## Algebra & Number Theory

### Complex group algebras of the double covers of the symmetric and alternating groups

#### Abstract

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n ≥ 5$ be an integer, $G$ a finite group, and let $Ân$ and $Ŝn±$ denote the double covers of $An$ and $Sn$, respectively. We prove that $ℂG≅ℂÂn$ if and only if $G≅Ân$, and $ℂG≅ℂŜn+≅ℂŜn−$ if and only if $G≅Ŝn+$ or $Ŝn−$. This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 3 (2015), 601-628.

Dates
Revised: 13 January 2015
Accepted: 23 February 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842313

Digital Object Identifier
doi:10.2140/ant.2015.9.601

Mathematical Reviews number (MathSciNet)
MR3340546

Zentralblatt MATH identifier
1321.20011

#### Citation

Bessenrodt, Christine; Nguyen, Hung; Olsson, Jørn; Tong-Viet, Hung. Complex group algebras of the double covers of the symmetric and alternating groups. Algebra Number Theory 9 (2015), no. 3, 601--628. doi:10.2140/ant.2015.9.601. https://projecteuclid.org/euclid.ant/1510842313

#### References

• A. Balog, C. Bessenrodt, J. B. Olsson, and K. Ono, “Prime power degree representations of the symmetric and alternating groups”, J. London Math. Soc. $(2)$ 64:2 (2001), 344–356.
• C. Bessenrodt and J. B. Olsson, “Prime power degree representations of the double covers of the symmetric and alternating groups”, J. London Math. Soc. $(2)$ 66:2 (2002), 313–324.
• R. Brauer, “Representations of finite groups”, pp. 133–175 in Lectures on Modern Mathematics, I, edited by T. L. Saaty, Wiley, New York, 1963.
• O. Brunat, “Counting $p'$-characters in finite reductive groups”, J. London Math. Soc. $(2)$ 81:3 (2010), 544–562.
• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, 1985.
• J. S. Frame, G. d. B. Robinson, and R. M. Thrall, “The hook graphs of the symmetric groups”, Canadian J. Math. 6 (1954), 316–324.
• H. Harborth and A. Kemnitz, “Calculations for Bertrand's postulate”, Math. Mag. 54:1 (1981), 33–34.
• P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Clarendon, New York, 1992.
• B. Huppert, “Some simple groups which are determined by the set of their character degrees, I”, Illinois J. Math. 44:4 (2000), 828–842.
• B. Huppert and W. Lempken, “Simple groups of order divisible by at most four primes”, Izv. Gomel. Gos. Univ. Im. F. Skoriny 2000:3(16) (2000), 64–75.
• I. M. Isaacs, Character theory of finite groups, Dover, New York, 1994. Corrected reprint of the 1976 original.
• A. S. Kleshchev and P. H. Tiep, “On restrictions of modular spin representations of symmetric and alternating groups”, Trans. Amer. Math. Soc. 356:5 (2004), 1971–1999.
• A. S. Kleshchev and P. H. Tiep, “Small-dimensional projective representations of symmetric and alternating groups”, Algebra Number Theory 6:8 (2012), 1773–1816.
• F. Lübeck, “Smallest degrees of representations of exceptional groups of Lie type”, Comm. Algebra 29:5 (2001), 2147–2169.
• I. G. Macdonald, “On the degrees of the irreducible representations of symmetric groups”, Bull. London Math. Soc. 3 (1971), 189–192.
• I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Clarendon, New York, 1995.
• J. McKay, “Irreducible representations of odd degree”, J. Algebra 20 (1972), 416–418.
• A. O. Morris, “The spin representation of the symmetric group”, Proc. London Math. Soc. $(3)$ 12 (1962), 55–76.
• H. Nagao, “On a conjecture of Brauer for $p$-solvable groups”, J. Math. Osaka City Univ. 13 (1962), 35–38.
• G. Navarro, “Problems in character theory”, pp. 97–125 in Character theory of finite groups, Contemp. Math. 524, Amer. Math. Soc., Providence, RI, 2010.
• H. N. Nguyen, “Low-dimensional complex characters of the symplectic and orthogonal groups”, Comm. Algebra 38:3 (2010), 1157–1197.
• H. N. Nguyen, “Quasisimple classical groups and their complex group algebras”, Israel J. Math. 195:2 (2013), 973–998.
• H. N. Nguyen and H. P. Tong-Viet, “Characterizing finite quasisimple groups by their complex group algebras”, Algebr. Represent. Theory 17:1 (2014), 305–320.
• J. B. Olsson, Combinatorics and representations of finite groups, Fachbereich Mathematik der Universität Essen, Heft 20, 1993.
• R. Rasala, “On the minimal degrees of characters of $S\sb{n}$”, J. Algebra 45:1 (1977), 132–181.
• I. Schur, “Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen”, J. Reine Angew. Math. 139 (1911), 155–250.
• P. H. Tiep and A. E. Zalesskii, “Minimal characters of the finite classical groups”, Comm. Algebra 24:6 (1996), 2093–2167.
• H. P. Tong-Viet, “Symmetric groups are determined by their character degrees”, J. Algebra 334 (2011), 275–284.
• H. P. Tong-Viet, “Alternating and sporadic simple groups are determined by their character degrees”, Algebr. Represent. Theory 15:2 (2012), 379–389.
• A. Wagner, “An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field”, Arch. Math. 29:6 (1977), 583–589.
• D. B. Wales, “Some projective representations of $S\sb{n}$”, J. Algebra 61:1 (1979), 37–57.