  We continue the study of the Inverse Participation Ratios (IPRs) of the XXZ
Heisenberg spin chain initiated by Misguich, Pasquier and Luck (2016) by
focusing on the case of the XX Heisenberg Spin Chain. For the ground state,
Misguich et al. note that calculating the IPR is equivalent to Dyson's constant
term ex-conjecture. We express the IPRs of excited states as an apparently new
"discrete" Hall inner product. We analyze this inner product using the theory
of symmetric functions (Jack polynomials, Schur polynomials, the standard Hall
inner product and $\omega_{q,t}$) to determine some exact expressions and
asymptotics for IPRs. We show that IPRs can be indexed by partitions, and
asymptotically the IPR of a partition is equal to that of the conjugate
partition. We relate the IPRs to two other models from physics, namely, the
circular symplectic ensemble of Dyson and the Dyson-Gaudin two-dimensional
Coulomb lattice gas. Finally, we provide a description of the IPRs in terms of
a signed count of diagonals of permutohedra.
