F. Labourie [arXiv:1212.5015] characterized the Hitchin components for PSL(n,R) for any n>1 by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank n swapping algebra, which is the quotient of the swapping algebra by the (n+1)×(n+1) determinant relations. The main results are the well-definedness of the rank n swapping algebra and the "cross-ratio" in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank n swapping algebra generated by these "cross-ratios" to characterize the PSL(n,R) Hitchin component for a fixed n>1. We also show the relation between the rank 2 swapping algebra and the cluster X space.
In [GS15], Goncharov and Shen introduce a family of mapping class group invariant regular functions on their A-moduli space to explicitly formulate a particular homological mirror symmetry conjecture. Using these regular functions, we obtain McShane identities general rank positive surface group representations with loxodromic boundary monodromy and (non-strict) McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple ratios. Moreover, we obtain McShane identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. We apply our identities to derive the simple spectral discreteness of unipotent bordered positive representations, collar lemmas, and generalizations of the Thurston metric.