The character of the alternating sum of the exterior powers of the standard representation

Question

Let $V$ denote the standard representation of $\mathfrak{S}_n$. Based on by-hand calculations when $n=4$ and $n=5$, I suspect that $$ \chi\left( \sum_{0\leq i < n} (-1)^i \textstyle{\bigwedge}^iV \right)(\tau) = \begin{cases} n, & \tau \textrm{ is an $n$-cycle} \\ 0, &\textrm{otherwise} \end{cases} $$ but I lack the techniques to prove this. Does anyone know a proof of or have a reference for this identity?

Answer 1

If $V$ is a representation of a group $G$ of degree $d$, $g\in G$ and $p\geq0$, then the character of $\Lambda^pV$ evaluated at $g$ is $$\chi_{\Lambda^pV}(g)=\sigma_p(\lambda_1,\dots,\lambda_m),$$ with $\lambda_1,\dots,\lambda_d$ the eigenvalues of $\rho_V(g)$ on $V$ listed with multiplicities and $\sigma_p$ the elementary symmetric function of degree $p$. It follows from this that if $\zeta$ is the character of the (virtual) representation $\sum_{p\geq0}(-1)^p\Lambda^pV$, then $$ \zeta(g) = \sum_{p\geq0}(-1)^p\sigma_p(\lambda_1,\dots,\lambda_d) = \prod_{i=1}^d(1-\lambda_i). $$ If $f_g=\det(t-\rho_V(g))\in\mathbb C[t]$ is the characteristic polynomial of $\rho_V(g)$, we thus have $$ \zeta(g) = f_g(1). $$

Now suppose that $G=S_n$, the symmetric group of degree $n$ and that $V$ is the standard, $(n-1)$-dimensional representation of $G$. If we add a copy of the trivial representation to get $W:=\mathbb C\oplus V$, this is isomorphic the permutation representation of $S_n$ on $\mathbb C^n$. If $g$ is a permutation and its disjoint cycles are of length $r_1,\dots,r_k$ (including all cycles of length $1$ corresponding to fixed points of $f$) then the characteristic polynomial of $\rho_W(g)$ is

$$\prod_{j=1}^k(t^{r_k}-1)$$

and it follows from this that the characteristic polynomial of $\rho_V(g)$ is

$$f_g=\frac{\prod_{j=1}^k(t^{r_k}-1)}{t-1},$$

which is a polynomial. This vanishes at $1$ unless $k=1$, that is, if $g$ is an $n$-cycle, and when it is such a cycle its value is clearly $n$.

Answer 2

Let $S^1$ be the unit circle in $\mathbb C$, let $T=(S^1)^n$ be the $n$-torus, and let $$U=\{(z_1,\dots,z_n)\in T:z_1\cdots z_n=1\}.$$ This is a subspace of $T$ homeorphic to an $(n-1)$-torus. The symmetric group $S_n$ acts on $T$ by permuting coordinates, and that action restricts to $U$, and passes on to the homology $H_*(U)$ (with complex coefficients). As an $S_n$-module $H_*(U)$ is the exterior algebra $W=\Lambda^*V$ of the standard representation of $S_n$, so the character of $W$ evaluated at a permutation $\pi$ is the Lefschetz number of the map $\pi:U\to U$, and is thus (see Lefschetz number equal to Euler Characteristic of Fixed Points) the Euler characteristic of the fixed point set of $\pi$ in $U$.

This fixed point set is a union of tori, all of the same dimension. That dimension is either zero or psotive — when it is positive, the Euler characeristic is zero, and when it is zero it is easy to see that the characteristic is $n$. You can probably guess for what permutations $\pi$ this happens.