ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

Authors : Susanne DANZ, Tommy HOFMANN

Pages : 129-170

Doi:10.24330/ieja.504139

View : 15 | Download : 11

Publication Date : 2019-01-08

Article Type : Research Paper

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 $insert ignore into journalissuearticles values(n-2,1^2);$. Moreover, we investigate the integral forms of the Steinberg module of finite special linear groups $\PSL_2insert ignore into journalissuearticles values(q);$ 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 $\PSL_2insert ignore into journalissuearticles values(q);$.
Keywords : Integral representation, lattice, JordanZassenhaus, symmetric group, Specht module, hook partition, projective special linear group, Steinberg module, Solomon zeta function