Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of this Cayley graph is a conjugacy class, it is not too difficult to use the representation theory of $S_n$ to elegantly count the number of nonzero eigenvalues of $A$:

$$ \text{rank}(A_n) = \binom{n-1}{0}^2 + \binom{n-1}{1}^2 + \cdots + \binom{n-1}{n-1}^2 = \binom{2(n-1)}{n-1}.$$

I am interested in the rank of $A_n$ modulo $p$ where $p$ is an odd prime for all $n$. One way to determine this would be to compute the Smith Normal Form of $A_n$ (over $\mathbb{Z}$). Let $D_n = \text{diag}(s_1,s_2,\cdots,s_r,0,\cdots,0)$ such that $s_i | s_{i+1}$ for all $1 \leq i < r := \text{rank}(A)$ be the Smith Normal Form of $A_n$. Computations for small $n$ show that the nonzero $s_i$'s are all powers of 2, which might suggest that $\text{rank}_p(A_n) = \text{rank}(A_n)$ for all $n$ and odd primes $p$.

It seems unlikely that one can divine unimodular matrices $U_n,V_n$ such that $D_n = U_nA_nV_n$ for all $n$, so I would like to think of $A_n$ as an endomorphism of the group algebra $\mathbb{F}_p[S_n]$ and perhaps use $p$-modular representation theory of $S_n$ to say something about the image of $A_n$. (Here, we are assuming $p$ is small, i.e., $p \mid n!$, so $\mathbb{F}_p[S_n]$ is not semisimple.)

Generally speaking, working with modular representations of $S_n$ is also difficult; however, the image of $A_n$ (in the characteristic 0 case) is the direct sum of the *hook-shaped* Specht modules, which are pretty well-understood, even in the modular case. In particular, Peel (1971) showed for odd primes $p$ that the hook-shaped Specht modules $S^{(n-k,1^k)}_{\mathbb{F}_p}$ are simple when $p \not \mid n$ and determined their composition series when $p \mid n$.

Experimentally, if one picks $b \in S^{(n-k,1^k)}_{\mathbb{F}_p}$ to be a $(n-k,1^k)$-*standard polytabloid* (which is a $\{0,\pm1 \}$-valued vector well-defined for any Specht module over any field), then $A_nx = b$ indeed has a solution over $\mathbb{F}_p$ for small $k$, odd primes $p$, and $n$. In the case that $p \not \mid n$, because $A_nx = b$ has a solution, it follows that $A_nx = b'$ for any $b' \in S^{(n-k,1^k)}_{\mathbb{F}_p}$, as $S^{(n-k,1^k)}_{\mathbb{F}_p}$ is irreducible by Peel's result. Here, we are "using the modular representation theory of $S_n$", but the problem is that showing a solution $x$ for $A_nx = b$ exists over $\mathbb{F}_p$ for all $0 \leq k < n$ and odd primes $p$ seems to involve similar row-operation-type calculations as putting $A_n$ into Smith Normal Form.

My (open-ended) question is whether there is a more clever way to leverage such information about the modular representation theory of $S_n$ that would circumvent row and column operations to say something about the Smith Normal Form of $A_n$ or $\text{rank}_p(A)$ for odd primes $p$.

**EDIT** Here's the SNF for small $n$ (thanks Dima for verifying these):

$n = 2$ the SNF is $1^2$

$n = 3$ the SNF is $1^42^2$

$n = 4$ the SNF is $1^{8}2^{12}$

$n = 5$ the SNF is $1^{16}2^{52}8^2$

$n = 6$ the SNF is $1^{32}2^{200}8^{20}$

$n = 7$ the SNF is $1^{64}2^{728}4^{2}8^{128}16^2$

(I have not gone beyond $n=7$, as this would take some time.)