Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:49:35.104Z Has data issue: false hasContentIssue false
  • English
  • Français

The multistep homology of the simplex and representations of symmetric groups

Part of: Representation theory of groups Homological algebra

Published online by Cambridge University Press:  20 June 2019

MARK WILDON
MARK WILDON*
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Eghan Hill, Eghan TW20 OEX, United Kingdom. e-mail: mark.wildon@rhul.qc.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

MSC classification

Primary: 20C30: Representations of finite symmetric groups
Secondary: 18G35: Chain complexes 20C20: Modular representations and characters
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 231 - 253
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, G. E.. Two theorems of Gauss and allied identities proved arithmetically. Pacific J. Math. 41 (1972), 563–578.CrossRefGoogle Scholar
Benson, D. J.. Representations and cohomology. I, second ed. Camb. Stud. Adv. Math., vol. 30 (Cambridge University Press, Cambridge, 1998), Basic representation theory of finite groups and associative algebras.Google Scholar
Benson, D.. Spin modules for symmetric groups. J. London Math. Soc. (2) 38 (1988), no. 2, 250–262.CrossRefGoogle Scholar
Boltje, R. and Hartmann, R.. Permutation resolutions for Specht modules. J. Algebraic Combin. 34 (2011), no. l, 141–162.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C.. The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993).CrossRefGoogle Scholar
Danz, S. and Külshammer, B.. The vertices and sources of the basic spin module for the symmetric group in characteristic 2. J. Pure Appl. Algebra 213 (2009), no. 7, 1264–1282.CrossRefGoogle Scholar
Doty, S., Erdmann, K. and Henke, A.. Endomorphism rings of permutation modules over maximal Young subgroups. J. Algebra 307 (2007), no. 1, 377–396.CrossRefGoogle Scholar
Giannelli, E., Jin Lim, K. and Wildon, M.. Sylow subgroups of symmetric and alternating groups and the vertex of S(kp-p ,1p) in characteristic. p. J. Algebra 455 (2016), 358–385.CrossRefGoogle Scholar
Gould, H. W.. Tables of combinatorial identities. http://www.math.wvu.edu/%20gould Vol.6.PDF 6 (May 2010), 26 pages.Google Scholar
Gupta, H.. The Andrews formula for Fibonacci numbers. Fibonacci Quart. 16 (1978), no. 6, 552–555.Google Scholar
Hamernik, W.. Specht modules and the radical of the group ring over the symmetric group γp. Comm. Algebra 4 (1976), 435–475.CrossRefGoogle Scholar
Henke, A.. On p-Kostka numbers and Young modules. European J. Combin. 26 (2005), no. 6, 923–942.CrossRefGoogle Scholar
James, G. and Kerber, A.. The representation theory of the symmetric group. Encyclopedia of Mathematics and its Applications, vol. 16 (Addison-Wesley Publishing Co., Reading, Mass., 1981).Google Scholar
James, G. D.. On the decomposition matrices of the symmetric groups. II. J. Algebra 43 (1976), no. 1, 45–54.CrossRefGoogle Scholar
James, G. D.. Representations of the symmetric groups over the field of order 2. J. Algebra 38 (1976), no. 2, 280–308.CrossRefGoogle Scholar
James, G. D.. The representation theory of the symmetric groups. Lecture Notes in Math., vol. 682 (Springer, Berlin, 1978).Google Scholar
Kleshchev, A.. Linear and projective representations of symmetric groups. Cambridge Tracts in Mathematics, vol. 163 (Cambridge University Press, Cambridge, 2005).Google Scholar
Müller, J. and Zimmermann, R.. Green vertices and sources of simple modules of the symmetric group labelled by hook partitions. Arch. Math. (Basel) 89 (2007), no. 2, 97–108.CrossRefGoogle Scholar
M, G.. Murphy. On decomposability of some Specht modules for symmetric groups. J. Algebra 66 (1980), no. 1, 156–168.Google Scholar
Peel, M. H.. Hook representations of the symmetric groups. Glasgow Math. J. 12 (1971), 136–149.CrossRefGoogle Scholar
Santana, A. P. and Yudin, I.. Characteristic-free resolutions of Weyl and Specht modules. Adv. Math. 229 (2012), no. 4, 2578–2601.CrossRefGoogle Scholar
Wildon, M.. Modular representations of symmetric groups. D. Phil. thesis, Oxford University, 2004.Google Scholar