Computing irreducible representations of groups
Author:
John D. Dixon
Journal:
Math. Comp. 24 (1970), 707712
MSC:
Primary 20.80; Secondary 65.00
MathSciNet review:
0280611
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Abstract: How can you find a complete set of inequivalent irreducible (ordinary) representations of a finite group? The theory is classical but, except when the group was very small or had a rather special structure, the actual computations were prohibitive before the advent of highspeed computers; and there remain practical difficulties even for groups of relatively small orders . The present paper describes three techniques to help solve this problem. These are: the reduction of a reducible unitary representation into its irreducible components; the construction of a complete set of irreducible unitary representations from a single faithful representation; and the calculation of the precise values of a group character from values which have only been computed approximately.
 [1]
C. Brott & J. Neubüser, A Program for the Calculation of Characters and Representations of Finite Groups, Proc. Conf. Comput. Algebra (Oxford 1967), Pergamon Press, New York, 1969.
 [2]
W.
Burnside, Theory of groups of finite order, Dover
Publications, Inc., New York, 1955. 2d ed. MR 0069818
(16,1086c)
 [3]
John
J. Cannon, Computers in group theory: A survey, Comm. ACM
12 (1969), 3–12. MR 0290613
(44 #7793)
 [4]
John
D. Dixon, Problems in group theory, Blaisdell Publishing Co.
Ginn and Co., Waltham, Mass.Toronto, Ont.London, 1967. MR 0218428
(36 #1514)
 [5]
John
D. Dixon, High speed computation of group characters, Numer.
Math. 10 (1967), 446–450. MR 0224726
(37 #325)
 [6]
Walter
Feit, Characters of finite groups, W. A. Benjamin, Inc., New
YorkAmsterdam, 1967. MR 0219636
(36 #2715)
 [7]
J. K. McKay, "A method for computing the simple character table of a finite group," in Computers in Mathematical Research, R. F. Churchhouse and J. C. Herz (Editors), NorthHolland, Amsterdam, 1968. MR 38 #1972.
 [8]
J. K. McKay, The Construction of the Character Table of a Finite Group from Generators and Relations, Proc. Conf. Comput. Algebra (Oxford 1967), Pergamon Press, New York, 1969.
 [9]
J. Neubüser, Investigations of Finite Groups on Computers, Proc. Conf. Comput. Algebra (Oxford 1967), Pergamon Press, New York, 1969.
 [10]
P. G. Rudd & E. R. Keown, The Computation of Irreducible Representations of Finite Groups of Order , Proc. Conf. Comput. Algebra (Oxford 1967) Pergamon Press, New York, 1969.
 [1]
 C. Brott & J. Neubüser, A Program for the Calculation of Characters and Representations of Finite Groups, Proc. Conf. Comput. Algebra (Oxford 1967), Pergamon Press, New York, 1969.
 [2]
 W. Burnside, Theory of Groups of Finite Order, 2nd ed., Dover. New York, 1955. MR 16, 1086. MR 0069818 (16:1086c)
 [3]
 J. Cannon, "Computers in group theory: a survey," Comm. ACM, v. 12, 1969, pp. 325. MR 0290613 (44:7793)
 [4]
 J. D. Dixon, Problems in Group Theory, Blaisdell, Waltham, Mass., 1967. MR 36 #1514. MR 0218428 (36:1514)
 [5]
 J. D. Dixon, "High speed computation of group characters," Numer. Math., v. 10, 1967, pp. 446450. MR 37 #325. MR 0224726 (37:325)
 [6]
 W. Feit, Characters of Finite Groups, Benjamin, New York, 1967. MR 36 #2715. MR 0219636 (36:2715)
 [7]
 J. K. McKay, "A method for computing the simple character table of a finite group," in Computers in Mathematical Research, R. F. Churchhouse and J. C. Herz (Editors), NorthHolland, Amsterdam, 1968. MR 38 #1972.
 [8]
 J. K. McKay, The Construction of the Character Table of a Finite Group from Generators and Relations, Proc. Conf. Comput. Algebra (Oxford 1967), Pergamon Press, New York, 1969.
 [9]
 J. Neubüser, Investigations of Finite Groups on Computers, Proc. Conf. Comput. Algebra (Oxford 1967), Pergamon Press, New York, 1969.
 [10]
 P. G. Rudd & E. R. Keown, The Computation of Irreducible Representations of Finite Groups of Order , Proc. Conf. Comput. Algebra (Oxford 1967) Pergamon Press, New York, 1969.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197002806116
PII:
S 00255718(1970)02806116
Keywords:
Computation of group representations,
computation of characters,
reduction of unitary representations,
irreducible components,
tensor products,
iterative processes,
finite Fourier analysis
Article copyright:
© Copyright 1970
American Mathematical Society
