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 $X$)?

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