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Masato Okado

Publications and source records attributed to Masato Okado.

69 records · Page 4Linked to original sources
arXiv2002

A crystal to rigged configuration bijection for nonexceptional affine algebras

Kerov, Kirillov, and Reshetikhin defined a bijection between highest weight vectors in the crystal graph of a tensor power of the vector representation, and combinatorial objects called rigged configurations, for type $A^{(1)}_n$. We define an analogous bijection for all nonexceptional affine types, thereby proving (in this special case) the fermionic formulas conjectured by Hatayama, Kuniba, Takagi, Tsuboi, Yamada, and the first author.

math.QA↗
arXiv2001

Virtual crystals and fermionic formulas of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$

We introduce ``virtual'' crystals of the affine types $g=D_{n+1}^{(2)}$, $A_{2n}^{(2)}$ and $C_n^{(1)}$ by naturally extending embeddings of crystals of types $B_n$ and $C_n$ into crystals of type $A_{2n-1}$. Conjecturally, these virtual crystals are the crystal bases of finite dimensional $U_q'(g)$-modules associated with multiples of fundamental weights. We provide evidence and in some cases proofs of this conjecture. Recently, fermionic formulas for the one dimensional configuration sums associated with tensor products of the finite dimensional $U_q'(g)$-modules were conjectured by Hatayama et al. We provide proofs of these conjectures in specific cases by exploiting duality properties of crystals and rigged configuration techniques. For type $A_{2n}^{(2)}$ we also conjecture a new fermionic formula coming from a different labeling of the Dynkin diagram.

math.QA↗
arXiv2001

Crystal bases and q-identities

The relation of crystal bases with $q$-identities is discussed, and some new results on crystals and $q$-identities associated with the affine Lie algebra $C_n^{(1)}$ are presented.

math.QA↗
arXiv1999

Energy Functions in Box Ball Systems

The box ball system is studied in the crystal theory formulation. New conserved quantities and the phase shift of the soliton scattering are obtained by considering the energy function (or $H$-function) in the combinatorial $R$-matrix.

math.QA↗
arXiv1999

Combinatorial R matrices for a family of crystals : C^{(1)}_n and A^{(2)}_{2n-1} cases

The combinatorial R matrices are obtained for a family {B_l} of crystals for U'_q(C^{(1)}_n) and U'_q(A^{(2)}_{2n-1}), where B_l is the crystal of the irreducible module corresponding to the one-row Young diagram of length l. The isomorphism B_l\otimes B_k \simeq B_k \otimes B_l and the energy function are described explicitly in terms of a C_n-analogue of the Robinson-Schensted-Knuth type insertion algorithm. As an application a C^{(1)}_n-analogue of the Kostka polynomials is calculated for several cases.

math.QA↗
arXiv1999

Remarks on Fermionic Formula

Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra U_q(X^{(1)}_n) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary X_n.

math.QA↗
arXiv1999

Finite Crystals and Paths

We consider a category of finite crystals of a quantum affine algebra whose objects are not necessarily perfect, and set of paths, semi-infinite tensor product of an object of this category with a certain boundary condition. It is shown that the set of paths is isomorphic to a direct sum of infinitely many, in general, crystals of integrable highest weight modules. We present examples from C_n^{(1)} and A_{n-1}^{(1)}, in which the direct sum becomes a tensor product as suggested from the Bethe Ansatz.

math.QA↗
arXiv1998

$q$-Wedge Modules for Quantized Enveloping Algebras of Classical Type

We use the fusion construction in the twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a byproduct we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the $q$-wedge products.

math.QA↗
arXiv1998

Character Formulae of $\hat{sl}_n$-Modules and Inhomogeneous Paths

Let B_{(l)} be the perfect crystal for the l-symmetric tensor representation of the quantum affine algebra U'_q(\hat{sl(n)}). For a partition mu = (mu_1,...,mu_m), elements of the tensor product B_{(mu_1)} \otimes ... \otimes B_{(mu_m)} can be regarded as inhomogeneous paths. We establish a bijection between a certain large mu limit of this crystal and the crystal of an (generally reducible) integrable U_q(\hat{sl(n)})-module, which forms a large family depending on the inhomogeneity of mu kept in the limit. For the associated one dimensional sums, relations with the Kostka-Foulkes polynomials are clarified, and new fermionic formulae are presented. By combining their limits with the bijection, we prove or conjecture several formulae for the string functions, branching functions, coset branching functions and spinon character formula of both vertex and RSOS types.

math.QA↗
arXiv1997

RSOS models and Jantzen-Seitz representations of Hecke algebras at roots of unity

A special family of partitions occurs in two apparently unrelated contexts: the evaluation of 1-dimensional configuration sums of certain RSOS models, and the modular representation theory of symmetric groups or their Hecke algebras $H_m$. We provide an explanation of this coincidence by showing how the irreducible $H_m$-modules which remain irreducible under restriction to $H_{m-1}$ (Jantzen-Seitz modules) can be determined from the decomposition of a tensor product of representations of affine $\sl_n$.

q-alg↗
arXiv1997

Combinatorics of solvable lattice models, and modular representations of Hecke algebras

We review and motivate recently-observed relationships between exactly solvable lattice models and modular representations of Hecke algebras. Firstly, we describe how the set of $n$-regular partitions label both of the following classes of objects: 1. The spectrum of unrestricted solid-on-solid lattice models based on level-1 representations of the affine algebras $\sl_n$, 2. The irreducible representations of type-A Hecke algebras at roots of unity: $H_m(\sqrt[n]{1})$. Secondly, we show that a certain subset of the $n$-regular partitions label both of the following classes of objects: 1. The spectrum of restricted solid-on-solid lattice models based on cosets of affine algebras $(sl(n)^_1 \times sl(n)^_1)/ sl(n)^_2$. 2. Jantzen-Seitz (JS) representations of $H_m(\sqrt[n]{1})$: irreducible representations that remain irreducible under restriction to $H_{m-1}(\sqrt[n]{1})$. Using the above relationships, we characterise the JS representations of $H_m(\sqrt[n]{1})$ and show that the generating series that count them are branching functions of affine $\sl_n$.

q-alg↗
arXiv1996

Demazure Modules and Perfect Crystals

We give a criterion for the Demazure crystal $B_w(λ)$ defined by Kashiwara to have a tensor product structure. We study the $\sln$ symmetric tensor case, and see some Demazure characters are expressed using Kostka-Foulkes polynomials.

q-alg↗
arXiv1996

Demazure modules and vertex models: the affine sl(2) case

We characterize, in the case of affine sl(2), the crystal base of the Demazure module E_w(\La) in terms of extended Young diagrams or paths for any dominant integral weight \La and Weyl group element w. Its character is evaluated via two expressions, 'bosonic' and 'fermionic'.

q-alg↗
arXiv1995

Excitation Spectra of Spin Models constructed from Quantized Affine Algebras of type $B_n^{(1)}$, $D_n^{(1)}$

The energy and momentum spectrum of the spin models constructed from the vector representation of the quantized affine algebras of type $\B$ and $\D$ are computed using the approach of Davies et al. \cite{DFJMN92}. The results are for the anti-ferromagnetic (massive) regime, and they agree with the mass spectrum found from the factorized S--matrix theory by Ogievetsky et al. \cite{ORW87}. The other new result is the explicit realization of the fusion construction for the quantized affine algebras of type $\B$ and $\D$.}

hep-th↗