The Bethe Equation at q=0, the Möbius Inversion Formula, and Weight Multiplicities. II. The Xn Case

Abstract

We study a family of power series characterized by a system of recursion Ž . relations Q-system with a certain convergence property. We show that the coefficients of the series are expressed by the numbers which formally count the Ž Ž1. . off-diagonal solutions of the U X Bethe equation at q s 0. The series are q n conjectured to be the X -characters of a certain family of irreducible finite-dimenn

Periodic Cellular Automata and Bethe Ansatz

“…The dynamical period and a state counting formula are proposed by the Bethe ansatz at q = 0 [11]. In this paper we review and generalize the results on the A (1) n case, where the associated automaton is known as the periodic box-ball system [14,19].…”

“…Back in the original indices, the determinants here can be simplified (cf. [11] (3.9)) to those of matrices indexed with H:…”

“…Let us turn to another Bethe ansatz result, the character formula called combinatorial completeness of the string hypothesis at q = 0 [11]: (12) and (13). We introduce T (P (m)) = n a=1 j≥1 {T (a) j (p) | p ∈ P (m)}, which is the subset of B consisting of all kinds of one step time evolutions of P (m).…”

“…We also recall an explicit weight multiplicity formula obtained by counting the off-diagonal solutions to the string center equation [11]. It is a version of the fermionic formula called the combinatorial completeness of the string hypothesis at q = 0.…”

“…First we relate the root of unity in the Bethe eigenvalue Λ (r) l to the dynamical period of the periodic A (1) n automaton under the time evolution T (r) l . Second we connect each summand in the weight multiplicity formula [11] to the number of states characterized by conserved quantities.…”