associative multiplication which formally resembles a convolution product, and he has shown that, if we denote by $e_i$ the Chevalley generators of $\n$, there is an algebra isomorphism $U(\n) \stackrel{\sim}{\rightarrow} \M$ mapping the product $e_{i_1}\cdots e_{i_d}$ to $\chi_\i$ for every $\i=(i_1,\ldots,i_d)$.

Now, following [@GLS1; @GLS2], we dualize the picture. Every $X\in\nil(\L)$ determines a linear form $\delta_X$ on $\M$ given by $$\delta_X(f) = f(X),\qquad (f\in\M).$$ Through the isomorphisms $\M_{\mathrm{gr}}^* \simeq U(\n)_{\mathrm{gr}}^* \simeq \C[N]$, the form $\delta_X$ corresponds to an element $\vph_X$ of $\C[N]$, and we have thus attached to every object $X$ in $\nil(\L)$ a polynomial function $\vph_X$ on $N$. Unwrapping this definition, we have an explicit formula for evaluating $\vph_X$ on an arbitrary product $\mathbf{x}_{\mathbf{j}}(\mathbf{t}) := x_{{j_1}}({t_1})\cdots x_{{j_r}}({t_r})$, namely $$\vph_X(\mathbf{x}_{\mathbf{j}}(\mathbf{t})) 
= 
\sum_{\mathbf{a}\in\N^r} \chi_{{\mathbf{j}}^{\mathbf{a}}}(X) \frac{\mathbf{t}^{\mathbf{a}}}{\mathbf{a}!},$$ where $\displaystyle\frac{\mathbf{t}^{\mathbf{a}}}{\mathbf{a}!} := \frac{t_1^{a_1}\cdots t_r^{a_r}}{a_1!\cdots a_r!}$, and ${\mathbf{j}}^{\mathbf{a}} := (j_1,\ldots,j_1,j_2, \ldots,j_2, \ldots, j_r\ldots,j_r)$ with each component $j_k$ repeated $a_k$ times.

If $\g$ is of type $A_3$, and if we denote by $P_i$ the projective cover of $S_i$, one has $$\vph_{P_1} = D_{123,234},\quad
 \vph_{P_2} = D_{12,34},\quad
 \vph_{P_3} = D_{1,4}.$$ More generally, the functions $\vph_X$ corresponding to the 12 indecomposable $\L$-modules are the 12 cluster variables of $\C[N]$ (see §\[sect2.2\]).

Via the correspondence $X \mapsto \vph_X$, the ring 