group elements {#sect4.3}
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Let $w\in W$ be of length $r$ and choose a reduced decomposition $w=s_{i_r}\cdots s_{i_1}$ (note that again, we number the factors from right to left). It is well-known that $\De_w$ consists of the roots $$\b_k:=s_{i_1}\cdots s_{i_{k-1}}(\a_{i_k}),\qquad  (1\le k \le r).$$ The following sequence of $R^+$ will also play an important role: $$\ga_k:=\varpi_{i_k} - s_{i_1}\cdots s_{i_{k}}(\varpi_{i_k}),\qquad  (1\le k \le r).$$ From now on, we shall freely identify dimension vectors $\d = (d_i)$ with elements of $R^+$ via $$\d \equiv \sum_{i\in I} d_i \a_i.$$ For $k = 1,\ldots, r$, one can show that there is a unique $V_k \in \nil(\L)$ (up to isomorphism) whose socle is $S_{i_k}$ and whose dimension vector is $\ga_k$. Let us write $\i = (i_r,\ldots,i_1)$, and define $$V_\i := \bigoplus_{k=1}^r V_k \in \nil(\L).$$ Up to duality, this $\L$-module is the same as the one introduced by Buan, Iyama, Reiten, and Scott [@BIRS]. Following [@BIRS], we consider the full subcategory of $\nil(\L)$ whose objects are factor modules of direct sums of finitely many copies of $V_\i$. It turns out that this category depends only on $w$ (and not on the choice of a reduced expression $\i$), so we may denote it by $\CC_w$.

For $j\in 