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The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. Flip invariance was proved for several particular cases of integrable quad-equations by Bazhanov, Mangazeev and Sergeev and by Lobb and Nijhoff. We provide a simple and case-independent proof for all integrable quad-equations. Moreover, we find a new relation for Lagrangians within one elementary quadrilateral which seems to be a fundamental building block of the various versions of flip invariance.

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincare equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincare equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.