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

TL;DR

This paper introduces generalized energies for U_q(D^{(1)}_n) crystals using piecewise linear functions, linking them to particle counting in an integrable cellular automaton, with conjectured formulas expressed as ultradiscrete tau functions.

Contribution

It defines a new class of generalized energies for affine crystals and connects them to particle dynamics in a D^{(1)}_n cellular automaton, including explicit conjectured formulas.

Findings

Generalized energies count particles and anti-particles in the automaton.

Explicit formulas for energies are conjectured as ultradiscrete tau functions.

The approach unifies crystal theory with integrable cellular automata.

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.