- #1

Tompson Lee

- 5

- 0

Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is homotopy equivalent to a wedge of spheres. A classical example is the action of groups of Lie type on spherical buildings. On homology this gives an integral form of the Steinberg representation.

One may ask if there exists a complex of lower dimension than the Tits building that realizes the (integral) Steinberg representation in this way. I am guessing that the answer is No, but how to prove it?

More generally, given an integral G-representation that can be realized as the homology of a spherical complex with an action of G, is there an effective lower bound on the dimension of such a complex? One obvious lower bound is given by the minimal length of a resolution by permutation representations. Is this something that has been studied?