number $\alpha$ we denote by $\bar{\alpha}$ the complex conjugate, and for a character ${\chi}$ of $G$ we denote by $\bar{\chi}$ the contragredient character.

Let $({\cal K}, {\cal O}, F)$ be a $p$-modular system such that ${\cal K}$ contains the field ${\bf Q}(\sqrt[|G|]{1})$. For a ($p$-)block $b$ of $G$, we denote by Irr$(b)$ the set of ordinary irreducible characters belonging to $b$ and IBr$(b)$ the set of irreducible Brauer characters belonging to $b$. Here $b$ denotes a block idempotent of ${\cal O}G$. We set $k(b) = |{\rm Irr}(b)|$, the cardinality of Irr$(b)$, and $l(b) = |{\rm IBr}(b)|$. Also we set ${\cal R}_{{\cal K}}(G, b) = \sum_{\chi \in {\rm Irr}(b)}\textrm{{\boldmath $Z$}}  \chi$ and we denote by $b(G)$ the principal block of $G$. We denote by $1_{G}$ the trivial irreducible character and the trivial Brauer character of $G$.

Let $(\pi, {\bf b})$ be a $b$-Brauer element, that is, $\bf b$ is a block of $C_{G}(\pi)$ associated with $b$. For $\chi \in {\rm Irr}(b)$, the class function $\chi^{(\pi, {\bf b})}$ of $G$ is defined as follows: $\chi^{(\pi, {\bf b})}$ vanishes outside of the $p$-section of $G$ containing $\pi$ and $\chi^{(\pi, {\bf b})}(\pi \rho) = \chi(\pi\rho {\bf b})$ for $\rho \in C_{G}(\pi)_{p'}$. Let $Bs({\bf b}) 