(q\)-tensor space and \(q\)-Weyl modules. Trans Am Math Soc 327(1):251–282

    Google Scholar 

  35. Drinfel’d V (1987) Quantum groups. In: Proceedings of the International Congress of Mathematicians, vol 1, 2 (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, pp 798–820

  36. Elias B (2014) Quantum Satake in type a: part I. arXiv:1403.5570

  37. Elias B (2015) Light ladders and clasp conjectures. arXiv:1510.06840

  38. Elias B (2016) Thicker Soergel calculus in type \(A\). Proc Lond Math Soc (3) 112(5):924–978

    Article  Google Scholar 

  39. Elias B (2016) The two-color Soergel calculus. Compos Math 152(2):327–398

    Article  Google Scholar 

  40. Elias B, Khovanov M (2010) Diagrammatics for Soergel categories. Int J Math Math Sci. Art. ID 978635, 58

  41. Elias B, Libedinsky N (2017) Indecomposable Soergel bimodules for universal Coxeter groups. Trans Am Math Soc 369(6):3883–3910 (With an appendix by Ben Webster)

    Article  Google Scholar 

  42. Elias B, Williamson G (2014) The Hodge theory of Soergel bimodules. Ann Math (2) 180(3):1089–1136

    Article  Google Scholar 

  43. Elias B, Williamson G (2016) Soergel calculus. Represent Theory 20:295–374

    Article  Google Scholar 

  44. Elias B, Williamson G (2021) Relative hard Lefschetz for Soergel bimodules. J Eur Math Soc (JEMS) 23(8):2549–2581

    Article  Google Scholar 

  45. Elias B, Makisumi S, Thiel U, Williamson G (2020) Introduction to Soergel bimodules, volume 5 of RSME springer series. Springer, Cham

    Book  Google Scholar 

  46. Freudenthal H, de Vries H (1969) Linear Lie groups. Pure and applied mathematics, vol 35. Academic Press, New York-London

    Google Scholar 

  47. Gerstenhaber M (1964) On the deformation of rings and algebras. Ann Math 2(79):59–103

    Article  Google Scholar 

  48. Gobet T, Thiel A-L (2020) A Soergel-like category for complex reflection groups of rank one. Math Z 295(1–2):643–665

    Article  Google Scholar 

  49. Gowers T, Barrow-GJ Leader I (eds) (2008) The Princeton companion to mathematics. Princeton University Press, Princeton

    Google Scholar 

  50. Howlett R, Lehrer G (1980) Induced cuspidal representations and generalised Hecke rings. Invent Math 58(1):37–64

    Article  Google Scholar 

  51. Huh J (2018) Combinatorial applications of the Hodge-Riemann relations. In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures, pp 3093–3111. World Sci. Publ., Hackensack

  52. Iwahori N (1964) On the structure of a Hecke ring of a Chevalley group over a finite field. J Fac Sci Univ Tokyo Sect I(10):215–236

    Google Scholar 

  53. Iwahori N, Matsumoto H (1965) On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups. Inst Hautes Études Sci Publ Math 25:5–48

    Article  Google Scholar 

  54. James G, Kerber A (1981) The representation theory of the symmetric group, volume 16 of encyclopedia of mathematics and its applications. Addison-Wesley Publishing Co., Reading, Mass. With a foreword by P. M. Cohn, With an introduction by Gilbert de B. Robinson

  55. Jensen LT (2017) The 2-braid group and Garside normal form. Math Z 286(1–2):491–520

    Article  Google Scholar 

  56. Jensen LT (2021) Correction of the Lusztig–Williamson billiards conjecture. arXiv:2105.04665

  57. Jensen LT, Williamson G (2017) The \(p\)-canonical basis for Hecke algebras. In: Categorification and higher representation theory, volume 683 of contemp. math., pp 333–361. Amer. Math. Soc., Providence

  58. Jimbo M (1986) A \(q\)-analogue of \(U({\mathfrak{gl}}(N+1))\), Hecke algebra, and the Yang–Baxter equation. Lett Math Phys 11(3):247–252

    Article  Google Scholar 

  59. Jones VFR (1985) A polynomial invariant for knots via von Neumann algebras. Bull Am Math Soc (NS) 12(1):103–111

    Article  Google Scholar 

  60. Joyal A, Street R (1993) Braided tensor categories. Adv Math 102(1):20–78

    Article  Google Scholar 

  61. Joyal A, Street R (1995) The category of representations of the general linear groups over a finite field. J Algebra 176(3):908–946

    Article  Google Scholar 

  62. Kamada S (2002) Braid and knot theory in dimension four, volume 95 of mathematical surveys and monographs. American Mathematical Society, Providence

    Book  Google Scholar 

  63. Kassel C, Turaev V (2008) Braid groups, volume 247 of graduate texts in mathematics. Springer, New York. With the graphical assistance of Olivier Dodane

  64. Kazhdan D, Lusztig G (1979) Representations of Coxeter groups and Hecke algebras. Invent Math 53(2):165–184

    Article  Google Scholar 

  65. Khovanov M (2007) Triply-graded link homology and Hochschild homology of Soergel bimodules. Int J Math 18(8):869–885

    Article  Google Scholar 

  66. Khovanov M, Seidel P (2002) Quivers, Floer cohomology, and braid group actions. J Am Math Soc 15(1):203–271

    Article  Google Scholar 

  67. Khovanov M, Sitaraman M, Tubbenhauer D (2022) Monoidal categories, representation gap and cryptography. arXiv:2201.01805

  68. Kostant B (1959) A formula for the multiplicity of a weight. Trans Am Math Soc 93:53–73

    Article  Google Scholar 

  69. Kurokawa N (1992) Multiple zeta functions: an example. In: Zeta functions in geometry (Tokyo, 1990), volume 21 of adv. stud. pure math.. Kinokuniya, Tokyo, pp 219–226

  70. Lascoux A, Schützenberger M-P (1978) Sur une conjecture de H. O. Foulkes. C R Acad Sci Paris Sér A-B 286(7):A323–A324

    Google Scholar 

  71. Lascoux A, Leclerc B, Thibon J-Y (1995) Crystal graphs and \(q\)-analogues of weight multiplicities for the root system \(A_n\). Lett Math Phys 35(4):359–374

    Article  Google Scholar 

  72. Libedinsky N (2008) Auteur de la catégorie des bimodules de Soergel. Thèse de Doctorat Université Paris 7

  73. Libedinsky N (2008) Équivalences entre conjectures de Soergel. J Algebra 320(7):2695–2705

    Article  Google Scholar 

  74. Libedinsky N (2008) Sur la catégorie des bimodules de Soergel. J Algebra 320(7):2675–2694

    Article  Google Scholar 

  75. Libedinsky N (2010) Presentation of right-angled Soergel categories by generators and relations. J Pure Appl Algebra 214(12):2265–2278

    Article  Google Scholar 

  76. Libedinsky N (2011) New bases of some Hecke algebras via Soergel bimodules. Adv Math 228(2):1043–1067

    Article  Google Scholar 

  77. Libedinsky N (2015) Light leaves and Lusztig’s conjecture. Adv Math 280:772–807

    Article  Google Scholar 

  78. Libedinsky N (2019) Gentle introduction to Soergel bimodules I: the basics. São Paulo J Math Sci 13(2):499–538

    Article  Google Scholar 

  79. Libedinsky N, Patimo L (2020) On the affine Hecke category for \(sl_3\). arXiv:2005.02647

  80. Libedinsky N, Williamson G (2014) Standard objects in 2-braid groups. Proc Lond Math Soc (3) 109(5):1264–1280

    Article  Google Scholar 

  81. Libedinsky N, Williamson G (2017) The anti-spherical category. arXiv:1702.00459

  82. Libedinsky N, Williamson G (2021) Kazhdan–Lusztig polynomials and subexpressions. J Algebra 568:181–192

    Article  Google Scholar 

  83. Libedinsky N, Patimo L, Plaza D (2021) Pre-canonical bases. arXiv:2103.06903

  84. Littelmann P (1995) Paths and root operators in representation theory. Ann Math (2) 142(3):499–525

    Article  Google Scholar 

  85. Loeb D (1992) Sets with a negative number of elements. Adv Math 91(1):64–74

    Article  Google Scholar 

  86. Lopez Peña J, Lorscheid O (2011) Mapping \({}_1\)-land: an overview of geometries over the field with one element. In: Noncommutative geometry, arithmetic, and related topics. Johns Hopkins Univ. Press, Baltimore, pp 241–265

  87. Lusztig G (2010) Bruhat decomposition and applications. arXiv:1006.5004

  88. Lusztig G, Williamson G (2018) Billiards and tilting characters for \({\rm SL}_3\). SIGMA Symmetry Integr Geom Methods Appl 14:Paper No. 015, 22

  89. Makisumi S (2019) On monoidal Koszul duality for the Hecke category. Rev Colombiana Mat 53(suppl.):195–222

    Article  Google Scholar 

  90. Manin Y (1992) Lectures on zeta functions and motives (according to Deninger and Kurokawa). vol 228, pp 121–163. 1995. Columbia University Number Theory Seminar (New York)

  91. Maschke H (1898) Über den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen. Math Ann 50(4):492–498

    Article  Google Scholar 

  92. Matsumoto H (1964) Générateurs et relations des groupes de Weyl généralisés. C R Acad Sci Paris 258:3419–3422

    Google Scholar 

  93. Mirković I, Vilonen K (2007) Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann Math (2) 166(1):95–143

    Article  Google Scholar 

  94. Palais R, Stewart T (1960) Deformations of compact differentiable transformation groups. Am J Math 82:935–937

    Article  Google Scholar 

  95. Patimo L (2021) Charges via the Affine Grassmannian. arXiv:2106.02564

  96. Plaza D (2017) Graded cellularity and the monotonicity conjecture. J Algebra 473:324–351

    Article  Google Scholar 

  97. Plaza D (2019) Diagrammatics for Kazhdan–Lusztig \(\widetilde{R}\)-polynomials. Eur J Combin 79:193–213

    Article  Google Scholar 

  98. Propp J (2003) Exponentiation and Euler measure. vol 49, pp 459–471. Dedicated to the memory of Gian-Carlo Rota

  99. References for Kazhdan-Lusztig theory. Accessed: 2021-11-15

  100. Riche S, Williamson G (2018) Tilting modules and the \(p\)-canonical basis. Astérisque (397):ix+184

  101. Rouquier R (2006) Categorification of \({\mathfrak{sl}}_2\) and braid groups. In: Trends in representation theory of algebras and related topics, volume 406 of contemp. math.. Amer. Math. Soc., Providence, pp 137–167

  102. Sahi S (2000) A new formula for weight multiplicities and characters. Duke Math J 101(1):77–84

    Article  Google Scholar 

  103. Schanuel SH (1991) Negative sets have Euler characteristic and dimension. In: Category theory (Como, 1990), volume 1488 of lecture notes in math.. Springer, Berlin, pp 379–385

  104. Schützer W (2012) A new character formula for Lie algebras and Lie groups. J Lie Theory 22(3):817–838

    Google Scholar 

  105. Senechal M (1981) Which tetrahedra fill space? Math Mag 54(5):227–243

    Article  Google Scholar 

  106. Sentinelli P (2021) Artin group injection in the Hecke algebra for right-angled groups. Geom Dedicata 214:193–210

    Article  Google Scholar 

  107. Serre J-P (1955) Géométrie algébrique et géométrie analytique. Ann Inst Fourier (Grenoble) 6:1–42 (56)

    Article  Google Scholar 

  108. Shimura G (1959) Sur les intégrales attachées aux formes automorphes. J Math Soc Jpn 11:291–311

    Google Scholar 

  109. Shimura G (1999) André Weil as I knew him. Not Am Math Soc 46(4):428–433

    Google Scholar 

  110. Soergel W (1992) The combinatorics of Harish–Chandra bimodules. J Reine Angew Math 429:49–74

    Google Scholar 

  111. Soergel W (1998) On the relation between intersection cohomology and representation theory in positive characteristic, vol 152, pp 311–335. 2000. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome)

  112. Soergel W (2007) Kazhdan–Lusztig–Polynome und unzerlegbare Bimoduln über Polynomringen. J Inst Math Jussieu 6(3):501–525

    Article  Google Scholar 

  113. Soulé C (2004) Les variétés sur le corps à un élément. Mosc Math J 4(1):217–244 (312)

    Article  Google Scholar 

  114. Steinberg R (1951) A geometric approach to the representations of the full linear group over a Galois field. Trans Am Math Soc 71:274–282

    Article  Google Scholar 

  115. Steinberg R (1963) Representations of algebraic groups. Nagoya Math J 22:33–56

    Article  Google Scholar 

  116. Sutton L, Tubbenhauer D, Wedrich P, Zhu J (2021) \({S}{L}_2\) tilting modules in the mixed case. arXiv:2105.07724

  117. Tits J (1957) Sur les analogues algébriques des groupes semi-simples complexes. In: Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, pp 261–289. Établissements Ceuterick, Louvain, Librairie Gauthier-Villars, Paris

  118. Weyl H (1925) Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I. Math Z 23(1):271–309

    Article  Google Scholar 

  119. Williamson G (2017) Schubert calculus and torsion explosion. J Am Math Soc 30(4):1023–1046 (With a joint appendix with Alex Kontorovich and Peter J. McNamara)

    Article  Google Scholar 

  120. Williamson G, Braden T (2012) Modular intersection cohomology complexes on flag varieties. Math Z 272(3–4):697–727

    Article  Google Scholar 

  121. Yale PB (1966) Automorphisms of the complex numbers. Math Mag 39(3):135–141

    Article  Google Scholar 

Download references


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.


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nicolas Libedinsky.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Libedinsky, N. IntroSurvey of Representation Theory. J Indian Inst Sci (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: