Dimer covering and Ising model on lattices homogeneous under the icosahedral group

Conclusions

The interest of present paper stems out, on the one hand, from the well-known relations

$$\begin{gathered} (3.1) (Pf A)^2 = \det A \hfill \\ and \hfill \\ (3.2) In det A = Tr ln A, \hfill \\ whereby \hfill \\ (3.3) In Pf A = \frac{1}{2} Tr ln A. \hfill \\ \end{gathered} $$

The left-hand side of (3.3) has been explicitly computed for tho two cases worked out in sect.1 and2. The right-hand side expanded in a formal power series results in a linear combination of all the possible words inG ₆₀, among which the trace operation picks out only those equivalent to the identity.

Comparing on both sides monomial terms of the general formx ^h y ^k gives then a complete solution of Dehn’s problem for that group.

On the other hand, finer and finer extensions (triangulations) of the lattices considered can be constructed by using new cosets, so as to obtain new lattices homogeneous under the same group with an increasing number of sites, thus gaining a different insight in the way the singularity inF (corresponding to the order-disorder phase transition) is built up when approaching the thermodynamic limit.

Work along these lines is in progress.