A Survey of the Shi Arrangement

  • Susanna FishelEmail author
Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)


In [58], Shi proved Lusztig’s conjecture that the number of two-sided cells for the affine Weyl group of type \(A_{n-1}\) is the number of partitions of n. As a byproduct, he introduced the Shi arrangement of hyperplanes and found a few of its remarkable properties. The Shi arrangement has since become a central object in algebraic combinatorics. This article is intended to be a fairly gentle introduction to the Shi arrangement, intended for readers with a modest background in combinatorics, algebra, and Euclidean geometry.



The author would like to thank Hélène Barcelo, Gizem Karaali, and Rosa Orellana for allowing her to write an article for this AWM series. She would like to thank Matthew Fayers, Sarah Mason, and Jian-Yi Shi for comments on the manuscript, and Nathan Williams and Brendon Rhoades for suggesting several related papers. She would like to thank Patrick Headley for help with his thesis. The anonymous referees’ comments helped enormously to improve exposition. This work was supported by a grant from the Simons Foundation (#359602, Susanna Fishel).


  1. 1.
    T. Abe, D. Suyama, S. Tsujie, The freeness of Ish arrangements. J. Comb. Theory Ser. A 146, 169–183 (2017). MR 3574228MathSciNetzbMATHGoogle Scholar
  2. 2.
    J. Anderson, Partitions which are simultaneously \(t_1\)- and \(t_2\)-core. Discret. Math. 248(1–3), 237–243 (2002). MR 1892698MathSciNetzbMATHGoogle Scholar
  3. 3.
    F. Ardila, Computing the Tutte polynomial of a hyperplane arrangement. Pac. J. Math. 230(1), 1–26 (2007). MR 2318445MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Memoirs of the American Mathematical Society (2009), x+159 p. MR 2561274Google Scholar
  5. 5.
    D. Armstrong, Hyperplane arrangements and diagonal harmonics. J. Comb. 4(2), 157–190 (2013). MR 3096132MathSciNetzbMATHGoogle Scholar
  6. 6.
    D. Armstrong, V. Reiner, B. Rhoades, Parking spaces. Adv. Math. 269, 647–706 (2015). MR 3281144MathSciNetzbMATHGoogle Scholar
  7. 7.
    D. Armstrong, B. Rhoades, The Shi arrangement and the Ish arrangement. Trans. Am. Math. Soc. 364(3), 1509–1528 (2012). MR 2869184MathSciNetzbMATHGoogle Scholar
  8. 8.
    C.A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields. Adv. Math. 122(2), 193–233 (1996). MR 1409420MathSciNetzbMATHGoogle Scholar
  9. 9.
    C.A. Athanasiadis, On free deformations of the braid arrangement. Eur. J. Comb. 19(1), 7–18 (1998). MR 1600259MathSciNetzbMATHGoogle Scholar
  10. 10.
    C.A. Athanasiadis, Deformations of Coxeter hyperplane arrangements and their characteristic polynomials, Arrangements-Tokyo 1998. Advanced Studies in Pure Mathematics, vol. 27 (Kinokuniya, Tokyo, 2000), pp. 1–26. MR 1796891Google Scholar
  11. 11.
    C.A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. Lond. Math. Soc. 36(3), 294–302 (2004). MR 2038717MathSciNetzbMATHGoogle Scholar
  12. 12.
    C.A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups. Trans. Am. Math. Soc. 357(1), 179–196 (2005). MR 2098091MathSciNetzbMATHGoogle Scholar
  13. 13.
    C.A. Athanasiadis, A combinatorial reciprocity theorem for hyperplane arrangements. Can. Math. Bull. 53(1), 3–10 (2010). MR 2583206MathSciNetzbMATHGoogle Scholar
  14. 14.
    C.A. Athanasiadis, S. Linusson, A simple bijection for the regions of the Shi arrangement of hyperplanes. Discret. Math. 204(1–3), 27–39 (1999). MR 1691861MathSciNetzbMATHGoogle Scholar
  15. 15.
    C.A. Athanasiadis, E. Tzanaki, On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements. J. Algebra. Comb. 23(4), 355–375 (2006). MR 2236611MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Beck, A. Berrizbeitia, M. Dairyko, C. Rodriguez, A. Ruiz, S. Veeneman, Parking functions, Shi arrangements, and mixed graphs. Am. Math. Mon. 122(7), 660–673 (2015). MR 3383893MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. Björner, F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231 (Springer, New York, 2005). MR 2133266Google Scholar
  18. 18.
    A. Blass, B.E. Sagan, Characteristic and Ehrhart polynomials. J. Algebra. Comb. 7(2), 115–126 (1998). MR 1609889MathSciNetzbMATHGoogle Scholar
  19. 19.
    N. Bourbaki, éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, vol. 1337 (Hermann, Paris, 1968). MR 0240238Google Scholar
  20. 20.
    P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra. J. Algebra 225(1), 130–141 (2000). MR 1743654MathSciNetzbMATHGoogle Scholar
  21. 21.
    P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra II. J. Algebra 258(1), 112–121 (2002). Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958899MathSciNetzbMATHGoogle Scholar
  22. 22.
    P. Cellini, P. Papi, Abelian ideals of Borel subalgebras and affine Weyl groups. Adv. Math. 187(2), 320–361 (2004). MR 2078340MathSciNetzbMATHGoogle Scholar
  23. 23.
    H.H. Crapo, G-C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, preliminary ed., (MIT Press, Cambridge, London, 1970). MR 0290980Google Scholar
  24. 24.
    C.-P. Dong, Ad-nilpotent ideals and the Shi arrangement. J. Comb. Theory Ser. A 120(8), 2118–2136 (2013). MR 3102177MathSciNetzbMATHGoogle Scholar
  25. 25.
    R. Duarte, A.G. de Oliveira, The braid and the Shi arrangements and the Pak-Stanley labelling. Eur. J. Comb. 50, 72–86 (2015). MR 3361413MathSciNetzbMATHGoogle Scholar
  26. 26.
    S. Fishel, M. Kallipoliti, E. Tzanaki, Facets of the generalized cluster complex and regions in the extended Catalan arrangement of type \(A\). Electron. J. Comb. 20(4), 7–21 (2013). MR 3139392MathSciNetzbMATHGoogle Scholar
  27. 27.
    S. Fishel, E. Tzanaki, M. Vazirani, Counting Shi regions with a fixed separating wall. Ann. Comb. 17(4), 671–693 (2013). MR 3129778MathSciNetzbMATHGoogle Scholar
  28. 28.
    S. Fishel, M. Vazirani, A bijection between dominant Shi regions and core partitions. Eur. J. Comb. 31(8), 2087–2101 (2010). MR 2718283MathSciNetzbMATHGoogle Scholar
  29. 29.
    D. Forge, T. Zaslavsky, Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements. J. Comb. Theory Ser. A 114(1), 97–109 (2007). MR 2275583MathSciNetzbMATHGoogle Scholar
  30. 30.
    A.M. Garsia, M. Haiman, A remarkable \(q, t\)-Catalan sequence and \(q\)-Lagrange inversion. J. Algebra. Comb. 5(3), 191–244 (1996). MR 1394305MathSciNetzbMATHGoogle Scholar
  31. 31.
    E. Gorsky, M. Mazin, M. Vazirani, Affine permutations and rational slope parking functions. Trans. Am. Math. Soc. 368(12), 8403–8445 (2016). MR 3551576MathSciNetzbMATHGoogle Scholar
  32. 32.
    P.E. Gunnells, Cells in Coxeter groups. Not. Am. Math. Soc. 53(5), 528–535 (2006). MR 2254399MathSciNetzbMATHGoogle Scholar
  33. 33.
    P.E. Gunnells, Automata and cells in affine Weyl groups. Represent. Theory 14, 627–644 (2010). MR 2726285MathSciNetzbMATHGoogle Scholar
  34. 34.
    P.E. Gunnells, E. Sommers, A characterization of Dynkin elements. Math. Res. Lett. 10(2–3), 363–373 (2003). MR 1981909MathSciNetzbMATHGoogle Scholar
  35. 35.
    J. Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics. University Lecture Series, vol. 41 (American Mathematical Society, Providence, 2008). With an appendix on the combinatorics of Macdonald polynomials. MR 2371044Google Scholar
  36. 36.
    M.D. Haiman, Conjectures on the quotient ring by diagonal invariants. J. Algebra. Comb. 3(1), 17–76 (1994). MR 1256101MathSciNetzbMATHGoogle Scholar
  37. 37.
    P.T. Headley, Reduced expressions in infinite Coxeter groups, Ph.D. Thesis, University of Michigan (ProQuest LLC, Ann Arbor, MI, 1994). MR 2691313Google Scholar
  38. 38.
    P. Headley, On a family of hyperplane arrangements related to the affine Weyl groups. J. Algebra. Comb. 6(4), 331–338 (1997). MR 1471893MathSciNetzbMATHGoogle Scholar
  39. 39.
    C. Hohlweg, P. Nadeau, N. Williams, Automata, reduced words and Garside shadows in Coxeter groups. J. Algebra 457, 431–456 (2016). MR 3490088MathSciNetzbMATHGoogle Scholar
  40. 40.
    S. Hopkins, D. Perkinson, Bigraphical arrangements. Trans. Am. Math. Soc. 368(1), 709–725 (2016). MR 3413881MathSciNetzbMATHGoogle Scholar
  41. 41.
    J.E. Humphreys, Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, 1990). MR 1066460Google Scholar
  42. 42.
    G. James, A. Kerber, The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16 (Addison-Wesley Publishing Co., Reading, 1981). With a foreword by P. M. Cohn, With an introduction by Gilbert de B. Robinson. MR 644144Google Scholar
  43. 43.
    V.G. Kac, Infinite-dimensional Lie Algebras, 3rd edn. (Cambridge University Press, Cambridge, 1990). MR 1104219zbMATHGoogle Scholar
  44. 44.
    D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979). MR 560412MathSciNetzbMATHGoogle Scholar
  45. 45.
    L. Lapointe, J. Morse, Tableaux on \(k+1\)-cores, reduced words for affine permutations, and \(k\)-Schur expansions. J. Comb. Theory Ser. A 112(1), 44–81 (2005). MR 2167475MathSciNetzbMATHGoogle Scholar
  46. 46.
    A. Lascoux, Ordering the affine symmetric group, Algebraic Combinatorics and Applications, Gößweinstein, 1999 (Springer, Berlin, 2001), pp. 219–231. MR 1851953Google Scholar
  47. 47.
    E. Leven, B. Rhoades, A.T. Wilson, Bijections for the Shi and Ish arrangements. Eur. J. Comb. 39, 1–23 (2014). MR 3168512MathSciNetzbMATHGoogle Scholar
  48. 48.
    G. Lusztig, Some examples of square integrable representations of semisimple \(p\)-adic groups. Trans. Am. Math. Soc. 277(2), 623–653 (1983). MR 694380MathSciNetzbMATHGoogle Scholar
  49. 49.
    J. McCammond, H. Thomas, N. Williams, Fixed points of parking functions (2017)Google Scholar
  50. 50.
    K. Mészáros, Labeling the regions of the type \(C_n\) Shi arrangement. Electron. J. Comb. 20(2), Paper 31, 12 (2013). MR 3066370Google Scholar
  51. 51.
    J.W. Moon, in Counting Labelled Trees, From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress, Vancouver, vol. 1969 (Que, Canadian Mathematical Congress, Montreal, 1970). MR 0274333Google Scholar
  52. 52.
    D.I. Panyushev, Ad-nilpotent ideals of a Borel subalgebra: generators and duality. J. Algebra 274(2), 822–846 (2004). MR 2043377MathSciNetzbMATHGoogle Scholar
  53. 53.
    T. Kyle Petersen, Eulerian Numbers, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks] (Birkhäuser/Springer, New York, 2015). With a foreword by Richard Stanley. MR 3408615zbMATHGoogle Scholar
  54. 54.
    A. Postnikov, Deformations of Coxeter hyperplane arrangements. J. Comb. Theory Ser. A 91(1–2), 544–597 (2000). In memory of Gian-Carlo Rota. MR 1780038MathSciNetzbMATHGoogle Scholar
  55. 55.
    M.J. Richards, Some decomposition numbers for Hecke algebras of general linear groups. Math. Proc. Camb. Philos. Soc. 119(3), 383–402 (1996). MR 1357053MathSciNetzbMATHGoogle Scholar
  56. 56.
    F. Rincón, A labelling of the faces in the Shi arrangement, Rose-Hulman Undergrad. Math. J. 8(1), 7 (2007)Google Scholar
  57. 57.
    J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups. J. Algebra 142(2), 441–455 (1991). MR 1127075MathSciNetzbMATHGoogle Scholar
  58. 58.
    J.Y. Shi, The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups. Lecture Notes in Mathematics, vol. 1179 (Springer, Berlin, 1986). MR 835214zbMATHGoogle Scholar
  59. 59.
    J.Y. Shi, Alcoves corresponding to an affine Weyl group. J. Lond. Math. Soc. (2) 35(1), 42–55 (1987). MR 871764MathSciNetzbMATHGoogle Scholar
  60. 60.
    J.Y. Shi, Sign types corresponding to an affine Weyl group. J. Lond. Math. Soc. (2) 35(1), 56–74 (1987). MR 871764MathSciNetzbMATHGoogle Scholar
  61. 61.
    J.Y. Shi, A two-sided cell in an affine Weyl group. J. Lond. Math. Soc. (2) 36(3), 407–420 (1987). MR 918633MathSciNetzbMATHGoogle Scholar
  62. 62.
    J.Y. Shi, A two-sided cell in an affine Weyl group II. J. Lond. Math. Soc. (2) 37(2), 253–264 (1988). MR 918633MathSciNetzbMATHGoogle Scholar
  63. 63.
    J.Y. Shi, Jian Yi Shi, Left cells in the affine Weyl group \(W_a({\widetilde{D}}_4)\). Osaka J. Math. 31(1), 27–50 (1994). MR 1262787MathSciNetzbMATHGoogle Scholar
  64. 64.
    J.-Y. Shi, The number of \(\oplus \)-sign types. Q. J. Math. Oxf. Ser. (2) 48(189), 93–105 (1997). MR 1439701MathSciNetzbMATHGoogle Scholar
  65. 65.
    S. Sivasubramanian, On the two variable distance enumerator of the Shi hyperplane arrangement. Eur. J. Comb. 29(5), 1104–1111 (2008). MR 2419213MathSciNetzbMATHGoogle Scholar
  66. 66.
    N. Eric, Sommers, \(B\)-stable ideals in the nilradical of a Borel subalgebra. Can. Math. Bull. 48(3), 460–472 (2005). MR 2154088zbMATHGoogle Scholar
  67. 67.
    P. Richard, Stanley, Hyperplane arrangements, interval orders, and trees. Proc. Natl. Acad. Sci. USA 93(6), 2620–2625 (1996). MR 1379568zbMATHGoogle Scholar
  68. 68.
    R.P. Stanley, Hyperplane arrangements, parking functions and tree inversions, Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996). Progress in Mathematics, vol. 161 (Birkhäuser, Boston, 1998), pp. 359–375. MR 1627378Google Scholar
  69. 69.
    R.P. Stanley, An introduction to hyperplane arrangements, Geometric Combinatorics, IAS/Park City Mathematics Series (American Mathematical Society, Providence, 2007), pp. 389–496. MR 2383131Google Scholar
  70. 70.
    R.P. Stanley, Enumerative Combinatorics, Volume 1, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 49 (Cambridge University Press, Cambridge, 2012). MR 2868112Google Scholar
  71. 71.
    R.P. Stanley, Catalan Numbers (Cambridge University Press, New York, 2015). MR 3467982zbMATHGoogle Scholar
  72. 72.
    R. Sulzgruber, Rational Shi tableaux and the skew length statistic (2015), arXiv:1512.04320
  73. 73.
    R. Suter, Abelian ideals in a Borel subalgebra of a complex simple Lie algebra. Invent. Math. 156(1), 175–221 (2004). MR 2047661MathSciNetzbMATHGoogle Scholar
  74. 74.
    M. Thiel, On floors and ceilings of the \(k\)-Catalan arrangement. Electron. J. Comb. 21(4), Paper 4.36, 15. MR 3292273Google Scholar
  75. 75.
    H. Thomas, N. Williams, Cyclic symmetry of the scaled simplex. J. Algebra. Comb. 39(2), 225–246 (2014). MR 3159251MathSciNetzbMATHGoogle Scholar
  76. 76.
    T. Zaslavsky, Counting the faces of cut-up spaces. Bull. Am. Math. Soc. 81(5), 916–918 (1975). MR 0400066MathSciNetzbMATHGoogle Scholar
  77. 77.
    Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Memoirs Am. Math. Soc. 1(154), vii+102 p. (1975). MR 0357135Google Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

Personalised recommendations