# Why do we want $p$-permutation modules in splendid equivalences?

First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences between (principal) blocks of algebras $A$ of $\mathbb{K}G$ and $B$ of $\mathbb{K}H$ with an isomorphic defect group $P$ this way:
$1)$ There exists a complex $X$ of finitely generated $(A,B)$-bimodules such that $\operatorname{Hom}_A (X,X) \simeq B$ and $\operatorname{Hom}_B(X,X) \simeq A$ in the homotopy category of complexes of $B$-modules (resp. $A$-modules), and all terms of $X$ are projective as left and right modules (this is called a split-endomorphism two sided tilting complex)
$2)$ All the terms of $X$, considered as modules of the group algebra of $G \times H$, are relatively projective with respect to the diagonal embedding of $P$ and are $p$-permutation modules (direct summands of permutation modules)

Now, reading both papers, I haven't really been able to understand the requirement that these modules have to be $p$-permutation modules, meaning I don't understand what we lose if we have a split-endomorphism two sided tilting complex made of $\operatorname{diag}(P)$-relatively projective modules that are not $p$-permutation modules.

I am kind of new to this theory, so I'm probably missing something huge. Thanks to anyone who will help me.

• Permutation (and $p$-permutation) modules encode combinatorial information. Perhaps that's why... Feb 13, 2016 at 11:22

Suppose $G$ is a finite group with abelian Sylow $p$-subgroup $P$, and $H=N_G(P)$ is the normalizer of $P$. Then Broué's Abelian Defect Group Conjecture predicts that the derived categories of the principal blocks of $kG$ and $kH$ (where $k$ is a sufficiently large field of characteristic $p$) should be equivalent. For any subgroup $Q\leq P$, the principal blocks of $kC_G(Q)$ and $kC_H(Q)$ should also have equivalent derived categories, and it seems reasonable to hope for some kind of compatibility between these equivalences for varying $Q$. This is what, at the level of character theory, Broué's notion of "isotypy" gave.
If you have a splendid equivalence between the principal blocks of $kG$ and $kH$, then you can obtain one between each pair $kC_G(Q)$ and $kC_H(Q)$ by applying a certain functor (the "Brauer construction") to the complex $X$. The main reason that it's important that the terms of $X$ are $p$-permutation modules is that, although it's possible to define the Brauer construction for general modules, it behaves much better for $p$-permutation modules, and this is needed to prove that it gives a tilting complex for $kC_G(Q)$ and $kC_H(Q)$.
One is that it's useful to be able to pass between characteristic $p$ and characteristic zero by considering representation theory over a complete discrete valuation ring $\mathcal{O}$ of characteristic zero with residue field $k$. The fact that $p$-permutation modules over $k$ lift to ($p$-permutation) modules over $\mathcal{O}G$, which is not true for modules in general, is needed to prove that a splendid equivalence lifts to characteristic zero.