I'm out of my depth here, but I'm imagining something like a path integral[0] between n particles. For example, with two fermions, A and B, you'd get a fraction f of A moving to B's location, and vice versa, giving:
d[AB] = f * ([B] - [A])
The fraction is real as f^2 is the action A -> B -> A which shouldn't get a phase or A would self-destruct. If you had a bunch of anyons circling around in a magnetic field[1], then f could be a root of unity or something more complicated. You can generalize the boundary operator
d[123...n] = Σ(-1)^k * [123...k-1,k+1...n]
to be
d[123...n] = Σ χ(k) * [123...k-1,k+1...n]
where χ(k) is the character of your group. For example, if you have n particles interacting in a circle 1 -> 2 -> 3 -> ... -> n -> 1, then χ(k) = e^2πik/n. This is where the immanant comes from.
I'm out of my depth here, but I'm imagining something like a path integral[0] between n particles. For example, with two fermions, A and B, you'd get a fraction f of A moving to B's location, and vice versa, giving:
d[AB] = f * ([B] - [A])
The fraction is real as f^2 is the action A -> B -> A which shouldn't get a phase or A would self-destruct. If you had a bunch of anyons circling around in a magnetic field[1], then f could be a root of unity or something more complicated. You can generalize the boundary operator
d[123...n] = Σ(-1)^k * [123...k-1,k+1...n]
to be
d[123...n] = Σ χ(k) * [123...k-1,k+1...n]
where χ(k) is the character of your group. For example, if you have n particles interacting in a circle 1 -> 2 -> 3 -> ... -> n -> 1, then χ(k) = e^2πik/n. This is where the immanant comes from.
[0]: https://en.wikipedia.org/wiki/Path_integral_formulation
[1]: https://en.wikipedia.org/wiki/Fractional_quantum_Hall_effect