Recognition of positive integral projections in a group algebra

Question

Let $G$ be any finite Group and $e \in \mathbb{C}G$ be a central idempotent element which decomposes $\mathbb{C}G = R \times S$ into a direct product of rings $R = \mathbb{C}Ge$ and $S = \mathbb{C}G(1-e)$. Let $\mathbb{N}G$ denote the additive submonoid of $\mathbb{C}G$ which consists of all nonnegative integral linear combinations of the group elements. I want to classify the projection of this monoid to $R$, i.e. $\mathbb{N}Ge$.

To be more specific, I am looking for a criterion which determines for any $x \in \mathbb{Z}G$ whether there is an $y \in \mathbb{N}G$ such that $xe = ye$ holds. It would be particularly nice if this recognition can be done by only knowing the character of $G$ corresponding to $e$ (and maybe also its decomposition into irreducible characters).

Thank you in advance!

Answer 1

Given a finite group $G$, a representation $V$ of it is a vector space equipped with a map $\Bbb CG\to{\rm End}(V)$; this induces a diagonal map $\Bbb CG\to\bigoplus {\rm End}(V)$ over all irreducible representations $V$. We know this map is an isomorphism, and the central primitive idempotent ${\rm id}_V$ (for a given irrep $V$) of the codomain corresponds to the central primitive idempotent $e(\chi_V)=\frac{1}{|G|}\sum_{g\in G}\chi_V(g^{-1})g$ of the group algebra.

The central idempotents of $\Bbb CG$ look like $e(\chi)=\sum_{V\in A}e(\chi_V)$ for subsets $A\subseteq{\rm Irr}(G)$.

We will assume $0\in\Bbb N$ (otherwise $\Bbb NG$ has no additive identity and isn't a monoid).

For $x\in\Bbb CG$ and $V\in{\rm Irr}(G)$ we have $x\,e(\chi_V)=0$ if and only if $xV=0$. Let $\eta=\sum_{g\in G}g$. Observe that $\eta V$ is $G$-invariant and so either $\eta V=0$ or $V=\Bbb C$ is the trivial representation.

• ($e$ contains no trivial rep): suppose $x\in\Bbb ZG$ is arbitrary. Pick $n\in\Bbb N$ big enough so $x+n\eta\in\Bbb NG$. Then $xe=(x+n\eta)e$. Hence $\Bbb ZGe=\Bbb NGe$.
• ($e=\eta$): since $g\eta=\eta$ for all $g\in G$, the map $x\mapsto xe$ takes $x$ to the sum of its coefficients times $\eta$. Hence $\Bbb NG\eta=\Bbb N\eta$ and $xe\in\Bbb NGe$ iff the sum of $x$'s coefficients is in $\Bbb N$.

Now we need to figure out what happens when $e$ contains both trivial and nontrivial reps...