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Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or disprove that

$\sum\limits_{g\in G} \frac{1}{\det\left(\mathrm{id}-T\rho\left(g\right)\right)} = 0$ as an equality between power series in $k\left[T\right]$.

Motivation:

Let $G$ be a finite group, and $k$ be any field. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. For every $d\geq 0$, let $n_d$ denote the dimension of the space $\left(\mathrm{Sym}^d V^{\ast}\right)^G$ of the $G$-invariant symmetric $d$-ary forms on $V$. Molien's formula states that

$\left|G\right|\cdot \sum\limits_{n=0}^{\infty} n_dT^d = \sum\limits_{g\in G} \frac{1}{\det\left(\mathrm{id}-T\rho\left(g\right)\right)}$ as an equality between power series in $k\left[T\right]$

if $\mathrm{char} k$ does not divide $\left|G\right|$. I am wondering whether this formula still holds if $\mathrm{char} k\mid \left|G\right|$. The standard proof, using the Maschke projection, does not make much sense in this case...

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# Molien for modular representations?

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or disprove that

$\sum\limits_{g\in G} \frac{1}{\det\left(\mathrm{id}-T\rho\left(g\right)\right)} = 0$ as an equality between power series in $k\left[T\right]$.

Motivation:

Let $G$ be a finite group, and $k$ be any field. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. For every $d\geq 0$, let $n_d$ denote the dimension of the space $\left(\mathrm{Sym}^d V^{\ast}\right)^G$ of the $G$-invariant symmetric $d$-ary forms on $V$. Molien's formula states that

$\left|G\right|\cdot \sum\limits_{n=0}^{\infty} n_dT^d = \sum\limits_{g\in G} \frac{1}{\det\left(\mathrm{id}-T\rho\left(g\right)\right)}$ as an equality between power series in $k\left[T\right]$

if $\mathrm{char} k$ does not divide $\left|G\right|$. I am wondering whether this formula still holds if $\mathrm{char} k\mid \left|G\right|$. The standard proof, using the Maschke projection, does not make much sense in this case...