Don Page derived the formula for the average entropy of a subsystem of a quantum system (assumed to be in a pure state), if the system is partitioned into two subsystems of dimensions $m$ and $n$, such that the total Hilbert space dimension is $mn$ (1). This is given by $S_{m,n} = \sum_{k = n+1}^{mn} \frac{1}{k} - \frac{m-1}{2n}$.
Let's say I have a spin system on N sites with $U(1)$ symmetry, and I restrict myself to the largest symmetry sector with $N/2$ up spins, of dimension ${N \choose N/2}$. In this sector, the Hilbert space dimension is no longer the product of subsystem Hilbert space dimensions, which Page assumes in his formula.
Then, what is the appropriate version of the average entropy of a subsystem within a symmetry sector? Could anyone point me to such a formula if it exists?
