Effect on homology of decorating vertices of a simplicial complex

Question

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 $X$)?

Answer 1

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 $k \leq m$.

Answer 2

This probably should just be a comment, but I do not have enough reputation to do that. I think your $X[m]$ are particular cases of the "inflated simplicial complexes" of

Björner, Anders; Wachs, Michelle L.; Welker, Volkmar Poset fiber theorems. Trans. Amer. Math. Soc. 357 (2005), no. 5, 1877–1899.

Answer 3

If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.

For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P > 0$, so $H_{d,P}(X[m])=C_d(X)\otimes U^{d,m}_P\otimes V_P$.