Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Invariants of pairs in $ SL(4, \mathbb{C})$ and $ SU(3, 1)$


Authors: Krishnendu Gongopadhyay and Sean Lawton
Journal: Proc. Amer. Math. Soc. 145 (2017), 4703-4715
MSC (2010): Primary 14D20, 20H10; Secondary 20C15, 14L30, 20E05, 30F40, 15B57, 51M10
DOI: https://doi.org/10.1090/proc/13638
Published electronically: June 16, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe a minimal global coordinate system of order 30 on the $ \textup {SL}(4,\mathbb{C})$-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in $ \textup {SU}(3,1)$.


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

  • [BH93] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
  • [CFLO14] Ana Casimiro, Carlos Florentino, Sean Lawton, and André Oliveira, Topology of moduli spaces of free group representations in real reductive groups, Forum Math. 28 (2016), no. 2, 275-294. MR 3466569, https://doi.org/10.1515/forum-2014-0049
  • [DLS09] Vesselin Drensky and Roberto La Scala, Defining relations of low degree of invariants of two $ 4\times 4$ matrices, Internat. J. Algebra Comput. 19 (2009), no. 1, 107-127. MR 2494472, https://doi.org/10.1142/S0218196709004981
  • [DS06] V. Drensky and L. Sadikova, Generators of invariants of two $ 4\times 4$ matrices, C. R. Acad. Bulgare Sci. 59 (2006), no. 5, 477-484. MR 2243395
  • [Dok07] Dragomir Ž. oković, Poincaré series of some pure and mixed trace algebras of two generic matrices, J. Algebra 309 (2007), no. 2, 654-671. MR 2303199, https://doi.org/10.1016/j.jalgebra.2006.09.018
  • [FL09] Carlos Florentino and Sean Lawton, The topology of moduli spaces of free group representations, Math. Ann. 345 (2009), no. 2, 453-489. MR 2529483, https://doi.org/10.1007/s00208-009-0362-4
  • [FL12] Carlos Florentino and Sean Lawton, Singularities of free group character varieties, Pacific J. Math. 260 (2012), no. 1, 149-179. MR 3001789, https://doi.org/10.2140/pjm.2012.260.149
  • [FL14] Carlos Florentino and Sean Lawton, Topology of character varieties of Abelian groups, Topology Appl. 173 (2014), 32-58. MR 3227204, https://doi.org/10.1016/j.topol.2014.05.009
  • [FLR17] Carlos Florentino, Sean Lawton, and Daniel Ramras,
    Homotopy groups of free group character varieties,
    Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5) 17 (2017), no. 1, 143-185, DOI 10.2422/2036-2145.201510_004.
  • [Fri96] R. Fricke,
    über die theorie der automorphen modulgrupper,
    Nachr. Akad.Wiss. Göttingen (1896), 91-101.
  • [Gol99] William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. MR 1695450
  • [Gol09] William M. Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 611-684. MR 2497777, https://doi.org/10.4171/055-1/16
  • [GP15] K. Gongopadhyay and S. Parsad, On Fenchel-Nielsen coordinates of surface group representations into $ \mathrm {SU}(3,1)$, Mathematical Proceedings of the Cambridge Philosophical Society, 1-23, DOI 10.1017/S0305004117000159.
  • [GPP15] Krishnendu Gongopadhyay, John R. Parker, and Shiv Parsad, On the classifications of unitary matrices, Osaka J. Math. 52 (2015), no. 4, 959-991. MR 3426624
  • [Law06] Sean Lawton, SL(3, C)-character varieties and RP2 -structures on a trinion, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-University of Maryland, College Park 2006. MR 2709451
  • [Law07] Sean Lawton, Generators, relations and symmetries in pairs of $ 3\times 3$ unimodular matrices, J. Algebra 313 (2007), no. 2, 782-801. MR 2329569, https://doi.org/10.1016/j.jalgebra.2007.01.003
  • [Law08] Sean Lawton, Minimal affine coordinates for $ {\rm SL}(3,\mathbb{C})$ character varieties of free groups, J. Algebra 320 (2008), no. 10, 3773-3810. MR 2457722, https://doi.org/10.1016/j.jalgebra.2008.06.031
  • [Law09] Sean Lawton, Obtaining the one-holed torus from pants: duality in an $ {\rm SL}(3,\mathbb{C})$-character variety, Pacific J. Math. 242 (2009), no. 1, 131-142. MR 2525506, https://doi.org/10.2140/pjm.2009.242.131
  • [Law10] Sean Lawton, Algebraic independence in $ {\rm SL}(3,\mathbb{C})$ character varieties of free groups, J. Algebra 324 (2010), no. 6, 1383-1391. MR 2671811, https://doi.org/10.1016/j.jalgebra.2010.06.023
  • [Par12] John R. Parker, Traces in complex hyperbolic geometry, Geometry, topology and dynamics of character varieties, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 23, World Sci. Publ., Hackensack, NJ, 2012, pp. 191-245. MR 2987619, https://doi.org/10.1142/9789814401364_0006
  • [PP08] John R. Parker and Ioannis D. Platis, Complex hyperbolic Fenchel-Nielsen coordinates, Topology 47 (2008), no. 2, 101-135. MR 2415771, https://doi.org/10.1016/j.top.2007.08.001
  • [Pro76] C. Procesi, The invariant theory of $ n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306-381. MR 0419491, https://doi.org/10.1016/0001-8708(76)90027-X
  • [PS00] Józef H. Przytycki and Adam S. Sikora, On skein algebras and $ {\rm Sl}_2({\bf C})$-character varieties, Topology 39 (2000), no. 1, 115-148. MR 1710996, https://doi.org/10.1016/S0040-9383(98)00062-7
  • [Raz74] Ju. P. Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723-756 (Russian). MR 0506414
  • [Sik13] Adam S. Sikora, Generating sets for coordinate rings of character varieties, J. Pure Appl. Algebra 217 (2013), no. 11, 2076-2087. MR 3057078, https://doi.org/10.1016/j.jpaa.2013.01.005
  • [Ter86] Yasuo Teranishi, The ring of invariants of matrices, Nagoya Math. J. 104 (1986), 149-161. MR 868442
  • [Ter87] Yasuo Teranishi, Linear Diophantine equations and invariant theory of matrices, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 259-275. MR 951208
  • [Vog89] H. Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Sup. (3) 6 (1889), 3-71 (French). MR 1508833
  • [Wen94] Zhi Xiong Wen, Relations polynomiales entre les traces de produits de matrices, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 2, 99-104 (French, with English and French summaries). MR 1260318
  • [Wil06] P. Will,
    Groupes libres, groupes triangulaires et tore épointé dans $ \mathrm {PU}(2, 1)$,
    Ph.D. thesis, University of Paris VI, 2006.
  • [Wil09] Pierre Will, Traces, cross-ratios and 2-generator subgroups of $ {\rm SU}(2,1)$, Canad. J. Math. 61 (2009), no. 6, 1407-1436. MR 2588430, https://doi.org/10.4153/CJM-2009-067-6

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14D20, 20H10, 20C15, 14L30, 20E05, 30F40, 15B57, 51M10

Retrieve articles in all journals with MSC (2010): 14D20, 20H10, 20C15, 14L30, 20E05, 30F40, 15B57, 51M10


Additional Information

Krishnendu Gongopadhyay
Affiliation: Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, S.A.S. Nagar, Sector 81, P. O. Manauli, Pin 140306, India
Email: krishnendug@gmail.com, krishnendu@iisermohali.ac.in

Sean Lawton
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, Virginia 22030
Email: slawton3@gmu.edu

DOI: https://doi.org/10.1090/proc/13638
Keywords: Character variety, $4\times 4$ invariants, complex hyperbolic space, two-generator subgroup, traces
Received by editor(s): March 3, 2016
Received by editor(s) in revised form: December 9, 2016
Published electronically: June 16, 2017
Communicated by: Michael Wolf
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society