The scalar product of XXZ spin chain revisited. Application to the ground state at $\Delta=-1/2$

For the scalar product $S_n$ of the XXZ $s=1/2$ spin chain we derive a new determinant expression which is symmetric in the Bethe roots. We consider an application of this formula to the inhomogeneous groundstate of the model with $\Delta=-1/2$ with twisted periodic boundary conditions. At this point the ground state eigenvalue $\tau_n$ of the transfer matrix is known and has a simple form that does not contain the Bethe roots. We use the knowledge of $\tau_n(\mu)$ to obtain a closed expression for the scalar product. The result is written in terms of Schur functions. The computations of the normalization of the ground state and the expectation value of $\sigma^z$ are also presented.