Linear relations among permutation matrices Given a permutation $\sigma\in S_n$, let $P_\sigma$ denote the corresponding $n\times n$ permutation matrix. It is easy to see that for $n=3$, there is only one linear relation up to scaling given by $\sum_\sigma sign(\sigma)P_\sigma = 0$. For larger $n$, what are all linear relations $\sum_\sigma c_\sigma P_\sigma=0$?
 A: The standard permutation representation $\mathbb{C}^n$ of $S_n$ breaks up as a direct sum of a trivial representation $1$ and an irreducible representation $V$ of dimension $n - 1$. Hence the natural map $\mathbb{C}[S_n] \to \text{End}(\mathbb{C}^n)$ has image in $\text{End}(1) \oplus \text{End}(V)$. In fact, in terms of the Peter-Weyl decomposition
$$\mathbb{C}[S_n] \cong \bigoplus_V \text{End}(V)$$
where $V$ runs over all complex irreps of $S_n$,the map above is precisely the projection onto the two summands given by $\text{End}(1)$ and $\text{End}(V)$. So the kernel of this projection consists of the summands $\text{End}(W)$ where $W$ is neither trivial nor isomorphic to $V$. Somewhat more explicitly, there is an element of the kernel for each matrix coefficient of $W$. For example, if $\chi_W$ is the character of $W$ then there are linear relations of the form
$$\sum_{\sigma \in S_n} \overline{\chi}_W(\sigma) \sigma = 0$$
and when $n = 3$ this is the only relation, coming from the only irrep of $S_3$ not appearing in $\mathbb{C}^3$, namely the sign rep. 
A: See https://arxiv.org/abs/1804.00916 for an answer in terms of the technology of cellular algebras (due to Graham and Lehrer [Invent. math. 1996]). If I understand the result, it shows that the set of linear relations is a certain cell ideal in the defining cell chain, with respect to Murphy's alternating basis of $\mathbb{C}[S_n]$.
