On the total character of finite groups

Abstract

For a finite group $G$‎, ‎we study the total character $tau_G$‎ ‎afforded by the direct sum of all the non-isomorphic irreducible‎ ‎complex representations of $G$‎. ‎We resolve for several classes of‎ ‎groups (the Camina $p$-groups‎, ‎the generalized Camina $p$-groups‎, ‎the groups which admit $(G,Z(G))$ as a generalized Camina pair)‎, ‎the problem of existence of a‎ ‎polynomial $f(x) in mathbb{Q}[x]$ such that $f(chi) = tau_G$ for‎ ‎some irreducible character $chi$ of $G$‎. ‎As a consequence‎, ‎we‎ ‎completely determine the $p$-groups of order at most $p^5$ (with $p$‎ ‎odd) which admit such a polynomial‎. ‎We deduce the characterization‎ ‎that these are the groups $G$ for which $Z(G)$ is cyclic and‎ ‎$(G,Z(G))$ is a generalized Camina pair and‎, ‎we conjecture that this‎ ‎holds good for $p$-groups of any order‎.