  We consider the question of what quantum spin chains naturally encode in
their Hilbert space. It turns out that quantum spin chains are rather rich
systems, naturally encoding solutions to various problems in combinatorics,
group theory, and algebraic geometry. In the case of the XX Heisenberg spin
chain these are given by skew Kostka numbers, skew characters of the symmetric
group, and Littlewood-Richardson coefficients. As we show, this is revealed by
a fermionic representation of the theory of "quantized" symmetric functions
formulated by Fomin and Greene, which provides a powerful framework for
constructing operators extracting this data from the Hilbert space of quantum
spin chains. Furthermore, these operators are diagonalized by the Bethe basis
of the quantum spin chain. Underlying this is the fact that quantum spin chains
are examples of "quantum integrable systems." This is somewhat analogous to
bosons encoding permanents and fermions encoding determinants. This points
towards considering quantum integrable systems, and the combinatorics
associated with them, as potentially interesting targets for quantum computers.
