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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [A1]
George
E. Andrews, A generalization of the classical
partition theorems, Trans. Amer. Math. Soc.
145 (1969),
205–221. MR 0250995
(40 #4226), http://dx.doi.org/10.1090/S00029947196902509951
 [A2]
George
E. Andrews, On a theorem of Schur and Gleissberg, Arch. Math.
(Basel) 22 (1971), 165–167. MR 0286767
(44 #3976)
 [A3]
George
E. Andrews, On the general RogersRamanujan theorem, American
Mathematical Society, Providence, R.I., 1974. Memiors of the American
Mathematical Society, No. 152. MR 0364082
(51 #337)
 [A4]
, qseries, CBMS Regional Conf. Ser. in Math., vol. 66, Amer. Math. Soc., Providence, RI, 1986.
 [A5]
George
E. Andrews, Physics, Ramanujan, and computer algebra, Computer
algebra (New York, 1984) Lecture Notes in Pure and Appl. Math.,
vol. 113, Dekker, New York, 1989, pp. 97–109. MR 1002979
(90d:05025)
 [BMO]
Christine
Bessenrodt, Alun
O. Morris, and Jørn
B. Olsson, Decomposition matrices for spin characters of symmetric
groups at characteristic 3, J. Algebra 164 (1994),
no. 1, 146–172. MR 1268331
(96c:20026), http://dx.doi.org/10.1006/jabr.1994.1058
 [F]
Walter
Feit, The representation theory of finite groups,
NorthHolland Mathematical Library, vol. 25, NorthHolland Publishing
Co., Amsterdam, 1982. MR 661045
(83g:20001)
 [JK]
Gordon
James and Adalbert
Kerber, The representation theory of the symmetric group,
Encyclopedia of Mathematics and its Applications, vol. 16,
AddisonWesley Publishing Co., Reading, Mass., 1981. With a foreword by P.
M. Cohn; With an introduction by Gilbert de B. Robinson. 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),
no. 12, 145–164. MR 931015
(89f:20017), http://dx.doi.org/10.1017/S0308210500026597
 [NT]
Hirosi
Nagao and Yukio
Tsushima, Representations of finite groups, Academic Press
Inc., Boston, MA, 1989. Translated from the Japanese. 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).
 [A1]
 G. E. Andrews, A generalization of the classical partition theorems, Trans. Amer. Math. Soc. 145 (1969), 205221. MR 0250995 (40:4226)
 [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).
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
11P83,
05A17,
20C25
Retrieve articles in all journals
with MSC:
11P83,
05A17,
20C25
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412209041
PII:
S 00029947(1994)12209041
Article copyright:
© Copyright 1994 American Mathematical Society
