Skip to main content Accessibility help
×
Home
Hostname: page-component-cf9d5c678-h2mp8 Total loading time: 0.194 Render date: 2021-08-01T23:34:40.893Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Twisted Dickson–Mui invariants and the Steinberg module multiplicity

Published online by Cambridge University Press:  18 March 2011

JINKUI WAN
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China. e-mail: wjk302@gmail.com
WEIQIANG WANG
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, U.S.A. e-mail: ww9c@virginia.edu
Corresponding

Abstract

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GLn(q) acting on the tensor product of the symmetric algebra S(V) and the exterior algebra ∧(V) of the natural GLn(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GLn(q) in S(V) ⊗ ∧(V), twisted by an arbitrary determinant power.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2011

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

[1]Crabb, M.Dickson–Mui invariants. Bull. London. Math. Soc. 37 (2005), 846856.CrossRefGoogle Scholar
[2]Curtis, C.The Steinberg character of a finite group with a (B, N)-pair. J. Algebra 4 (1966), 433441.CrossRefGoogle Scholar
[3]Dickson, L.A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. Amer. Math. Soc. 12 (1911), 7598.CrossRefGoogle Scholar
[4]Hewett, T.Modular invariant theory of parabolic subgroups of GL n(q) and the associated Steenrod modules. Duke Math. J. 82 (1996), 91102; Erratum, Duke Math. J. 97 (1999), 217.CrossRefGoogle Scholar
[5]Humphreys, J.Modular representations of finite groups of Lie type. LMS Lecture Note Series 326 (Cambridge University Press, 2006).Google Scholar
[6]Kuhn, N. and Mitchell, S.The multiplicity of the Steinberg representation of GL n(q) in the symmetric algebra. Proc. Amer. Math. Soc. 96 (1986), 16.Google Scholar
[7]Minh, P. and Tùng, V.Modular invariants of parabolic subgroups of general linear groups. J. Algebra 232 (2000), 197208.CrossRefGoogle Scholar
[8]Mitchell, S.Finite complexes with A(n)-free cohomolgy. Topology 24 (1985), 227248.CrossRefGoogle Scholar
[9]Mitchell, S. and Priddy, S.Stable splittings derived from the Steinberg module. Topology 22 (1983), 285298.CrossRefGoogle Scholar
[10]Mui, H.Modular invariant theory and chomomogy algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 22 (1975), 319369.Google Scholar
[11]Smith, L.Polynomial invariants of finite groups. Res. Notes in Math. 6 (AK Peters, Ltd., Wellesley, MA, 1995).Google Scholar
[12]Solomon, L. The Steinberg character of a finite group with BN-pair. Theory of Finite Groups (Harvard Symposium) (Benjamin, Elmsford, N.Y., 1969), 213221.Google Scholar
[13]Wilkerson, C.A primer on the Dickson invariants. Contemp. Math. 19 (1983), 421434.CrossRefGoogle Scholar
[14]Wan, J. and Wang, W. The GL n(q)-module structure of the symmetric algebra around the Steinberg module. arXiv:1012.0406 (2010).Google Scholar
2
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Twisted Dickson–Mui invariants and the Steinberg module multiplicity
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Twisted Dickson–Mui invariants and the Steinberg module multiplicity
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Twisted Dickson–Mui invariants and the Steinberg module multiplicity
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *