대한수학회지 (Journal of the Korean Mathematical Society)

  • 제60권6호
  • /
  • Pages.1255-1302
  • /
  • 2023
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

FINITE QUOTIENTS OF SINGULAR ARTIN MONOIDS AND CATEGORIFICATION OF THE DESINGULARIZATION MAP

  • Helena Jonsson (Department of Mathematics Uppsala University) ;
  • Volodymyr Mazorchuk (Department of Mathematics Uppsala University) ;
  • Elin Persson Westin (Department of Mathematics Uppsala University) ;
  • Shraddha Srivastava (Department of Mathematics Uppsala University, Department of Mathematics Indian Institute of Technology) ;
  • Mateusz Stroinski (Department of Mathematics Uppsala University) ;
  • Xiaoyu Zhu (School of Mathematical Sciences, Tongji University, School of Mathematics and Statistics Ningbo University)
  • 투고 : 2023.01.11
  • 심사 : 2023.08.18
  • 발행 : 2023.11.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We study various aspects of the structure and representation theory of singular Artin monoids. This includes a number of generalizations of the desingularization map and explicit presentations for certain finite quotient monoids of diagrammatic nature. The main result is a categorification of the classical desingularization map for singular Artin monoids associated to finite Weyl groups using BGG category 𝒪.

키워드

과제정보

This research is partially supported by the Swedish Research Council. We thank Hankyung Ko for very helpful discussions. We thank James East for helpful comments. We thank the referee for helpful comments.

참고문헌

  1. N. Antony, On the injectivity of the Vassiliev homomorphism of singular Artin monoids, Bull. Austral. Math. Soc. 70 (2004), no. 3, 401-422. https://doi.org/10.1017/S0004972700034651 
  2. N. Antony, On singular Artin monoids and contributions to Birman's conjecture, Comm. Algebra 33 (2005), no. 11, 4043-4056. https://doi.org/10.1080/00927870500261355 
  3. E. Backelin, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267-283. https://doi.org/10.1090/S1088-4165-01-00099-1 
  4. J. C. Baez, Link invariants of finite type and perturbation theory, Lett. Math. Phys. 26 (1992), no. 1, 43-51. https://doi.org/10.1007/BF00420517 
  5. A. Beilinson and J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981), no. 1, 15-18. 
  6. J. Bernstein and S. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245-285. 
  7. I. Bernstein, I. Gelfand, and S. Gelfand, Differential operators on the base affine space and a study of g-modules, in Lie groups and their representations (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971), pp. 21-64. Halsted, New York, 1975. 
  8. I. Bernstein, I. Gelfand, and S. Gelfand, A certain category of g-modules, (Russian) Funkcional. Anal. i Prilozen. 10 (1976), no. 2, 1-8.  https://doi.org/10.1007/BF01075764
  9. S. Bigelow, Braid groups and Iwahori-Hecke algebras, in Problems on mapping class groups and related topics, 285-299, Proc. Sympos. Pure Math., 74, Amer. Math. Soc., Providence, RI, 2006. https://doi.org/10.1090/pspum/074/2264547 
  10. J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253-287. https://doi.org/10.1090/S0273-0979-1993-00389-6 
  11. R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857-872. https://doi.org/10.2307/1968843 
  12. J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387-410.  https://doi.org/10.1007/BF01389272
  13. K. J. Carlin, Extensions of Verma modules, Trans. Amer. Math. Soc. 294 (1986), no. 1, 29-43. https://doi.org/10.2307/2000116 
  14. C.-W. Chen, B. Frisk Dubsky, H. Jonsson, V. Mazorchuk, E. Persson Westin, X. Zhang, and J. Zimmermann, Extreme representations of semirings, Serdica Math. J. 44 (2018), no. 3-4, 365-412. 
  15. C. Clark and J. East, Presentations for wreath products involving symmetric inverse monoids and categories, J. Algebra 620 (2023), 630-668.  https://doi.org/10.1016/j.jalgebra.2022.12.032
  16. A. Cohen and S. Liu, Brauer algebras of type B, Forum Math. 27 (2015), no. 2, 1163-1202.  https://doi.org/10.1515/forum-2012-0041
  17. R. Corran, A normal form for a class of monoids including the singular braid monoids, J. Algebra 223 (2000), no. 1, 256-282. https://doi.org/10.1006/jabr.1999.8044 
  18. R. Corran, Conjugacy in singular Artin monoids, J. Aust. Math. Soc. 79 (2005), no. 2, 183-212. https://doi.org/10.1017/S1446788700010442 
  19. K. Coulembier and V. Mazorchuk, Dualities and derived equivalences for category O, Israel J. Math. 219 (2017), no. 2, 661-706.  https://doi.org/10.1007/s11856-017-1494-y
  20. D. Easdown, J. East, and D. FitzGerald, Presentations of factorizable inverse monoids, Acta Sci. Math. (Szeged) 71 (2005), no. 3-4, 509-520. 
  21. J. East, Birman's conjecture is true for I2(p), J. Knot Theory Ramifications 15 (2006), no. 2, 167-177. https://doi.org/10.1142/S0218216506004385 
  22. J. East, The factorizable braid monoid, Proc. Edinb. Math. Soc. (2) 49 (2006), no. 3, 609-636. https://doi.org/10.1017/S0013091504001452 
  23. J. East, Singular braids and partial permutations, Acta Sci. Math. (Szeged) 81 (2015), no. 1-2, 55-77. https://doi.org/10.14232/actasm-014-012-7 
  24. B. Everitt and J. Fountain, Partial mirror symmetry, lattice presentations and algebraic monoids, Proc. Lond. Math. Soc. (3) 107 (2013), no. 2, 414-450.  https://doi.org/10.1112/plms/pds068
  25. D. G. FitzGerald, A presentation for the monoid of uniform block permutations, Bull. Austral. Math. Soc. 68 (2003), no. 2, 317-324. https://doi.org/10.1017/S0004972700037692 
  26. D. G. FitzGerald and J. Leech, Dual symmetric inverse monoids and representation theory, J. Austral. Math. Soc. Ser. A 64 (1998), no. 3, 345-367.  https://doi.org/10.1017/S1446788700039227
  27. O. Ganyushkin and V. Mazorchuk, Factor powers of semigroups of transformations, Dokl. Akad. Nauk Ukraini 1993 (1993), no. 12, 5-9. 
  28. O. Ganyushkin and V. Mazorchuk, Factor powers of finite symmetric groups, (Russian) Mat. Zametki 58 (1995), no. 2, 176-188; translation in Math. Notes 58 (1995), no. 1-2, 794-802 (1996)  https://doi.org/10.1007/BF02304101
  29. O. Ganyushkin and V. Mazorchuk, The structure of subsemigroups of factor powers of finite symmetric groups, (Russian) Mat. Zametki 58 (1995), no. 3, 341-354, 478; translation in Math. Notes 58 (1995), no. 3-4, 910-920 (1996)  https://doi.org/10.1007/BF02304767
  30. O. Ganyushkin and V. Mazorchuk, On the radical of FP+(Sn), Mat. Stud. 20 (2003), no. 1, 17-26. 
  31. O. Ganyushkin and V. Mazorchuk, Classical finite transformation semigroups: An introduction, Algebra and Applications, 9. Springer-Verlag London, Ltd., London, 2009. 
  32. E. Godelle and L. Paris, On singular Artin monoids. Geometric methods in group theory, 43-57, Contemp. Math., 372, Amer. Math. Soc., Providence, RI, 2005. 
  33. S. V. Hudzenko, Automorphisms of a finitary factor power of an infinite symmetric group, Ukrainian Math. J. 62 (2010), no. 7, 1158-1162; translated from Ukrain. Mat. Zh. 62 (2010), no. 7, 997-1001. https://doi.org/10.1007/s11253-010-0420-9 
  34. J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer, New York, 1972. 
  35. J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, 94, Amer. Math. Soc., Providence, RI, 2008. https://doi.org/10.1090/gsm/094 
  36. M. Jab lonowski, Presentations and representations of surface singular braid monoids, J. Korean Math. Soc. 54 (2017), no. 3, 749-762. https://doi.org/10.4134/JKMS.j160142 
  37. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184.  https://doi.org/10.1007/BF01390031
  38. M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1, 203-271.  https://doi.org/10.1090/S0894-0347-01-00374-5
  39. M. Kosuda, Irreducible representations of the party algebra, Osaka J. Math. 43 (2006), no. 2, 431-474. http://projecteuclid.org/euclid.ojm/1152203949 
  40. G. Kudryavtseva and V. Mazorchuk, On presentations of Brauer-type monoids, Cent. Eur. J. Math. 4 (2006), no. 3, 413-434.  https://doi.org/10.2478/s11533-006-0017-6
  41. P. Martin, Temperley-Lieb algebras for nonplanar statistical mechanics-the partition algebra construction, J. Knot Theory Ramifications 3 (1994), no. 1, 51-82. https://doi.org/10.1142/S0218216594000071 
  42. P. Martin and V. Mazorchuk, Partitioned binary relations, Math. Scand. 113 (2013), no. 1, 30-52.  https://doi.org/10.7146/math.scand.a-15480
  43. P. Martin and V. Mazorchuk, On the representation theory of partial Brauer algebras, Q. J. Math. 65 (2014), no. 1, 225-247.  https://doi.org/10.1093/qmath/has043
  44. V. Mazorchuk, On the structure of Brauer semigroup and its partial analogue, Problems in Algebra 13 (1998), 29-45. 
  45. V. Mazorchuk, All automorphisms of FP +(Sn) are inner, Semigroup Forum 60 (2000), no. 3, 486-490. https://doi.org/10.1007/s002339910040 
  46. V. Mazorchuk, Some homological properties of the category O, Pacific J. Math. 232 (2007), no. 2, 313-341. https://doi.org/10.2140/pjm.2007.232.313 
  47. V. Mazorchuk, Applications of the category of linear complexes of tilting modules associated with the category O, Algebr. Represent. Theory 12 (2009), no. 6, 489-512. https://doi.org/10.1007/s10468-008-9108-3 
  48. V. Mazorchuk, Simple modules over factorpowers, Acta Sci. Math. (Szeged) 75 (2009), no. 3-4, 467-485. 
  49. V. Mazorchuk and S. Srivastava, Jucys-Murphy elements and Grothendieck groups for generalized rook monoids, J. Comb. Algebra 6 (2022), no. 1-2, 185-222.  https://doi.org/10.4171/JCA/65
  50. V. Mazorchuk and S. Srivastava, Multiparameter colored partition category and the product of reduced Kronecker coefficients, Preprint arXiv:2204.13564. To appear in J. Pure Appl. Alg. 
  51. V. Mazorchuk and B. Steinberg, Double Catalan monoids, J. Algebraic Combin. 36 (2012), no. 3, 333-354.  https://doi.org/10.1007/s10801-011-0336-y
  52. V. Mazorchuk and C. Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Tran. Amer. Math. Soc. 357 (2005), no. 7, 2939-2973.  https://doi.org/10.1090/S0002-9947-04-03650-5
  53. V. Mazorchuk and C. Stroppel, On functors associated to a simple root, J. Algebra 314 (2007), no. 1, 97-128.  https://doi.org/10.1016/j.jalgebra.2007.03.015
  54. L. Paris, The proof of Birman's conjecture on singular braid monoids, Geom. Topol. 8 (2004), 1281-1300. https://doi.org/10.2140/gt.2004.8.1281 
  55. R. Plemmons and M. West, On the semigroup of binary relations, Pacific J. Math. 35 (1970), 743-753.  https://doi.org/10.2140/pjm.1970.35.743
  56. R. Rouquier, Categorification of sl2 and braid groups, in Trends in representation theory of algebras and related topics, 137-167, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, 2006. https://doi.org/10.1090/conm/406/07657 
  57. W. Soergel, Kategorie O, perverse Garben und Modulnuber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421-445. https://doi.org/10.2307/1990960 
  58. W. Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimodulnuber Polynomringen, J. Inst. Math. Jussieu 6 (2007), no. 3, 501-525. https://doi.org/10.1017/S1474748007000023 
  59. C. Stroppel, Category O: gradings and translation functors, J. Algebra 268 (2003), no. 1, 301-326. https://doi.org/10.1016/S0021-8693(03)00308-9 
  60. V. Vershinin, On the singular braid monoid, St. Petersburg Math. J. 21 (2010), no. 5, 693-704; translated from Algebra i Analiz 21 (2009), no. 5, 19-36. https://doi.org/10.1090/S1061-0022-2010-01112-9