can compare the coefficient of $x_1\otimes x_1^{m-1}x_{\gs_1}\cdots x_{\gs_q}$ of the two sides of and find the relation . Finally, if $m=0$ and $n>0$ then we may similarly consider the coefficient of $x_{\gs_1}\cdots x_{\gs_p}x_0^{n-1}\otimes x_0$ in . This finishes the proof of the proposition.

For any divisor $d$ of $N$ let $\gG^d=\{\gs^d:\gs\in \gG\}$, $i_d:\gG^d\hookrightarrow \gG$ the embedding, and $p^d:\gG\twoheadrightarrow \gG^d$ the $d$th power map. They induce two algebra homomorphisms: $$\begin{aligned}
{6}
p^d_\ast: \Q\langle\bfX_\gG\rangle & \lra\Q\langle\bfX_{\gG^d}\rangle & {} & &\hskip1cm
i_d^\ast: \Q\langle\bfX_\gG\rangle & \lra\Q\langle\bfX_{\gG^d}\rangle \\
x_\gs &\lms \begin{cases} d x_0, \  &\text{ if }\gs=0,\\
x_{\gs^d}, \  &\text{ if }\gs\in \gG.\end{cases}
 &\qquad\overset{\displaystyle\text{ and }}{\phantom{\sum}} & &
x_\gs &\lms \begin{cases} x_0, \  &\text{ if }\gs=0,\\
x_\gs, \  &\text{ if }\gs\in \gG^d,\\
0, \  &\text{ otherwise}.\end{cases}\end{aligned}$$

It is easy to see that both $i_d^\ast$ and $p^d_\ast$ are $\gD$-coalgebra morphisms such that $i_d^\ast(\calI)$ and $p^d_\ast(\calI)$ have the same image under the map $\pi_\cv$. By the standard Lie-algebra mechanism one has

\[prop:RDist\] *([@Rac Prop.2.26])* For every divisor $d$ of $N$ $$\label{equ:RDist}
   p^d_\ast(\calI)=\exp\left(\sum_{\gs^d=1,\gs\ne 1} Li_1(\gs) x_1\right) i_d^\ast(\calI).$$

Combined with Proposition \[prop:CoeffI\] the above result provides the so-called *regularized distribution relations* (RDT) which of course include all the FDT of MPVs given by .

Computation suggests the the following 