ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

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 $\PSL_2(q)$.