Generalized energies and integrable D(1)n cellular automaton
Abstract
We introduce generalized energies for a class of Uq(D(1)n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a special case. It is shown that the generalized energies count the particles and anti-particles in a quadrant of the two dimensional lattice generated by time evolutions of an integrable D(1)n cellular automaton. Explicit formulas are conjectured for some of them in the form of ultradiscrete tau functions.
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