What is the minimal dimension of a complex realising a group representation?

[Tompson Lee]

This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex).

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?

 

[jvicens]

I find this question very intriguing. It is fascinating to see how group actions on graphs can lead to integral representations of groups. In response to your question, I would like to suggest a possible approach to prove that there is no complex of lower dimension than the Tits building that can realize the Steinberg representation.

Firstly, we can consider the concept of "essential dimension" in algebraic geometry. Essential dimension is a numerical invariant that measures the complexity of an algebraic structure, such as a group or a representation. It is defined as the minimal number of parameters needed to define the structure up to isomorphism. In the case of groups, it can be thought of as the minimal number of generators needed to generate the group. In the case of representations, it can be thought of as the minimal number of variables needed to define the representation.

We can apply this concept to our problem by considering the essential dimension of the Steinberg representation. Since the Steinberg representation arises from the action of a group on a simplicial complex, we can define its essential dimension as the minimal number of parameters needed to define the action up to isomorphism. If we can show that the essential dimension of the Steinberg representation is greater than the dimension of the Tits building, then we can conclude that there is no complex of lower dimension that can realize the Steinberg representation.

To prove this, we can use the fact that the Tits building is a highly symmetric structure. It is known that the essential dimension of a highly symmetric structure is equal to its dimension. Therefore, if we can show that the essential dimension of the Steinberg representation is greater than the dimension of the Tits building, then we can conclude that there is no complex of lower dimension that can realize the Steinberg representation.

In conclusion, I believe that the concept of essential dimension can provide a possible approach to prove that there is no complex of lower dimension than the Tits building that can realize the Steinberg representation. As for your second question, I am not aware of any specific studies on finding an effective lower bound for the dimension of a complex realizing an integral G-representation. However, I believe that investigating the essential dimension of the representation may provide some useful insights.