On projective representations of finite wreath products
Authors: John R. Durbin and K. Bolling Farmer
Journal: Math. Comp. 31 (1977), 527-535
MSC: Primary 20C25
MathSciNet review: 0453855
Full-text PDF Free Access
Abstract: The theory of induced projective representations is applied to finite wreath products, yielding algorithms which add to the collection of groups for which projective representations can be computed systematically. For finite Abelian and Abelian-wreath-cyclic groups, the factor sets are determined explicitly by establishing a one-to-one correspondence between certain lower triangular matrices and the inequivalent factor sets of these two classes of groups. This correspondence is used to determine the number and degrees of the inequivalent, irreducible projective representations.
- N. B. Backhouse, Projective representations of space groups. II. Factor systems, Quart. J. Math. Oxford Ser. (2) 21 (1970), 277–295. MR 281803, DOI https://doi.org/10.1093/qmath/21.3.277
- N. B. Backhouse, Projective representations of space groups. III. Symmorphic space groups, Quart. J. Math. Oxford Ser. (2) 22 (1971), 277–290. MR 292947, DOI https://doi.org/10.1093/qmath/22.2.277
- Norman Blackburn, Some homology groups of wreathe products, Illinois J. Math. 16 (1972), 116–129. MR 291294
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979
- John D. Dixon, Computing irreducible representations of groups, Math. Comp. 24 (1970), 707–712. MR 280611, DOI https://doi.org/10.1090/S0025-5718-1970-0280611-6
- John R. Durbin, On locally compact wreath products, Pacific J. Math. 57 (1975), no. 1, 99–107. MR 376950 R. FRUCHT, "Über die Darstellung endlicher abelscher Gruppen durch Kollineationen," J. Reine Angew. Math., v. 166, 1931, pp. 16-29.
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- Adalbert Kerber, Zur Darstellungstheorie von Kranzprodukten, Canadian J. Math. 20 (1968), 665–672 (German). MR 233902, DOI https://doi.org/10.4153/CJM-1968-064-6
- Adalbert Kerber, Representations of permutation groups. I, Lecture Notes in Mathematics, Vol. 240, Springer-Verlag, Berlin-New York, 1971. MR 0325752
- Laurens Jansen and Michael Boon, Theory of finite groups. Applications in physics. (Symmetry groups of quantum mechanical systems.), North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR 0223442
- George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI https://doi.org/10.1007/BF02392428
- È. M. Žmud′, Symplectic geometries and projective representations of finite abelian groups, Mat. Sb. (N.S.) 87(129) (1972), 3–17 (Russian). MR 0292963
N. B. BACKHOUSE, "Projective representations of space groups. II, Factor systems," Quart. J. Math. Oxford Ser. (2), v. 21, 1970, pp. 277-295. MR 43 #7517.
N. B. BACKHOUSE, "Projective representations of space groups. III, Symmorphic space groups," Quart. J. Math. Oxford Ser. (2), v. 22, 1971, pp. 277-290. MR 45 #2028.
N. BLACKBURN, "Some homology groups of wreathe products," Illinois J. Math., v. 16, 1972, pp. 116-129. MR 45 #388.
C. W. CURTIS & I. REINER, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York and London, 1962. MR 26 #2519.
J. D. DIXON, "Computing irreducible representations of groups," Math. Comp., v. 24, 1970, pp. 707-712. MR 43 #6330.
J. R. DURBIN, "On locally compact wreath products," Pacific J. Math., v. 57, 1975, pp. 99-107. MR 51 #13125.
R. FRUCHT, "Über die Darstellung endlicher abelscher Gruppen durch Kollineationen," J. Reine Angew. Math., v. 166, 1931, pp. 16-29.
B. HUPPERT, Endliche Gruppen. I, Springer-Verlag, Berlin and New York, 1967. MR 37 #302.
A. KERBER, "Zur Darstellungstheorie von Kranzprodukten," Canad. J. Math., v. 20, 1968, pp. 665-672. MR 38 #2223.
A. KERBER, Representations of Permutation Groups. I, Lecture Notes in Math., vol. 240, Springer-Verlag, Berlin and New York, 1971. MR 48 #4098.
L. JANSEN & M. BOON, Theory of Finite Groups. Applications in Physics, North-Holland, Amsterdam; Interscience, New York, 1967. MR 36 #6490.
G. W. MACKEY, "Unitary representations of group extensions. I," Acta Math., v. 99, 1958, pp. 265-311. MR 20 #4789.
È. M. ŽMUD’, "Symplectic geometric and projective representations of finite abelian groups," Mat. Sb. (N.S.), v. 87 (129), 1972, pp. 3-17 = Math. USSR Sb.,v. 16, 1972, pp. 1-16. MR 45 #2044.
Retrieve articles in Mathematics of Computation with MSC: 20C25
Retrieve articles in all journals with MSC: 20C25
Keywords: Wreath products, Abelian groups, projective representations, factor sets, induced representations, algorithm
Article copyright: © Copyright 1977 American Mathematical Society