In my research, the following construction came up.


Let $X$ be an $n$-dimensional simplicial complex. For an integer $m \geq 1$, let $X[m]$ denote the following simplicial complex. The vertices of $X[m]$ are pairs $(v,i)$, where $v \in X^{(0)}$ and $i \in \{1,\ldots,m\}$. A set $\{(v\_0,i\_0),\ldots,(v\_k,i\_k)\}$ of vertices of $X[m]$ forms a $k$-simplex if the $v\_j$ are all distinct and the set $\{v\_0,\ldots,v\_k\}$ is a $k$-simplex of $X$.


I'm interested in the relationship between the homology groups of $X$ and $X[m]$.


I suspect that the answer is pretty complicated in general, so let me describe a more specific situation/question. The complexes $X$ I'm interested in are spherical buildings. For those that don't know what these are, the precise definition is not too important; rather I think the following property is the key. By the Solomon-Tits theorem, such an $X$ is $(n-1)$-connected, and thus homotopy equivalent to a wedge of $n$-spheres. In fact, even more is true. Namely, such an $X$ is a *Cohen-Macaulay complex*, that is, for a $k$-dimensional simplex $\sigma$ of $X$, the link of $\sigma$ is an $(n-k-1)$-dimensional complex which is $(n-k-2)$-connected.


If $X$ is as above, then $X[m]$ is still an $n$-dimensional simplicial complex. Moreover, it is an easy exercise to show that $X[m]$ is also still $(n-1)$-connected, so the interesting homology group is $H\_n(X[m])$. There is a natural action of the symmetric group $S\_m$ on $X[m]$, so the homology group $H\_n(X[m];\mathbb{Q})$ is a representation of $S\_m$.


This brings me to my more specific question. Assume that $X$ is as above. How can one describe the $S\_m$-representation $H\_n(X[m];\mathbb{Q})$ in terms of the homology groups of $X$ (and possibly other topological data about $X$)?



Effect on homology of decorating vertices of a simplicial complex
If I understand correctly, $w \in S\_m$ acts by sending $(v,i)$ to $(v,w(i))$. If so, you are in the nice situation where the stabilizer of a face fixes every point in that face. So, for $w \in S\_m$, the set of faces of $X[m]$ fixed by $w$ is a subcomplex of $X[m]$. By what is often called (if my memory is correct) the Hopf trace formula, the value at $w$ of the character of your representation is $(-1)^n$ times the reduced Euler characteristic of this fixed point subcomplex, where $n=dim(X)$ as in your question.


Now $w$ fixes a face $\{(v\_0,i\_0),\ldots,(v\_k,i\_k)\}$ if and only if $w$ fixes each $i\_j$. Therefore, the fixed point subcomplex for $w$ is isomorphic with $X[fix(w)]$, where $fix(w)$ is the number of fixed points of $w$ on $[m]$.


So, if you know the reduced Euler characteristic of $X[k]$ (which is the same as knowing the rank of $\widetilde{H}\_n(X[k])$) for each $k \leq m$ then you know the character values for your representation. You can use the formula of Bj\"orner-Wachs\_Welker to get this Euler characteristic. I hope this is of use to you. 


If what I have said so far is correct, your complexes provide nice examples of the type requested in problem 7.65(b) of Richard Stanley's ``Enumerative Combinatorics, Volume 2". That is, you have a sequence $\theta\_1,\theta\_2,\ldots$, where $\theta\_m$ is a character of $S\_m$, and for each $w \in S\_m$, $|\theta\_m(w)|=\theta\_k(1)$ for some $k \leq m$.


