On a counting formula of Djoković for elements of finite order in compact Lie groups
Authors:
F. Destrempes and A. Pianzola
Journal:
Proc. Amer. Math. Soc. 121 (1994), 943950
MSC:
Primary 22E40; Secondary 22C05
MathSciNet review:
1185259
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Abstract: Given a compact connected simple Lie group and a positive integer N relatively prime to the order of the Weyl group we give a counting formula for the number of conjugacy classes of elements x of order N in with the property that the Ncyclotonic field when viewed as a Galois extension of the field of characters of x has Galois group containing a fixed chosen cyclic group . The case recovers a formula, due to Djoković, which counts the number of conjugacy classes of elements of order dividing N in .
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W. Carter, Conjugacy classes in the Weyl group, Compositio
Math. 25 (1972), 1–59. MR 0318337
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Destrempes and A.
Pianzola, Elements of compact connected simple Lie groups with
prime power order and given field of characters, Geom. Dedicata
45 (1993), no. 2, 225–235. MR 1202101
(94e:22013), http://dx.doi.org/10.1007/BF01264522
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Dragomir
Ž. Djoković, On conjugacy classes of elements of
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(81h:20052), http://dx.doi.org/10.1090/S00029939198005745327
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Dragomir
Ž. Djoković, On conjugacy classes of elements of
finite order in complex semisimple Lie groups, J. Pure Appl. Algebra
35 (1985), no. 1, 1–13. MR 772157
(86h:22010), http://dx.doi.org/10.1016/00224049(85)90026X
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V. Moody and J.
Patera, Characters of elements of finite order in Lie groups,
SIAM J. Algebraic Discrete Methods 5 (1984), no. 3,
359–383. MR
752042 (86e:22023), http://dx.doi.org/10.1137/0605037
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Pianzola, On the arithmetic of the representation ring and elements
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no. 1, 1–33. MR 887189
(88h:22012), http://dx.doi.org/10.1016/00218693(87)901190
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Pianzola, On the regularity and rationality of certain elements of
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(1987), 40–48. MR 887398
(88h:22013), http://dx.doi.org/10.1515/crll.1987.377.40
 [PW]
A.
Pianzola and A.
Weiss, The rationality of elements of prime order in compact
connected simple Lie groups, J. Algebra 144 (1991),
no. 2, 510–521. MR 1140619
(92k:22014), http://dx.doi.org/10.1016/00218693(91)90119S
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Louis
Solomon, Invariants of finite reflection groups, Nagoya Math.
J. 22 (1963), 57–64. MR 0154929
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A. Springer, Regular elements of finite reflection groups,
Invent. Math. 25 (1974), 159–198. MR 0354894
(50 #7371)
 [Ctr]
 R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 159. MR 0318337 (47:6884)
 [DP]
 F. Destrempes and A. Pianzola, Elements of compact connected simple Lie groups with prime power order and given field of characters, Geom. Dedicata 45 (1993), 225235. MR 1202101 (94e:22013)
 [Djk1]
 D. Djoković, On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups, Proc. Amer. Math. Soc. 80 (1980), 181184. MR 574532 (81h:20052)
 [Djk2]
 , On the conjugacy classes of elements of finite order in complex semisimple Lie groups, J. Pure Appl. Algebra 35 (1985), 113. MR 772157 (86h:22010)
 [MPt]
 R. V. Moody and J. Patera, Characters of elements of finite order in Lie groups, SIAM J. Algebra Discrete Methods 5 (1984), 359383. MR 752042 (86e:22023)
 [Pzl1]
 A. Pianzola, On the arithmetic of the representation ring and elements of finite order in Lie groups, J. Algebra 108 (1987), 133. MR 887189 (88h:22012)
 [Pzl2]
 , On the rationality and regularity of certain elements of finite order in Lie groups, J. Reine Angew. Math. 377 (1987), 4048. MR 887398 (88h:22013)
 [PW]
 A. Pianzola and A. Weiss, The rationality of elements of prime order in compact connected simple Lie group, J. Algebra 144 (1991), 510521. MR 1140619 (92k:22014)
 [Slm]
 L. Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 5764. MR 0154929 (27:4872)
 [Spg]
 T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159198. MR 0354894 (50:7371)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199411852595
PII:
S 00029939(1994)11852595
Article copyright:
© Copyright 1994 American Mathematical Society
