I$, let $k_j := \max\{1\le k\le r\mid i_k = j\}$. The submodule $\bigoplus_{j\in I} V_{k_j}$ of $V_\i$ also depends only on $w$, and we denote it by $I_w$.

\[Example4.4\] We consider again $\g = \slchap_2$ and $w=s_2s_1s_2s_1$, so $\i=(2,1,2,1)$. We take for $Q$ the Kronecker quiver: $$\xymatrix{1 \ar@<0.5ex>[r]^{a}\ar@<-0.5ex>[r]_b & 2}$$ The following pictures describe the indecomposable direct summands of $V_\i$. The numbers 1 and 2 in the pictures are basis vectors of the modules $V_k$. The solid edges show how the arrows $a$ and $b$ of $\overline{Q}$ act on these vectors, and the dotted edges illustrate the actions of $a^*$ and $b^*$. The arrows $a$ and $b^*$ are pointing south east, and the arrow $a^*$ and $b$ are pointing south west. $$\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
V_1=
\xymatrix@-1.5pc{
1
}
\qquad
\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
V_2=
\xymatrix@-1.5pc{
1\ar@{->}[rd]& &\ar@{->}[ld]1 \\
 & 2
}$$ $$\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
V_3=
\xymatrix@-1.5pc{
 1\ar@{->}[rd]& &\ar@{->}[ld]1\ar@{->}[rd]& &\ar@{->}[ld]1\\
  &2\ar@{.>}[rd]& &\ar@{.>}[ld]2 \\
 & & 1
}
\qquad
\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
V_4=
\xymatrix@-1.5pc{
1\ar@{->}[rd]&&\ar@{->}[ld]1\ar@{->}[rd]&&\ar@{->}[ld]1\ar@{->}[rd]&&\ar@{->}[ld]1\\
 &2\ar@{.>}[rd]& &\ar@{.>}[ld]2\ar@{.>}[rd]& &\ar@{.>}[ld]2\\
 & &1\ar@{->}[rd]& &\ar@{->}[ld]1 \\
& & & 2
}$$ Here, $I_w = V_3\oplus V_4$. The modules $$\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
M_3=
\xymatrix@-1.5pc{
 1\ar@{->}[rd]& &\ar@{->}[ld]1\ar@{->}[rd]& &\ar@{->}[ld]1\\
  &2& &2
}
\qquad
\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
M_4=
\xymatrix@-1.5pc{
1\ar@{->}[rd]&&\ar@{->}[ld]1\ar@{->}[rd]&&\ar@{->}[ld]1\ar@{->}[rd]&&\ar@{->}[ld]1\\
 &2& &2& &2\\
}$$ are factor modules of $V_3$ and $V_4$ respectively, so 