On some integral representations of groups and global irreducibility

Document Type : Ischia Group Theory 2016


UWI, Mona, Kingston


Arithmetic aspects of integral representations of finite groups and their irreducibility are considered with a focus on globally irreducible representations and their generalizations to arithmetic rings. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. Let $K$ be a finite extension of the rational number field and $O_K$ the ring of integers of $K$. Let $G$ be a finite subgroup of $GL(2,K)$, the group of $(2 \times 2)$-matrices over $K$. We obtain some conditions on $K$ for $G$ to be conjugate to a subgroup of $GL(2,O_K)$.


Main Subjects

[1] M. Auslander and O. Goldman, Maximal Orders, Trans. Amer. Math. Soc., 97 (1960) 1–24.
[2] G. Cliff, J. Ritter and A. Weiss, Group representations and integrality, J. Reine Angew. Math., 426 (1992) 193–202.
[3] J. Conway, R. Curtis, S. Norton, R. Parker and R. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985.
[4] Ch. Curtis and I. Reiner, Represenation theory of finite groups and associative algebras, Pure and Applied Mathe-matics, 11, Interscience Publishers, a division of John Wiley Sons, New York-London, 1962.
[5] F. Destrempes, Deformations of Galois representations: the flat case, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), Canad. Math. Soc. Conf. Proc., Amer. Math. Soc., Providence, RI, 17 (1995) 209–231.
[6] F. Destrempes and D. Malinin, The maximal tamely ramified extension over a local field with applications to torsion points of formal groups or elliptic curves, a preprint.
[7] M. Deuring, Algebren, Springer-Verlag, Berlin-New York, 1968.
[8] J. D. Dixon and A. E. Zalesskii, Finite primitive linear groups of prime degree, J. London Math. Soc. (2), 57 (1998) 126–134, errata: in J. London Math. Soc. June, 77 (2008) 808–812.
[9] D. K. Faddeev, On generalized integral representations over Dedekind rings, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 227, 1995, Voprosy Teor. Predstav. Algebr i Grupp. 4, 113–118; English translation in J. Math. Sci., (New York), 89 (1998) 1154–1158.
[10] D. K. Faddeev, Tables of the fundamental unitary representations of the Fedorov groups, Trudy Mat. Inst. Steklov, 56 (1961) pp. 174.
[11] D. K. Faddeev, An introduction to the multiplicative theory of modules of integral representations, (Russian) Trudy Mat. Inst. Steklov, 80 (1965) 145–182.
[12] B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. (3), 2 (1990) 929–960.
[13] H. Hasse, Number Theory, Translated from the third German edition of 1969 by H. Gnter Zimmer, Akademie-Verlag, Berlin, 1979.
[14] H. Hasse, Zur Geschlechtertheorie in quadratischen Zahlkörpern, J. Math. Soc. Japan, 3 (1951) 45–51.
[15] H. Hasse,Über die Klassenzahl Abelscher Zahlkörper, Berlin, Akademie Verlag, Berlin, 1952.
[16] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Leipzig, Akademische Verlagsgesellschaft, 1923.
[17] D. Hilbert,Über den Dirichletischen Zahlkörper, Math. Ann., 45 (1894) 309-340.
[18] I. M. Isaacs, Character Theory of finite groups, Academic Press, 1976.
[19] V. V. Ishkhanov, B. B. Lurje and D. K. Faddeev, The Embedding Problem in Galois Theory, Translations of Mathe-matical Monographs, 165, American Mathematical Society, Providence, 1997.
[20] H. Jacobinski, Genera and decompositions of lattices over orders, Acta Math., 121 (1968) 1–29.
[21] M. Knebusch and W. Scharlau, Algebraic theory of quadratic forms: Generic methods and Pfister Forms, DMV Seminar 1, Birkhauser, 1980.
[22] D. Malinin and F. Van Oystaeyen, Realizability of two-dimensional linear groups over rings of integers of algebraic number fields, Algebr. Represent. Theory, 14 (2011) 201–211.
[23] D. Malinin, Integral representations of p-groups of given nilpotency class over local fields, St. Petersburg Math. J., 10 (1999) 45–52.
[24] G. A. Miller, H. F. Blichfeldt and L. E. Dickson, Theory and applications of nite groups, Dover, New York, 1916.
[25] F. Van Oystaeyen and A. E. Zalesski╦ś ─▒, Finite groups over arithmetic rings and globally irreducible representations, J. Algera, 215 (1999) 418–436.
[26] A. Pfister, Zur Darstellung von-1 als Summe von Quadraten in einem Korper, J. London Math. Soc., 40 (1965) 159–165.
[27] I. Reiner, Maximal Orders, Academic Press, 1975.
[28] J.-P. Serre, Cours d’arithmétique, Paris, 1970.
[29] J.-P. Serre, Three letters to Walter Feit on group representations and quaternions, J. Algebra, 319 (2008) 549–557.
[30] J.-P. Serre, Propriétés galoisiennes des points dordre fini des courbes elliptiques, Invent. Math., 15 (1972) 259–331.
[31] M. F. Vigneras, Arithmetique des algebres de quaterniones, Springer Lect. Notes in Math., 397, 1974.