On the irreducible representations of the alternating group which remain irreducible in characteristic
Author:
Matthew Fayers
Journal:
Represent. Theory 14 (2010), 601626
MSC (2010):
Primary 20C30, 20C20; Secondary 05E10
Published electronically:
September 1, 2010
MathSciNet review:
2685098
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider the problem of which ordinary irreducible representations of the alternating group remain irreducible modulo a prime . We solve this problem for , and present a conjecture for odd , which we prove in one direction.
 [B]
Dave
Benson, Spin modules for symmetric groups, J. London Math.
Soc. (2) 38 (1988), no. 2, 250–262. MR 966297
(89k:20020), http://dx.doi.org/10.1112/jlms/s238.2.250
 [BK]
Jonathan
Brundan and Alexander
Kleshchev, Representation theory of symmetric groups and their
double covers, Groups, combinatorics & geometry (Durham, 2001)
World Sci. Publ., River Edge, NJ, 2003, pp. 31–53. MR 1994959
(2004i:20016), http://dx.doi.org/10.1142/9789812564481_0003
 [CT]
Joseph
Chuang and Kai
Meng Tan, Some canonical basis vectors in the basic
𝑈_{𝑞}(̂𝔰𝔩_{𝔫})module,
J. Algebra 248 (2002), no. 2, 765–779. MR 1882121
(2002k:20011), http://dx.doi.org/10.1006/jabr.2001.9030
 [F1]
Matthew
Fayers, Reducible Specht modules, J. Algebra
280 (2004), no. 2, 500–504. MR 2089249
(2005h:20019), http://dx.doi.org/10.1016/j.jalgebra.2003.09.053
 [F2]
Matthew
Fayers, Irreducible Specht modules for Hecke algebras of type
𝐴, Adv. Math. 193 (2005), no. 2,
438–452. MR 2137291
(2006e:20007), http://dx.doi.org/10.1016/j.aim.2004.06.001
 [G]
James
A. Green, Polynomial representations of
𝐺𝐿_{𝑛}, Lecture Notes in Mathematics,
vol. 830, SpringerVerlag, BerlinNew York, 1980. MR 606556
(83j:20003)
 [J1]
G.
D. James, On the decomposition matrices of the symmetric groups.
II, J. Algebra 43 (1976), no. 1, 45–54. MR 0430050
(55 #3057b)
 [J2]
G.
D. James, The representation theory of the symmetric groups,
Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
(80g:20019)
 [JM1]
Gordon
James and Andrew
Mathas, A 𝑞analogue of the JantzenSchaper theorem,
Proc. London Math. Soc. (3) 74 (1997), no. 2,
241–274. MR 1425323
(97j:20013), http://dx.doi.org/10.1112/S0024611597000099
 [JM2]
Gordon
James and Andrew
Mathas, The irreducible Specht modules in characteristic 2,
Bull. London Math. Soc. 31 (1999), no. 4,
457–462. MR 1687552
(2000f:20018), http://dx.doi.org/10.1112/S0024609399005822
 [LM]
Bernard
Leclerc and Hyohe
Miyachi, Some closed formulas for canonical
bases of Fock spaces, Represent. Theory 6 (2002), 290–312
(electronic). MR
1927956 (2004a:17022), http://dx.doi.org/10.1090/S108841650200136X
 [L]
Sinéad
Lyle, Some reducible Specht modules, J. Algebra
269 (2003), no. 2, 536–543. MR 2015852
(2004h:20015), http://dx.doi.org/10.1016/S00218693(03)005374
 [P]
M.
H. Peel, Hook representations of the symmetric groups, Glasgow
Math. J. 12 (1971), 136–149. MR 0308249
(46 #7363)
 [R]
Matthew
J. Richards, Some decomposition numbers for Hecke algebras of
general linear groups, Math. Proc. Cambridge Philos. Soc.
119 (1996), no. 3, 383–402. MR 1357053
(97d:20009), http://dx.doi.org/10.1017/S0305004100074296
 [S]
Joanna
Scopes, Cartan matrices and Morita equivalence for blocks of the
symmetric groups, J. Algebra 142 (1991), no. 2,
441–455. MR 1127075
(92h:20023), http://dx.doi.org/10.1016/00218693(91)903194
 [T]
W.
Turner, Rock blocks, Mem. Amer. Math. Soc.
202 (2009), no. 947, viii+102. MR 2553536
(2011c:20021), http://dx.doi.org/10.1090/S0065926609005626
 [B]
 D. Benson, Spin modules for symmetric groups, J. London Math. Soc. (2) 38 (1988), 250262. MR 0966297 (89k:20020)
 [BK]
 J. Brundan & A. Kleshchev, Representation theory of the symmetric groups and their double covers, Groups, combinatorics & geometry (Durham, 2001), 3153, World Sci. Publishing, River Edge, NJ, 2003. MR 1994959 (2004i:20016)
 [CT]
 J. Chuang & K. M. Tan, Some canonical basis vectors in the basic module, J. Algebra 248 (2002), 765779. MR 1882121 (2002k:20011)
 [F1]
 M. Fayers, Reducible Specht modules, J. Algebra 280 (2004), 500504. MR 2089249 (2005h:20019)
 [F2]
 , Irreducible Specht modules for Hecke algebras of type , Adv. Math. 193 (2005), 438452. MR 2137291 (2006e:20007)
 [G]
 J. Green, Polynomial representations of , Lecture Notes in Mathematics, 830, SpringerVerlag, New York/Berlin, 1980. MR 0606556 (83j:20003)
 [J1]
 G. James, On the decomposition matrices of the symmetric groups II, J. Algebra 43 (1976), 4554. MR 0430050 (55:3057b)
 [J2]
 , The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, SpringerVerlag, New York/Berlin, 1978. MR 0513828 (80g:20019)
 [JM1]
 G. James & A. Mathas, A analogue of the JantzenSchaper theorem, Proc. London Math. Soc. (3) 74 (1997), 241274. MR 1425323 (97j:20013)
 [JM2]
 , The irreducible Specht modules in characteristic , Bull. London Math. Soc. 31 (1999), 457462. MR 1687552 (2000f:20018)
 [LM]
 B. Leclerc & H. Miyachi, Some closed formulas for canonical bases of Fock spaces, Represent. Theory 6 (2002), 290312. MR 1927956 (2004a:17022)
 [L]
 S. Lyle, Some reducible Specht modules, J. Algebra 269 (2003), 536543. MR 2015852 (2004h:20015)
 [P]
 M. Peel, Hook representations of the symmetric groups, Glasgow Math. J. 12 (1971), 136149. MR 0308249 (46:7363)
 [R]
 M. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), 383402. MR 1357053 (97d:20009)
 [S]
 J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441455. MR 1127075 (92h:20023)
 [T]
 W. Turner, Rock blocks, Mem. Amer. Math. Soc. 202 (2009), no. 947, viii+102 pp. MR 2553536
Similar Articles
Retrieve articles in Representation Theory of the American Mathematical Society
with MSC (2010):
20C30,
20C20,
05E10
Retrieve articles in all journals
with MSC (2010):
20C30,
20C20,
05E10
Additional Information
Matthew Fayers
Affiliation:
Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
Email:
m.fayers@qmul.ac.uk
DOI:
http://dx.doi.org/10.1090/S108841652010003908
PII:
S 10884165(2010)003908
Received by editor(s):
February 15, 2007
Received by editor(s) in revised form:
July 3, 2010
Published electronically:
September 1, 2010
Additional Notes:
Part of this research was undertaken with the support of a Research Fellowship from the Royal Commission for the Exhibition of 1851. The author is very grateful to the Commission for its generous support.
Part of this research was undertaken while the author was visiting the Massachusetts Institute of Technology as a Postdoctoral Fellow. He is very grateful to Professor Richard Stanley for the invitation, and to M.I.T. for its hospitality.
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
