Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Decompositions of small tensor powers and Larsen's conjecture


Authors: Robert M. Guralnick and Pham Huu Tiep
Journal: Represent. Theory 9 (2005), 138-208
MSC (2000): Primary 20C15, 20C20, 20C33, 20C34, 20G05, 20G40
DOI: https://doi.org/10.1090/S1088-4165-05-00192-5
Published electronically: February 2, 2005
MathSciNet review: 2123127
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify all pairs $(G,V)$ with $G$ a closed subgroup in a classical group $\mathcal G$ with natural module $V$ over $\mathbb C$, such that $\mathcal G$ and $G$have the same composition factors on $V^{\otimes k}$ for a fixed $k\in \{2,3,4\}$. In particular, we prove Larsen's conjecture stating that for $\dim(V)>6$ and $k=4$ there are no such $G$ aside from those containing the derived subgroup of $\mathcal G$. We also find all the examples where this fails for $\dim(V)\le 6$. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz's recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.


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

  • [A] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469-514. MR 0746539 (86a:20054)
  • [AGM] M. Aschbacher, R. M. Guralnick, and K. Magaard, (in preparation).
  • [BL] Y. Barnea and M. Larsen, Random generation in semisimple algebraic groups over local fields, J. Algebra 271 (2004), 1-10. MR 2022476 (2004k:60015)
  • [Atlas] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985. MR 0827219 (88g:20025)
  • [C] R. Carter, Finite Groups of Lie type: Conjugacy Classes and Complex Characters, Wiley, Chichester, 1985. MR 0794307 (87d:20060)
  • [D] P. Deligne, La conjecture de Weil II, Publ. Math. I.H.E.S. 52 (1980), 137-252. MR 0601520 (83c:14017)
  • [Di] J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199-205. MR 0251758 (40 #4985)
  • [Dyn] E. B. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Translations 6 (1960), 245-378.
  • [FH] W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991. MR 1153249 (93a:20069)
  • [G] R. M. Guralnick, Some applications of subgroup structure to probabilistic generation and covers of curves, in: Algebraic groups and their representations, Cambridge, 1997, pp. 301-320, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 517, Kluwer Acad. Publ., Dordrecht, 1998. MR 1670777 (2000d:20062)
  • [GLSS] R. M. Guralnick, M. W. Liebeck, J. Saxl, and A. Shalev, Random generation of finite simple groups, J. Algebra 219 (1999), 345-355. MR 1707675 (2000g:20139)
  • [GLS] R. M. Guralnick, F. Lübeck and A. Shalev, 0,1 generation laws for Chevalley groups, (preprint).
  • [GMST] R. M. Guralnick, K. Magaard, J. Saxl, and Pham Huu Tiep, Cross characteristic representations of symplectic and unitary groups, J. Algebra 257 (2002), 291-347. MR 1947325 (2004b:20022)
  • [GS] R. M. Guralnick and J. Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), 519-571. MR 2009321
  • [GT1] R. M. Guralnick and Pham Huu Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), 4969-5023. MR
  • [GT2] R. M. Guralnick and Pham Huu Tiep, The non-coprime $k(GV)$problem, (submitted).
  • [He] C. Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, II, J. Algebra 93 (1985), 151-164. MR 0780488 (86k:20046)
  • [H] G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, RWTH Aachen, Germany, 1993.
  • [HM] G. Hiss and G. Malle, Corrigenda: Low-dimensional representations of quasi-simple groups, LMS J. Comput. Math. 5 (2002), 95-126. MR 1942256 (2003k:20010)
  • [Him] F. Himstedt, Notes on the 2-modular irreducible representations of Steinberg's triality groups $\hspace{0.5mm}^{3}\hspace*{-0.2mm}D_{4}(q)$, $q$ odd, (preprint).
  • [Ho] C. Hoffman, Projective representations for some exceptional finite groups of Lie type, in: Modular Representation Theory of Finite Groups, M. J. Collins, B. J. Parshall, L. L. Scott, eds., Walter de Gruyter, Berlin et al., 2001, 223-230. MR 1889347 (2003a:20019)
  • [Hu] J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, N.Y. et al., 1975. MR 0396773 (53 #633)
  • [J] G. James, On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cam. Phil. Soc. 94 (1983), 417-424. MR 0720791 (86c:20018)
  • [Jans] C. Jansen, Minimal degrees of faithful representations for sporadic simple groups and their covering groups, (in preparation).
  • [JLPW] C. Jansen, K. Lux, R. A. Parker, and R. A. Wilson, An atlas of Brauer Characters, Oxford University Press, Oxford, 1995. MR 1367961 (96k:20016)
  • [Jan] J. C. Jantzen, Representations of Algebraic Groups, Academic Press, Boston, 1987. MR 0899071 (89c:20001)
  • [Ka1] N. Katz, Larsen's alternative, moments, and the monodromy of Lefschetz pencils, in: Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins University Press, pp. 521-560. MR 2058618
  • [Ka2] N. Katz, Moments, Monodromy, and Perversity: a Diophantine Perspective, Annals of Math. Study, Princeton Univ. Press (to appear).
  • [KaL] W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67-87. MR 1065213 (91j:20041)
  • [KL] P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. no. 129, Cambridge University Press, 1990. MR 1057341 (91g:20001)
  • [Kle] P. B. Kleidman, The Subgroup Structure of Some Finite Simple Groups, Ph.D. Thesis, Trinity College, 1986.
  • [KT] A. S. Kleshchev and Pham Huu Tiep, On restrictions of modular spin representations of symmetric and alternating groups, Trans. Amer. Math. Soc. 356 (2004), 1971-1999. MR 2031049 (2005a:20020)
  • [LS] V. Landazuri and G. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418-443. MR 0360852 (50 #13299)
  • [Lars] M. Larsen, A characterization of classical groups by invariant theory, (preprint, mid-1990's).
  • [LP] M. Larsen and R. Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. (to appear).
  • [LST] W. Lempken, B. Schröder, Pham Huu Tiep, Symmetric squares, spherical designs, and lattice minima, J. Algebra, 240 (2001), 185-208. MR 1830550 (2002e:05035)
  • [Li] M. W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. 54 (1987), 477-516. MR 0879395 (88m:20004)
  • [LiS] M. W. Liebeck and G. M. Seitz, Subgroups of exceptional algebraic groups which are irreducible on an adjoint or minimal module, J. Group Theory 7 (2004), 347-372. MR 2063402
  • [LSh] M. W. Liebeck and A. Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103-113. MR 1338320 (96h:20116)
  • [Lu1] F. Lübeck, Smallest degrees of representations of exceptional groups of Lie type, Comm. Algebra 29 (2001), 2147-2169. MR 1837968 (2002g:20029)
  • [Lu2] F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135-169. MR 1901354 (2003e:20013)
  • [Mag] K. Magaard, On the irreducibility of alternating powers and symmetric squares, Arch. Math. 63 (1994), 211-215. MR 1287249 (95k:20067)
  • [MM] K. Magaard and G. Malle, Irreducibility of alternating and symmetric squares, Manuscripta Math. 95 (1998), 169-180. MR 603309 (99a:20011)
  • [MMT] K. Magaard, G. Malle and Pham Huu Tiep, Irreducibility of tensor squares, symmetric squares, and alternating squares, Pacific J. Math. 202 (2002), 379-427. MR 1888172 (2002m:20025)
  • [MT1] K. Magaard and Pham Huu Tiep, Irreducible tensor products of representations of finite quasi-simple groups of Lie type, in: Modular Representation Theory of Finite Groups, M. J. Collins, B. J. Parshall, L. L. Scott, eds., Walter de Gruyter, Berlin et al., 2001, pp. 239-262. MR 1889349 (2002m:20024)
  • [MT2] K. Magaard and Pham Huu Tiep, The classes $\mathcal{C}_{6}$ and $\mathcal{C}_{7}$ of maximal subgroups of finite classical groups, (in preparation).
  • [Mal] G. Malle, Almost irreducible tensor squares, Comm. Algebra 27 (1999), 1033-1051. MR 1669100 (99m:20007)
  • [NRS] G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, Des. Codes Cryptogr. 24 (2001), 99-121. MR 1845897 (2002h:20066)
  • [Pre] A. A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Math. USSS-Sb. 61 (1988), 167-183. MR 0905003 (88h:20051)
  • [Sch] M. Schönert and others, GAP-Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 3rd edn., (1993-1997).
  • [S1] G. Seitz, Some representations of classical groups, J. London Math. Soc. 10 (1975), 115-120. MR 0369556 (51 #5789)
  • [S2] G. M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365. MR 0888704 (88g:20092)
  • [SZ] G. Seitz and A. E. Zalesskii, On the minimal degrees of projective representations of the finite Chevalley groups, II, J. Algebra 158 (1993), 233-243. MR 1223676 (94h:20017)
  • [St] R. Steinberg, Generators for simple groups, Canad. J. Math. 14 (1962), 277-283. MR 143801 (26 #1351)
  • [T] Pham Huu Tiep, Dual pairs and low-dimensional representations of finite classical groups, (in preparation).
  • [TZ1] Pham Huu Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), 2093-2167. MR 1386030 (97f:20018)
  • [TZ2] Pham Huu Tiep and A. E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), 130-165. MR 1449955 (99d:20074)
  • [Wa1] D. Wales, Some projective representations of $S_{n}$, J. Algebra 61 (1979), 37-57. MR 0554850 (81f:20015)
  • [Wa2] D. Wales, Quasiprimitive linear groups with quadratic elements, J. Algebra 245 (2001), 584-606. MR 1863893 (2002j:20098)
  • [We] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, 1946.
  • [Zs] K. Zsigmondy, Zur Theorie der Potenzreste, Monath. Math. Phys. 3 (1892), 265-284.

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20C15, 20C20, 20C33, 20C34, 20G05, 20G40

Retrieve articles in all journals with MSC (2000): 20C15, 20C20, 20C33, 20C34, 20G05, 20G40


Additional Information

Robert M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
Email: guralnic@math.usc.edu

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: tiep@math.ufl.edu

DOI: https://doi.org/10.1090/S1088-4165-05-00192-5
Received by editor(s): March 31, 2003
Received by editor(s) in revised form: December 8, 2018, and December 15, 2004
Published electronically: February 2, 2005
Additional Notes: The authors gratefully acknowledge the support of the NSF (grants DMS-0236185 and DMS-0070647), and of the NSA (grant H98230-04-0066)
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society