Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generalized nil-Coxeter algebras over discrete complex reflection groups

Author: Apoorva Khare
Journal: Trans. Amer. Math. Soc. 370 (2018), 2971-2999
MSC (2010): Primary 20F55; Secondary 20F05, 20C08
Published electronically: November 28, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the ``usual'' nil-Coxeter algebras: a novel $ 2$-parameter type $ A$ family that we call $ NC_A(n,d)$. We explore several combinatorial properties of $ NC_A(n,d)$, including its Coxeter word basis, length function, and Hilbert-Poincaré series, and show that the corresponding generalized Coxeter group is not a flat deformation of $ NC_A(n,d)$. These algebras yield symmetric semigroup module categories that are necessarily not monoidal; we write down their Tannaka-Krein duality.

Further motivated by the Broué-Malle-Rouquier (BMR) freeness conjecture [J. Reine Angew. Math. 1998], we define generalized nil-Coxeter algebras $ NC_W$ over all discrete real or complex reflection groups $ W$, finite or infinite. We provide a complete classification of all such algebras that are finite dimensional. Remarkably, these turn out to be either the usual nil-Coxeter algebras or the algebras $ NC_A(n,d)$. This proves as a special case--and strengthens--the lack of equidimensional nil-Coxeter analogues for finite complex reflection groups. In particular, generic Hecke algebras are not flat deformations of $ NC_W$ for $ W$ complex.

References [Enhancements On Off] (What's this?)

  • [AnSc] Nicolás Andruskiewitsch and Hans-Jürgen Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2) 171 (2010), no. 1, 375-417. MR 2630042,
  • [As] Joachim Assion, A proof of a theorem of Coxeter, C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 1, 41-44. MR 511837
  • [Ba] Tathagata Basak, On Coxeter diagrams of complex reflection groups, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4909-4936. MR 2922614,
  • [BSS] Chris Berg, Franco Saliola, and Luis Serrano, Pieri operators on the affine nilCoxeter algebra, Trans. Amer. Math. Soc. 366 (2014), no. 1, 531-546. MR 3118405,
  • [BGG] Joseph N. Bernstein, Israel M. Gelfand, and Sergei I. Gelfand, Schubert cells, and the cohomology of the spaces $ G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3-26. MR 0429933
  • [BS] Joseph Bernstein and Ossip Schwarzman, Complex crystallographic Coxeter groups and affine root systems, J. Nonlinear Math. Phys. 13 (2006), no. 2, 163-182. MR 2238956,
  • [Be] David Bessis, Zariski theorems and diagrams for braid groups, Invent. Math. 145 (2001), no. 3, 487-507. MR 1856398,
  • [BMR1] Michel Broué, Gunter Malle, and Raphaël Rouquier, On complex reflection groups and their associated braid groups, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 1-13. MR 1357192
  • [BMR2] Michel Broué, Gunter Malle, and Raphaël Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127-190. MR 1637497
  • [Ca] Roger W. Carter, Representation theory of the 0-Hecke algebra, J. Algebra 104 (1986), no. 1, 89-103. MR 865891,
  • [Ch] Eirini Chavli, The Broué-Malle-Rouquier conjecture for the exceptional groups of rank 2, Ph.D. thesis, 2016.
  • [Coh] Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379-436. MR 0422448
  • [Cox1] Harold S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588-621. MR 1503182,
  • [Cox2] Harold S. M. Coxeter,Factor groups of the braid group, Proceedings of the 4th Canadian Mathematical Congress (Banff, 1957), University of Toronto Press, 95-122, 1959.
  • [Dr] Vladimir G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69-70. MR 831053
  • [Et] Pavel Etingof,Proof of the Broué-Malle-Rouquier conjecture in characteristic zero (after I. Losev and I. Marin-G. Pfeiffer), Arnold Mathematical Journal, 3 (2017), no. 3, 445-449.
  • [EG] Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243-348. MR 1881922,
  • [ES] Pavel Etingof and Olivier Schiffmann, Lectures on quantum groups, Lectures in Mathematical Physics, International Press, Boston, MA, 1998. MR 1698405
  • [FS] Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196-207. MR 1265793,
  • [GHV] Matias Graña, István Heckenberger, and Leandro Vendramin, Nichols algebras of group type with many quadratic relations, Adv. Math. 227 (2011), no. 5, 1956-1989. MR 2803792,
  • [HV1] István Heckenberger and Leandro Vendramin, A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 2, 299-356. MR 3605018,
  • [HV2] István Heckenberger and Leandro Vendramin, The classification of Nichols algebras over groups with finite root system of rank two, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 7, 1977-2017. MR 3656477,
  • [Hug1] Mervyn C. Hughes, Complex reflection groups, Comm. Algebra 18 (1990), no. 12, 3999-4029. MR 1084439,
  • [Hug2] Mervyn C. Hughes, Extended root graphs for complex reflection groups, Comm. Algebra 27 (1999), no. 1, 119-148. MR 1668224,
  • [Hum] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
  • [Kha] Apoorva Khare, Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property, Groups, rings, group rings, and Hopf algebras, Contemp. Math., vol. 688, Amer. Math. Soc., Providence, RI, 2017, pp. 139-168. MR 3649174
  • [Kho] Mikhail Khovanov, Nilcoxeter algebras categorify the Weyl algebra, Comm. Algebra 29 (2001), no. 11, 5033-5052. MR 1856929,
  • [KL] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347. MR 2525917,
  • [KK] Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of $ G/P$ for a Kac-Moody group $ G$, Adv. Math. 62 (1986), no. 3, 187-237. MR 866159,
  • [Ko] David W. Koster, Complex reflection groups, ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)-The University of Wisconsin - Madison. MR 2625485
  • [LS] Alain Lascoux and Marcel-Paul Schützenberger, Fonctorialité des polynômes de Schubert, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 585-598. MR 1000001,
  • [LT] Gustav I. Lehrer and Donald E. Taylor, Unitary reflection groups, Australian Mathematical Society Lecture Series, vol. 20, Cambridge University Press, Cambridge, 2009. MR 2542964
  • [Lo] Ivan Losev, Finite-dimensional quotients of Hecke algebras, Algebra Number Theory 9 (2015), no. 2, 493-502. MR 3320850,
  • [Lu] George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599-635. MR 991016,
  • [Mal] Gunter Malle, Presentations for crystallographic complex reflection groups, Transform. Groups 1 (1996), no. 3, 259-277. MR 1417713,
  • [Mar] Ivan Marin, The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group $ G_{26}$, J. Pure Appl. Algebra 218 (2014), no. 4, 704-720. MR 3133700,
  • [MP] Ivan Marin and Götz Pfeiffer, The BMR freeness conjecture for the 2-reflection groups, Math. Comp. 86 (2017), no. 306, 2005-2023. MR 3626546,
  • [No] P. N. Norton, 0-Hecke algebras, J. Austral. Math. Soc. Ser. A 27 (1979), no. 3, 337-357. MR 532754
  • [ORS] Peter Orlik, Victor Reiner, and Anne V. Shepler, The sign representation for Shephard groups, Math. Ann. 322 (2002), no. 3, 477-492. MR 1895703,
  • [Pop1] Vladimir L. Popov, Discrete complex reflection groups, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, vol. 15, Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, 1982. MR 645542
  • [Pop2] Vladimir L. Popov, personal communication, Feb. 2016.
  • [ReS] Victor Reiner and Anne V. Shepler, Invariant derivations and differential forms for reflection groups, preprint, arXiv:1612.01031.
  • [ST] Geoffrey C. Shephard and John A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274-304. MR 0059914
  • [SW1] Anne V. Shepler and Sarah Witherspoon, A Poincaré-Birkhoff-Witt theorem for quadratic algebras with group actions, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6483-6506. MR 3267016,
  • [SW2] Anne V. Shepler and Sarah Witherspoon, Poincaré-Birkhoff-Witt theorems, Commutative algebra and noncommutative algebraic geometry. Vol. I, Math. Sci. Res. Inst. Publ., vol. 67, Cambridge Univ. Press, New York, 2015, pp. 259-290. MR 3525474

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F55, 20F05, 20C08

Retrieve articles in all journals with MSC (2010): 20F55, 20F05, 20C08

Additional Information

Apoorva Khare
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India

Keywords: Complex reflection group, generalized Coxeter group, generalized nil-Coxeter algebra, length function
Received by editor(s): April 6, 2017
Received by editor(s) in revised form: May 30, 2017
Published electronically: November 28, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society