Book Review
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MathSciNet review:
3364933
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Book Information:
Authors:
Martin W. Liebeck and
Gary M. Seitz
Title:
Unipotent and nilpotent classes in simple algebraic groups and Lie algebras
Additional book information:
Mathematical Surveys and Monographs, 180,
American Mathematical Society,
Providence, RI,
2012,
xii+380 pp.,
ISBN 978-0-8218-6920-8,
US$76.80
Jason Fulman and Robert Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012), no. 6, 3023–3070. MR 2888238, DOI 10.1090/S0002-9947-2012-05427-4
Robert M. Guralnick, Gunter Malle, and Pham Huu Tiep, Products of conjugacy classes in finite and algebraic simple groups, Adv. Math. 234 (2013), 618–652. MR 3003939, DOI 10.1016/j.aim.2012.11.005
Wim H. Hesselink, Nilpotency in classical groups over a field of characteristic $2$, Math. Z. 166 (1979), no. 2, 165–181. MR 525621, DOI 10.1007/BF01214043
D. F. Holt and N. Spaltenstein, Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic, J. Austral. Math. Soc. Ser. A 38 (1985), no. 3, 330–350. MR 779199
James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976, DOI 10.1090/surv/043
Jean-Philippe Anker and Bent Orsted (eds.), Lie theory, Progress in Mathematics, vol. 228, Birkhäuser Boston, Inc., Boston, MA, 2004. Lie algebras and representations. MR 2042688, DOI 10.1007/978-0-8176-8192-0
Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 114875, DOI 10.2307/2372999
John F. Kurtzke, Centers of centralizers in reductive algebraic groups, Comm. Algebra 19 (1991), no. 12, 3393–3410. MR 1135632, DOI 10.1080/00927879108824323
R. Lawther, Elements in reductive algebraic groups with abelian connected centralizers, J. Algebra 359 (2012), 1–34. MR 2914622, DOI 10.1016/j.jalgebra.2012.03.015
Ross Lawther, Martin Liebeck, and Gary Seitz, Outer unipotent classes in automorphism groups of simple algebraic groups, preprint.
Betty Lou, The centralizer of a regular unipotent element in a semi-simple algebraic group, Bull. Amer. Math. Soc. 74 (1968), 1144–1146. MR 231826, DOI 10.1090/S0002-9904-1968-12085-3
G. Lusztig, On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), no. 3, 201–213. MR 419635, DOI 10.1007/BF01403067
G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, III, Represent. Theory 16 (2012), 450–488. MR 2968566, DOI 10.1090/S1088-4165-2012-00422-8
G. Lusztig, Remarks on Springer’s representations, Represent. Theory 13 (2009), 391–400. MR 2540702, DOI 10.1090/S1088-4165-09-00358-6
Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups $E_{7}$ and $E_{8}$, Tokyo J. Math. 3 (1980), no. 2, 391–461. MR 605099, DOI 10.3836/tjm/1270473003
Gopal Prasad, Weakly-split spherical Tits systems in quasi-reductive groups, Amer. J. Math. 136 (2014), no. 3, 807–832. MR 3214277, DOI 10.1353/ajm.2014.0017
R. W. Richardson Jr., Conjugacy classes in Lie algebras and algebraic groups, Ann. of Math. (2) 86 (1967), 1–15. MR 217079, DOI 10.2307/1970359
Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
Robert Steinberg, Classes of elements of semisimple algebraic groups, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 277–284. MR 0238856
Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
John Williamson, On the Normal Forms of Linear Canonical Transformations in Dynamics, Amer. J. Math. 59 (1937), no. 3, 599–617. MR 1507266, DOI 10.2307/2371583
References
- Jason Fulman and Robert Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012), no. 6, 3023–3070. MR 2888238, DOI 10.1090/S0002-9947-2012-05427-4
- Robert M. Guralnick, Gunter Malle, and Pham Huu Tiep, Products of conjugacy classes in finite and algebraic simple groups, Adv. Math. 234 (2013), 618–652. MR 3003939, DOI 10.1016/j.aim.2012.11.005
- Wim H. Hesselink, Nilpotency in classical groups over a field of characteristic $2$, Math. Z. 166 (1979), no. 2, 165–181. MR 525621 (82d:14030), DOI 10.1007/BF01214043
- D. F. Holt and N. Spaltenstein, Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic, J. Austral. Math. Soc. Ser. A 38 (1985), no. 3, 330–350. MR 779199 (86g:17007)
- James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976 (97i:20057)
- Jens Carsten Jantzen and Karl-Hermann Neeb, Lie theory, Progress in Mathematics, vol. 228, Birkhäuser Boston, Inc., Boston, MA, 2004. Lie algebras and representations; Edited by Jean-Philippe Anker and Bent Orsted. MR 2042688 (2004j:22001), DOI 10.1007/978-0-8176-8192-0
- Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 0114875 (22 \#5693)
- John F. Kurtzke, Centers of centralizers in reductive algebraic groups, Comm. Algebra 19 (1991), no. 12, 3393–3410. MR 1135632 (92j:20043), DOI 10.1080/00927879108824323
- R. Lawther, Elements in reductive algebraic groups and abelian connected centralizers, J. Algebra 359 (2012), 1–34. MR 2914622, DOI 10.1016/j.jalgebra.2012.03.015
- Ross Lawther, Martin Liebeck, and Gary Seitz, Outer unipotent classes in automorphism groups of simple algebraic groups, preprint.
- Betty Lou, The centralizer of a regular unipotent element in a semi-simple algebraic group, Bull. Amer. Math. Soc. 74 (1968), 1144–1146. MR 0231826 (38 \#154)
- G. Lusztig, On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), no. 3, 201–213. MR 0419635 (54 \#7653)
- G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, III, Represent. Theory 16 (2012), 450–488. MR 2968566, DOI 10.1090/S1088-4165-2012-00422-8
- G. Lusztig, Remarks on Springer’s representations, Represent. Theory 13 (2009), 391–400. MR 2540702 (2011a:17010), DOI 10.1090/S1088-4165-09-00358-6
- Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups $E_{7}$ and $E_{8}$, Tokyo J. Math. 3 (1980), no. 2, 391–461. MR 605099 (82m:20046), DOI 10.3836/tjm/1270473003
- Gopal Prasad, Weakly-split spherical Tits systems in quasi-reductive groups, Amer. J. Math. 136 (2014), no. 3, 807–832. MR 3214277, DOI 10.1353/ajm.2014.0017
- R. W. Richardson Jr., Conjugacy classes in Lie algebras and algebraic groups, Ann. of Math. (2) 86 (1967), 1–15. MR 0217079 (36 \#173)
- Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610 (84a:14024)
- Robert Steinberg, Classes of elements of semisimple algebraic groups, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 277–284. MR 0238856 (39 \#216)
- Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728 (37 \#6288)
- John Williamson, On the Normal Forms of Linear Canonical Transformations in Dynamics, Amer. J. Math. 59 (1937), no. 3, 599–617. MR 1507266, DOI 10.2307/2371583
Review Information:
Reviewer:
Robert M. Guralnick
Affiliation:
Department of Mathematics, University of Southern California
Email:
guralnic@usc.edu
Journal:
Bull. Amer. Math. Soc.
52 (2015), 353-356
DOI:
https://doi.org/10.1090/S0273-0979-2014-01478-8
Published electronically:
December 11, 2014
Review copyright:
© Copyright 2014
American Mathematical Society
