A non-Abelian version of the Hirota-Miwa equation is considered. In an earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. 39, 5053-5065] it was shown how solutions expressed as quasideterminants could be constructed for this system by means of Darboux transformations. In this paper we discuss these solutions from a different perspective and show that the solutions are quasi-Plücker coordinates and that the non-Abelian Hirota-Miwa equation may be written as a quasi-Plücker relation. The special case of the matrix Hirota-Miwa equation is also considered using a more traditional, bilinear approach and the techniques are compared.
We present a geometric description, based on the affine Weyl group E (1) 6 , of two discrete analogues of the Painlevé VI equation, known as the asymmetric q-P V and asymmetric d-P IV . This approach allows us to describe in a unified way the evolution of the mapping along the independent variable and along the various parameters (the latter evolution being the one induced by the Schlesinger transformations). It turns out that both discrete Painlevé equations exhibit the property of self-duality: the same equation governs the evolution along any direction in the space of E (1) 6 .
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