ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

Year 2019, , 129 - 170, 08.01.2019

Susanne Danz Tommy Hofmann

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 $\PSL_2(q)$.

Keywords

Integral representation, lattice, Jordan–Zassenhaus, symmetric group, Specht module, hook partition, projective special linear group, Steinberg module, Solomon zeta function

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.