Symmetric group
9
Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or
greater than n then by Maschke's theorem the group algebra KS
n
is semisimple. In these cases the irreducible
representations defined over the integers give the complete set of irreducible representations (after reduction modulo
the characteristic if necessary).
However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this
context it is more usual to use the language of modules rather than representations. The representation obtained from
an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be
irreducible. The modules so constructed are called Specht modules, and every irreducible does arise inside some such
module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For
example, even their dimensions are not known in general.
The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as
one of the most important open problems in representation theory.
References
[1] Jacobson (2009), p. 31.
[2] Jacobson (2009), p. 32. Theorem 1.1.
• Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge
University Press, ISBN€978-0-521-65378-7
• Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, 163, Berlin, New
York: Springer-Verlag, ISBN€978-0-387-94599-6, MR1409812
• Jacobson, Nathan (2009), Basic algebra, 1 (2nd ed.), Dover, ISBN€978-0-486-47189-1.
• Kaloujnine, Léo (1948), "La structure des p-groupes de Sylow des groupes symétriques finis" (http:/ / www.
numdam. org/ item?id=ASENS_1948_3_65__239_0), Annales Scientifiques de l'École Normale Supérieure.
Troisième Série 65: 239–276, ISSN€0012-9593, MR0028834
• Kerber, Adalbert (1971), Representations of permutation groups. I, Lecture Notes in Mathematics, Vol. 240, 240,
Berlin, New York: Springer-Verlag, doi:10.1007/BFb0067943, MR0325752
• Liebeck, M.W.; Praeger, C.E.; Saxl, J. (1988), "On the O'Nan-Scott theorem for finite primitive permutation
groups", J. Austral. Math. Soc. 44: 389–396
• Nakaoka, Minoru (March 1961), "Homology of the Infinite Symmetric Group" (http:/ / www. jstor. org/ stable/
1970333), The Annals of Mathematics, 2 (Annals of Mathematics) 73 (2): 229–257, doi:10.2307/1970333
• Netto, E. (1882) (in German), Substitutionentheorie und ihre Anwendungen auf die Algebra., Leipzig. Teubner,
JFM€14.0090.01
• Scott, W.R. (1987), Group Theory, New York: Dover Publications, pp.€45–46, ISBN€978-0-486-65377-8
• Schur, Issai (1911), "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene
lineare Substitutionen", Journal für die reine und angewandte Mathematik 139: 155–250
• Schreier, J.; Ulam, Stanislaw (1936), "Über die Automorphismen der Permutationsgruppe der natürlichen
Zahlenfolge." (http:/ / matwbn. icm. edu. pl/ ksiazki/ fm/ fm28/ fm28128. pdf) (in German), Fundam. Math. 28:
258–260, Zbl:€0016.20301
External links
• Marcus du Sautoy: Symmetry, reality's riddle (http:/ / www. ted. com/ talks/
marcus_du_sautoy_symmetry_reality_s_riddle. html) (video of a talk)