On projective representations of finite wreath products

Durbin, John R.; Farmer, K. Bolling

doi:10.1090/S0025-5718-1977-0453855-4

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
DOI: https://doi.org/10.1090/S0025-5718-1977-0453855-4
MathSciNet review: 0453855
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

[1] N. B. Backhouse, Projective representations of space groups. II. Factor systems, Quart. J. Math. Oxford Ser. (2) 21 (1970), 277–295. MR 281803, https://doi.org/10.1093/qmath/21.3.277
[2] N. B. Backhouse, Projective representations of space groups. III. Symmorphic space groups, Quart. J. Math. Oxford Ser. (2) 22 (1971), 277–290. MR 292947, https://doi.org/10.1093/qmath/22.2.277
[3] Norman Blackburn, Some homology groups of wreathe products, Illinois J. Math. 16 (1972), 116–129. MR 0291294
[4] 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
[5] John D. Dixon, Computing irreducible representations of groups, Math. Comp. 24 (1970), 707–712. MR 280611, https://doi.org/10.1090/S0025-5718-1970-0280611-6
[6] John R. Durbin, On locally compact wreath products, Pacific J. Math. 57 (1975), no. 1, 99–107. MR 376950
[7] R. FRUCHT, "Über die Darstellung endlicher abelscher Gruppen durch Kollineationen," J. Reine Angew. Math., v. 166, 1931, pp. 16-29.
[8] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
[9] Adalbert Kerber, Zur Darstellungstheorie von Kranzprodukten, Canadian J. Math. 20 (1968), 665–672 (German). MR 233902, https://doi.org/10.4153/CJM-1968-064-6
[10] Adalbert Kerber, Representations of permutation groups. I, Lecture Notes in Mathematics, Vol. 240, Springer-Verlag, Berlin-New York, 1971. MR 0325752
[11] 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
[12] George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, https://doi.org/10.1007/BF02392428
[13] È. M. Žmud′, Symplectic geometries and projective representations of finite abelian groups, Mat. Sb. (N.S.) 87(129) (1972), 3–17 (Russian). MR 0292963