Reiho Sakamoto

Tau functions in combinatorial Bethe ansatz

Abstract We introduce ultradiscrete tau functions associated with rigged configurations for A n ( 1 ) . They satisfy an ultradiscrete version of the Hirota bilinear equation and play a role analogous to a corner transfer matrix for the box–ball system. As an application, we establish a piecewise linear formula for the Kerov–Kirillov–Reshetikhin bijection in the combinatorial Bethe ansatz. They also lead to general N-soliton solutions of the box–ball system.

Correspondence between conformal field theory and Calogero-Sutherland model

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators Ln. We calculate explicitly the matrix elements of Ln with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagrammatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions.

Crystal interpretation of Kerov---Kirillov---Reshetikhin bijection II. Proof for

Abstract In proving the Fermionic formulae, a combinatorial bijection called the Kerov–Kirillov–Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial R matrices on rigged configurations.

\mathfrak{sl}_{n}

Vertex models with quantum group symmetry give rise to integrable cellular automata at q = 0. We study a prototype example known as the periodic box–ball system. The initial value problem is solved in terms of an ultradiscrete analogue of the Riemann theta function whose period matrix originates in the Bethe ansatz at q = 0.

case

The Kirillov-Schilling-Shimozono (KSS) bijection appearing in theory of the Fermionic formula gives an one-to-one correspondence between the set of elements of tensor products of the Kirillov-Reshetikhin crystals (called paths) and the set of rigged configurations. It is a generalization of Kerov-Kirillov-Reshetikhin bijection and plays inverse scattering formalism for the box-ball systems. In this paper, we give an algebraic reformulation of the KSS map from the paths to rigged configurations, using the combinatorial R and energy functions of crystals. It gives a characterization of the KSS bijection as an intrinsic property of tensor products of crystals.

The Bethe ansatz in a periodic box?ball system and the ultradiscrete Riemann theta function

We provide a conjecture for the following two quantities related to the spin- isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.

Kirillov–Schilling–Shimozono Bijection as Energy Functions of Crystals

Extending the work in Schilling (Int. Math. Res. Not. 2006:97376, 2006), we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov–Reshetikhin crystal Br,s of type

Singular solutions to the Bethe ansatz equations and rigged configurations

D_{n}^{(1)}

Affine crystal structure on rigged configurations of type

for any r,s. We also introduce a representation of Br,s (r≠n−1,n) in terms of tableaux of rectangular shape r×s, which we coin Kirillov–Reshetikhin tableaux (using a nontrivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov–Reshetikhin crystals and rigged configurations.

D_{n}^{(1)}

For types A (1) and D (1) n we prove that the rigged configuration bijection in- tertwines the classical Kashiwara operators on tensor products of the arbitrary Kirillov{ Reshetikhin crystals and the set of the rigged configurations.