= \{ \varphi^{(\pi)}_j \ | 1 \leq j \leq l({\bf b}) \}$ be a basic set for ${\bf b}$ in the sense of Brauer [@Brauer], that is, $Bs({\bf b})$ is a $\textrm{{\boldmath $Z$}}$-basis of the $\textrm{{\boldmath $Z$}}$-module $$\bigoplus_{\varphi \in {\rm IBr}({\bf b})} \textrm{{\boldmath $Z$}}  \varphi.$$ Then there exist ${d}_{\chi {\varphi}_j^{(\pi)}}^{\pi} \in {\cal O}$, what we call the generalized decomposition numbers with respect to $Bs({\bf b})$, which satisfy $$\chi^{(\pi,{\bf  b})}(\pi \rho) = \sum_{j = 1}^{l({\bf b})} 
{d}_{\chi {\varphi}_{j}^{(\pi)}}^{\pi}
{\varphi}_{j}^{(\pi)}(\rho) \ \ (\forall \rho \in C_{G}(\pi)_{p'}).$$ Set $${D}^{(\pi, {\bf b})}  = \Big( {d}_{\chi {\varphi}_{j}^{(\pi)}}^{\pi}\Big)_
{\chi \in {\rm Irr}(b), \varphi_{j}^{(\pi)} \in Bs({\bf b})}.$$ By [@NT], Theorem 5.4.11, the matrix ${^t}({D}^{(\pi, {\bf b})})\overline{D^{(\pi, {\bf b})}}$ is similar to the Cartan matrix of ${\bf  b}$ [w.r.t.]{} the basic set $Bs({\bf b})$, where $\overline{D^{(\pi, {\bf b})}}$ is the complex conjugate matrix of ${D}^{(\pi, {\bf b})}$ (see [@Navarro], Lemma 7.5). The following plays an important role in this paper.

Let $b$ $(resp. \  b')$ be a block of a finite group $G$ $(resp. \ 
G')$ such that $b$ and $b'$ satisfy [(i) - (iv)]{} below.

[(i)]{} $b$ and $b'$ has a common defect group $P$. Let $(P, b_P)$ $(resp. \ (P, b'_P))$ be a maximal $b$ $resp. 