Consider the Cayley graph G = Cay$(S_n,T_n)$ where $S_n$ is the symmetric group and $T_n = \{(i,i+1) | 1 \leq i \leq n-1\}$ is the set of adjacent transpositions. G is sometimes called the permutahedron (two vertices, indexed by permutations $\pi, \sigma \in S_n$, are adjacent iff $\pi = \tau \circ \sigma$ for some $\tau \in T_n$).
I have found some eigenvectors for the adjacency matrix of G here: https://msp.org/pjm/1990/143-1/pjm-v143-n1-p06-p.pdf (Section 6). Is there a full characterization of all the eigenspaces?