Partition identities and labels for some modular characters
Authors:
G. E. Andrews, C. Bessenrodt and J. B. Olsson
Journal:
Trans. Amer. Math. Soc. 344 (1994), 597615
MSC:
Primary 11P83; Secondary 05A17, 20C25
MathSciNet review:
1220904
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Abstract: In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups of the finite symmetric groups in characteristic 5. One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. O. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture , is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels.
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George
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145 (1969),
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George
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 [A1]
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 [A2]
 , On a theorem of Schur and Gleissberg, Arch. Math. 22 (1971), 165167. MR 0286767 (44:3976)
 [A3]
 , On the general RogersRamanujan theorem, Mem. Amer. Math. Soc. 152 (1974). MR 0364082 (51:337)
 [A4]
 , qseries, CBMS Regional Conf. Ser. in Math., vol. 66, Amer. Math. Soc., Providence, RI, 1986.
 [A5]
 , Physics, Ramanujan and computer algebra, Computer Algebra (D. V. Chudnovsky and R. D. Jenks, eds.), Dekker, New York, 1989, pp. 97109. MR 1002979 (90d:05025)
 [BMO]
 C. Bessenrodt, A. O. Morris, and J. B. Olsson, Decomposition matrices for spin characters of symmetric groups at characteristic 3, preprint, 1992. MR 1268331 (96c:20026)
 [F]
 W. Feit, The representation theory of finite groups, NorthHolland, Amsterdam, 1982. MR 661045 (83g:20001)
 [JK]
 G. James and A. Kerber, The representation theory of the symmetric group, AddisonWesley, London, 1981. MR 644144 (83k:20003)
 [MY]
 A. O. Morris and A. K. Yaseen, Decomposition matrices for spin characters of symmetric groups, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 145164. MR 931015 (89f:20017)
 [NT]
 H. Nagao and Y. Tsushima, Representations of finite groups, Academic Press, San Diego, 1989. MR 998775 (90h:20008)
 [S1]
 I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew Math. 139 (1911), 155250 (Ges. Abhandlungen 1, SpringerVerlag, 1973, pp. 346441).
 [S2]
 I. Schur, Zur additiven Zahlentheorie, Sitzungsber. Preuss. Akad. Wiss., Phys.Math. Kl. (1926), 488496 (Ges. Abhandlungen 3, SpringerVerlag, 1973, pp. 4350).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412209041
PII:
S 00029947(1994)12209041
Article copyright:
© Copyright 1994
American Mathematical Society
