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ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

Year 2019, Volume 25, Issue 25, 129 - 170, 08.01.2019
https://doi.org/10.24330/ieja.504139

Abstract

We investigate integral forms of certain simple modules over group algebras in characteristic 0 whose
$p$-modular reductions have precisely three composition factors. As a consequence we, in particular, complete
the description of the integral forms of the simple $\QQ\mathfrak{S}_n$-module labelled by the hook partition $(n-2,1^2)$.
Moreover, we investigate the integral forms of the Steinberg module of finite special linear groups $\PSL_2(q)$
over suitable fields of characteristic 0. In the second part of the paper we explicitly determine the Solomon zeta functions
of various families of modules and lattices over group algebra, including Specht modules of symmetric groups labelled by
hook partitions and the Steinberg module of $\PSL_2(q)$.

References

  • C. Bonnafé, Representations of SL2(Fq), Algebra and Applications, 13, Springer-Verlag London, Ltd., London, 2011.
  • C. J. Bushnell and I. Reiner, Solomon’s conjectures and the local functional equation for zeta functions of orders, Bull. Amer. Math. Soc. (N.S.), 2(2) (1980), 306-310.
  • M. Craig, A characterization of certain extreme forms, Illinois J. Math., 20(4) (1976), 706-717.
  • C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I. With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981.
  • S. Danz and T. Hofmann, On integral forms of Specht modules labelled by hook partitions, Preprint, arXiv:1706.02860v2, (2018).
  • T. Hofmann, Zeta functions of lattices of the symmetric group, Comm. Algebra, 44(5) (2016), 2243-2255.
  • B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967.
  • G. D. James, The irreducible representations of the symmetric groups, Bull. London Math. Soc., 8(3) (1976), 229-232.
  • G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978.
  • G. D. James, Representations of General Linear Groups, London Mathematical Society Lecture Note Series, 94, Cambridge University Press, Cambridge, 1984.
  • J. Müller and J. Orlob, On the structure of the tensor square of the natural module of the symmetric group, Algebra Colloq., 18(4) (2011), 589-610.
  • W. Plesken, Beiträge zur Bestimmung der endlichen irreduziblen Untergruppen von GL(n,Z) und ihrer ganzzahligen Darstellungen. PhD thesis, RWTH Aachen, 1974.
  • W. Plesken, On absolutely irreducible representations of orders, In Hans Zassenhaus, editor, Number theory and algebra, Academic Press, New York, (1977), 241-262.
  • W. Plesken, Gruppenringe über lokalen Dedekindbereichen, Habilitation, RWTH Aachen, 1980.
  • W. Plesken, Group Rings of Finite Groups over p-adic Integers, Lecture Notes in Mathematics, 1026, Springer-Verlag, Berlin, 1983.
  • L. Solomon, Zeta functions and integral representation theory, Advances in Math., 26(3) (1977), 306-326.

Year 2019, Volume 25, Issue 25, 129 - 170, 08.01.2019
https://doi.org/10.24330/ieja.504139

Abstract

References

  • C. Bonnafé, Representations of SL2(Fq), Algebra and Applications, 13, Springer-Verlag London, Ltd., London, 2011.
  • C. J. Bushnell and I. Reiner, Solomon’s conjectures and the local functional equation for zeta functions of orders, Bull. Amer. Math. Soc. (N.S.), 2(2) (1980), 306-310.
  • M. Craig, A characterization of certain extreme forms, Illinois J. Math., 20(4) (1976), 706-717.
  • C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I. With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981.
  • S. Danz and T. Hofmann, On integral forms of Specht modules labelled by hook partitions, Preprint, arXiv:1706.02860v2, (2018).
  • T. Hofmann, Zeta functions of lattices of the symmetric group, Comm. Algebra, 44(5) (2016), 2243-2255.
  • B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967.
  • G. D. James, The irreducible representations of the symmetric groups, Bull. London Math. Soc., 8(3) (1976), 229-232.
  • G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978.
  • G. D. James, Representations of General Linear Groups, London Mathematical Society Lecture Note Series, 94, Cambridge University Press, Cambridge, 1984.
  • J. Müller and J. Orlob, On the structure of the tensor square of the natural module of the symmetric group, Algebra Colloq., 18(4) (2011), 589-610.
  • W. Plesken, Beiträge zur Bestimmung der endlichen irreduziblen Untergruppen von GL(n,Z) und ihrer ganzzahligen Darstellungen. PhD thesis, RWTH Aachen, 1974.
  • W. Plesken, On absolutely irreducible representations of orders, In Hans Zassenhaus, editor, Number theory and algebra, Academic Press, New York, (1977), 241-262.
  • W. Plesken, Gruppenringe über lokalen Dedekindbereichen, Habilitation, RWTH Aachen, 1980.
  • W. Plesken, Group Rings of Finite Groups over p-adic Integers, Lecture Notes in Mathematics, 1026, Springer-Verlag, Berlin, 1983.
  • L. Solomon, Zeta functions and integral representation theory, Advances in Math., 26(3) (1977), 306-326.

Details

Primary Language English
Journal Section Articles
Authors

Susanne DANZ This is me (Primary Author)


Tommy HOFMANN This is me

Publication Date January 8, 2019
Published in Issue Year 2019, Volume 25, Issue 25

Cite

Bibtex @research article { ieja504139, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2019}, volume = {25}, number = {25}, pages = {129 - 170}, doi = {10.24330/ieja.504139}, title = {ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS}, key = {cite}, author = {Danz, Susanne and Hofmann, Tommy} }
APA Danz, S. & Hofmann, T. (2019). ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS . International Electronic Journal of Algebra , 25 (25) , 129-170 . DOI: 10.24330/ieja.504139
MLA Danz, S. , Hofmann, T. "ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS" . International Electronic Journal of Algebra 25 (2019 ): 129-170 <https://dergipark.org.tr/en/pub/ieja/issue/41930/504139>
Chicago Danz, S. , Hofmann, T. "ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS". International Electronic Journal of Algebra 25 (2019 ): 129-170
RIS TY - JOUR T1 - ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS AU - Susanne Danz , Tommy Hofmann Y1 - 2019 PY - 2019 N1 - doi: 10.24330/ieja.504139 DO - 10.24330/ieja.504139 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 129 EP - 170 VL - 25 IS - 25 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.504139 UR - https://doi.org/10.24330/ieja.504139 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS %A Susanne Danz , Tommy Hofmann %T ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS %D 2019 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 25 %N 25 %R doi: 10.24330/ieja.504139 %U 10.24330/ieja.504139
ISNAD Danz, Susanne , Hofmann, Tommy . "ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS". International Electronic Journal of Algebra 25 / 25 (January 2019): 129-170 . https://doi.org/10.24330/ieja.504139
AMA Danz S. , Hofmann T. ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS. IEJA. 2019; 25(25): 129-170.
Vancouver Danz S. , Hofmann T. ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS. International Electronic Journal of Algebra. 2019; 25(25): 129-170.
IEEE S. Danz and T. Hofmann , "ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS", International Electronic Journal of Algebra, vol. 25, no. 25, pp. 129-170, Jan. 2019, doi:10.24330/ieja.504139