Generalized energies and integrable D^{(1)}_n cellular automaton

Abstract

We introduce generalized energies for a class of U_q(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.

For proceedings of "Infinite Analysis 09: New Trends in Quantum Integrable Systems"

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