Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z\_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ over $R$, i.e., $R[G]$-modules of the form $R[G/H]$, where $H\subset G$ is a subgroup, and direct sums of such modules. These are free $R$-modules where $G$ acts so that there exists a basis of the module preserved (as a whole) by the action.


I am interested in finite exact sequences of representations of the above type, particularly in those of them that are not very long. There are some beautiful examples, e.g., for $G=\mathbb Z/2$ and $R=\mathbb Z/2$ there is an exact sequence
$$
 0\rightarrow R\rightarrow R[G]\rightarrow R\rightarrow 0,
$$
while for $G=\mathbb Z/n$ and $R=\mathbb Z$ there is an exact sequence
$$
 0\rightarrow R\rightarrow R[G]\rightarrow R[G]\rightarrow R\rightarrow 0.
$$
For the fourth symmetric group $G=\mathbb S\_4$ and $R=\mathbb Z$ there is an exact sequence
$$
 0\rightarrow R\rightarrow R[\mathbb X\_4]\oplus R\rightarrow
 R[\mathbb X\_6]\oplus R\rightarrow R[\mathbb X\_3]\rightarrow0,
$$
where $\mathbb X\_4$ is the four-element set that $\mathbb S\_4$ permutes, $\mathbb X\_6$ is the set of all two-element subsets of $\mathbb X\_4$, and $\mathbb X\_3$ is the quotient set of $\mathbb X\_6$ by the obvious involution. Dihedral groups also have some four-term exact sequences of permutational representations.


Where is one supposed to get such exact sequences? There are some obvious ways, like e.g. one can take cones of morphisms of exact sequences of this type, or one can do restriction or induction from one group to another one. Are there any other constructions?


For constructions to be interesting, they should of course be removed far enough from the trivial case when $|G|$ is invertible in $R$. E.g., to have $R=\mathbb Z\_p$ and $|G|$ a large $p$-group would be perhaps most highly nontrivial.


EDIT: One of the commenters asked where does the sequence for $\mathbb S\_4$ come from, so let me say a few words about this. Not that I really understand it, but there is a geometric construction using a CW complex, and not quite of the kind that Greg suggests in his second answer below.


Represent the group $\mathbb S\_4$ as the group of rotations of the $3$-dimensional cube. Consider the quotient CW complex of the cube's surface by the central symmetry involution. The group $\mathbb S\_4$ still acts on the quotient. The set of vertices of the quotient is the $\mathbb S\_4$-set $\mathbb X\_4$, the set of edges is the $\mathbb S\_4$-set $\mathbb X\_6$, and the set of faces is the $\mathbb S\_4$-set $\mathbb X\_3$.


Now consider the map $R[\mathbb X\_4]\rightarrow R[\mathbb X\_6]$ assigning to a vertex the sum of the three edges ending in it (without any signs!) and also the map $R[\mathbb X\_6]\rightarrow R[\mathbb X\_3]$ assigning to an edge the sum of the two faces bordering on it (also without any signs!). The composition of these two maps is not zero, of course; what it does is taking every vertex to twice the sum of all the three faces. One somehow transforms this pair of arrows into a four-term exact sequence by adding the trivial $\mathbb S\_4$-module direct summands $R$ in several degrees.



Exact sequences of permutational representations?
[Here is a preprint](http://math.ucsc.edu/~boltje/publications/p09y.pdf) by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S\_n$ (over $\mathbb{Z}$) by Young modules. This is presumably a related tool.


(An earlier version of this answer had some out-of-step comments about resolutions that were either not helpful or already addressed in the original question.)


