Paths, Crystals and Fermionic Formulae

Abstract

We introduce a fermionic formula associated with any quantum affine algebra Uq(X (r) N ). Guided by the interplay between corner transfer matrix and Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowski-Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one dimensional sums conjecturally give rise to branching functions of an integrable G

Existence of Crystal Bases for Kirillov–Reshetikhin Modules of Type $D$

“…The second one is a derivation of the multiplicity formula, called fermionic formula, in [16] for type A and in [7] for all non-twisted cases, by using the Q-systems. However, it deserves to emphasize that there is also a q-analogue of the conjecture, called X = M conjecture [7,6]. The definition of X requires the existence of the crystal base of a KR module.…”

Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

“…There are certain generalizations of these formulas for twisted Yangians [11], and the KirillovReshetikhin characters in these cases were shown to satisfy a Q-system [12]. We have not addressed these systems in this paper, although it is clear that the same methods should apply to these cases.…”

A geometric realization of the periodic discrete Toda lattice and its tropicalization

“….). In general, C i has a complicated form because it is the ultradiscretization of c i (see (8)). For small or large i, however, C i has a relatively simple form:…”

“…In recent years, many studies on ultradiscrete integrable systems, or integrable cellular automata have intensively been made by using various mathematical tools such as combinatorics [24,3,37,31], ultradiscretization procedure [35,27,34,14], crystal bases of quantum groups [9,7,8,11], and so forth. In particular, among many studies on geometric aspects of ultradiscrete integrable systems, several remarkable works have successively been made by Kimijima-Tokihiro [21], Inoue-Takenawa [12,13], and Iwao [17] in terms of tropical geometry and ultradiscretization procedure.…”