# Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?

A $$\mathbb Z_+$$-algebra is an algebra $$A$$ over $$\mathbb C$$ with given basis $$\{v_i\}$$ such that $$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$ An example of such an object is the character ring of a group $$G$$, given by $$\mathbb C[\chi_\rho\ |\ \rho\in\mathrm{Irr} G]$$.

A $$\mathbb Z_+$$-module over $$A$$ is a module $$M$$ over $$A$$ with given basis $$\{m_i\}$$ such that $$v_im_j=\sum_k n_{ijk}'m_k,\hspace{10mm}n_{ijk}'\in\mathbb Z_{\geq0}.$$ If $$H\subset G$$ is a subgroup, then the character ring of $$H$$ becomes a $$\mathbb Z_+$$-module over the character ring of $$G$$, by restriction to $$H$$. (so for $$\rho\in\mathrm{Irr} G$$ and $$\sigma\in\mathrm{Irr} H$$, we have $$\chi_\rho\cdot\chi_\sigma=\chi_\rho\vert_{H}\cdot\chi_\sigma$$)

Direct sums of $$\mathbb Z_+$$-modules is just the direct sum of modules, and the basis is the union of the bases. A $$\mathbb Z_+$$-module is said to be indecopmosable if it is not the direct sum of nonzero $$\mathbb Z_+$$-modules.

Now my question is this:

Are all indecomposable $$\mathbb Z_+$$-modules of the character ring of $$G$$ of the form above: so the character ring of a subgroup $$H$$ of $$G$$?

The reason I believe this is true is because a more categorial version of this is proved in EGNO's Tensor Categories, Corollary 7.12.20, but I don't understand their proof as it refers to 3 other theorems, which themselves each refer to other theorems and so on. I also think that a proof just in terms of $$\mathbb Z_+$$-algebras would be easier, but I don't know how to do it.

I do not think this is true. Let $$G = A_{5}$$, let $$A$$ denote the character ring of $$G$$. Let $$M$$ denote the $$\mathbb{C}$$-span of the Brauer characters of $$G$$ for the prime $$2$$, (extended to take the value $$0$$ on all $$2$$-singular elements).
Then $$M$$ has a $$\mathbb{C}$$-basis containing 4 class functions $$\alpha, \beta, \gamma, \delta$$, defined as follows: $$\alpha$$ takes the value $$1$$ on elements of odd order, and $$0$$ on elements of even order. The function $$\beta$$ takes value $$2$$ on the identity, $$-1$$ on elements of order $$3$$, $$\omega + \omega^{-1}$$ ( where $$\omega = e^{\frac{2 \pi i}{5}}$$) on one conjugacy class of elements of order $$5$$, $$\omega^{2} + \omega^{-2}$$ on the other class of elements of order $$5$$, and $$0$$ on elements of even order. The function $$\gamma$$ is algebraically conjugate to (and different from) $$\beta$$. The function $$\delta$$ takes the value $$4$$ on the identity, $$1$$ on elements of order $$3$$, $$-1$$ on elements of order $$5$$, $$0$$ on elements of even order.
Because $$\alpha, \beta, \gamma, \delta$$ are essentially (irreducible) Brauer characters, it follows that for any irreducible character $$\chi$$ of $$G$$, each of $$\chi \alpha, \chi \beta, \chi \gamma, \chi \delta$$ is a non-negative integer combination of $$\{\alpha, \beta, \gamma, \delta\}.$$ Then $$M$$ is an indecomposable $$\mathbb{Z}_{+}$$-module for $$A$$, but is not the character ring of a subgroup of $$G$$.