Let $G$ be a finite group and $k$ a field. In modular representation theory the Brauer correspondence establishes a bijection between the isomorphism classes of indecomposable $kG$-modules with trivial source and vertex $P$ and the isomorphism classes of non-zero projective indecomposable $k[N\_G(P)/P]$-modules. This bijection is induced by the *Brauer map* 
$$M \mapsto \operatorname{Br}\_P(M) := \frac{M^P}{\sum\_Q tr\_{Q}^P (M^Q)}$$
$(Q < P)$ where $M^P$ are the $P$-invariants and $tr^P\_Q: M^Q \to M^P,\;\; m \mapsto \sum\_{g \in P/Q}gm$ is the trace map. This was proved in Theorem 3.2 of the paper 


M. Broué: On Scott Modules and p-Permutation Modules: An Approach through the Brauer Morphism. Proc. Amer. Math. Soc. 93(1985), 401-408


and the statement can also be found in [this paper](ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/wildon%3Adecomposing_permutation_representations_of_symmetric_groups.pdf) of M. Wildon (end of p. 12). 


However, to my understanding, Broué's proof doesn't show that if $M\_1,M\_2$ aren't ismorphic then $\operatorname{Br}\_P(M\_1), \operatorname{Br}\_P(M\_2)$ aren't isomorphic as well (as $k[N\_G(P)/P]$-modules). 


**Question:** Does someone know, if the Brauer correspondence $[M] \mapsto [\operatorname{Br}\_P(M)]$ ($[-]$ denotes isomorphism classes) is injective ? If so, a proof or a reference which contains a proof is also welcome.




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Let me just explain why I think Broué's proof doesn't show injectivity in full strenght: He decomposes $kG \otimes\_{kP}k = \bigoplus\_{i=1}^n M\_i$ into indecomposable $kG$-Modules whereby the ones with vertex $P$ are exactly those such that $\operatorname{Br}\_P(M\_i) \neq 0$. Assume this holds for $i=1,...m$. Then 
$$k[N\_G(P)/P] = \operatorname{Br}\_P(kG \otimes\_Pk)= \bigoplus\_{i=1}^m \operatorname{Br}\_P(M\_i)$$
and the $\operatorname{Br}\_P(M\_i),\;(i=1,...m)$ can be shown to be indecomposable. Since every indecomposable projective $k[N\_G(P)/P]$-module is isomorphic to a direct summand of $k[N\_G(P)/P]$, the Brauer correspondence is surely surjective. 


However, to my understanding, it may happen that $M\_i,M\_j$ aren't isomorphic, but $\operatorname{Br}\_P(M\_i) \cong \operatorname{Br}\_P(M\_j)$ (i.e. this summand has multiplicity $>1$ in $k[N\_G(P)/P])$. 



Is the Brauer correspondence injective ? 
Actually the answer seems to be in the paper of Wildon you linked. Right after Lemma 1.3.7. he shows that $Br\_P$ coincides with the Green correspondence on indecomposable direct summands of $1\uparrow \_P^G$ (which are exactly your modules $M\_i$). Since Green corrensondence establishes a bijection between isomorphism classes of $kG$-modules with vertex $P$ and isomorphism classes of $kN\_G(P)$-modules with vertex $P$, injectivity of $Br\_P$ follows (surjectivity doesn't follow immediately from this argument, because we restricted the domain of definition of the Green correspondence to trivial source modules).


