  Thermoelectric transport is traditionally analyzed using relations imposed by
time-reversal symmetry, ranging from Onsager's results to fluctuation relations
in counting statistics. In this paper, we show that a recently discovered
duality relation for fermionic systems -- deriving from the fundamental
fermion-parity superselection principle of quantum many-particle systems --
provides new insights into thermoelectric transport. Using a master equation,
we analyze the stationary charge and heat currents through a weakly coupled,
but strongly interacting single-level quantum dot subject to electrical and
thermal bias. In linear transport, the fermion-parity duality shows that
features of thermoelectric response coefficients are actually dominated by the
average and fluctuations of the charge in a dual quantum dot system, governed
by attractive instead of repulsive electron-electron interaction. In the
nonlinear regime, the duality furthermore relates most transport coefficients
to much better understood equilibrium quantities. Finally, we naturally
identify the fermion-parity as the part of the Coulomb interaction relevant for
both the linear and nonlinear Fourier heat. Altogether, our findings hence
reveal that next to time-reversal, the duality imposes equally important
symmetry restrictions on thermoelectric transport. As such, it is also expected
to simplify computations and clarify the physical understanding for more
complex systems than the simplest relevant interacting nanostructure model
studied here.
