A tantalizing open question in computational complexity is to understand the 'behavioral differences' between the determinant and the permanent. While the former is computable in polynomial time with Gauss pivot, the second is $\# P$-hard by Valiant's result. There have been attempts to generalize the notions of determinant/permanent, but I was wondering if the following notion of subgroup-restricted determinant had been considered.
Fix a series $\Sigma$ of subgroups $G_1 \leq G_2 \leq \ldots \leq G_k$ of $S_n$, and for an index $p \leq k$ let $\Sigma_p$ denote the truncated series $G_1 \leq G_2 \leq \ldots \leq G_p$. Say that $\Sigma$ is normal if (i) $G_1$ is simple, (ii) for each $1 \leq i < k$, $G_i$ is a maximal normal subgroup of $G_{i+1}$. We may then define, for an $n \times n$ matrix $A$:
Given $K \subseteq S_n$, let $Sum_{K}(A) = \sum_{\sigma \in K} a_{1,\sigma(1)} \ldots a_{n,\sigma(n)}$;
Let $SgrDet_{\Sigma}(A) = Sum_{G_k}(A) - \sum_{i = 1}^{k-1} [G_i : G_k] SgrDet_{\Sigma_i}(A)$.
Observe that when $G_{k}$ is isomorphic to $Z_n$, any maximal series $\Sigma$ corresponds to a prime factorization of $n = p_1^{i_1} \ldots p_k^{i_k}$, and then $SgrDet_{\Sigma}(I_n)$ corresponds to the 'signed divisor function' $\zeta'(n) = [i_1]_{-p_1} \ldots [i_k]_{-p_k}$ where by convention $[n]_q = \sum_{i = 0}^{n-1} q^i$.
Observe that we can recover the permanent and determinant with the series $(S_n)$ and $(A_n \lhd S_n)$, respectively. Note that the second series is normal (by the simplicity of $A_n$) while the first is not. This leads to the following questions, probably difficult:
(1) does the formula always have exponential algebraic complexity when the series is not normal?
(2) are there other examples of a normal series for which the formula has polynomial complexity?
(NOTE: I updated the definition of $SgrDet$ to have a better-behaved operator, but it's still unclear whether it is the 'right' definition).
