I am following Wall’s paper *A note on symmetry of singularities* and I have some questions regarding representation theory and the homology of some objects:

Consider an action of $Sigma_k$ on a finite simplicial complex $X$ of dimension $n$, such that the action is simplicial and a simplex is fixed iff it is point-wise fixed. Consider $sigmainSigma_k$, then

$$chi_{Top}(X^sigma)=chi_{Sigma_k}(X)(sigma):= sum_i(-1)^i text{trace } sigma^*:H_i(X)rightarrow H_i(X), $$

where $X^sigma$ is the fixed subcomplex fixed by $sigma$ and $chi_{Top}$ the Euler-PoincarĂ© characteristic.

My questions are:

**1.-** Is the sum of characters $chi_{Sigma_k}(X)$ called the equivariant Euler characteristic? Because I believe the equivariant Euler characteristic is the Euler characteristic of $H_*(X)^G$ in other texts.

**2.-** Can one say anything about the character on a **single** homology group instead of the alternated sum of all the groups? For example, compare the trace of $sigma$ on one $H_i$ and $H_i(X^sigma)$. I have a lot of troubles trying to isolate one homology group from this equality (with the extra structure that I have in my problem).

**3.-** Can one say something about the isotypes corresponding to a **single** irreducible representation of $Sigma_k$? For example, compare the number of copies of the $sign$ representations and $H_*(X^sigma)$ for some $sigma$.

If it is useful, in my problem I know that every $X^sigma$ has lower dimension than $X$, if $sigmaneq 1_{Sigma_k}$.