中島 俊樹

Let G = Sp2r (C) be a simply connected simple algebraic group over C of type Cr , B and B its two opposite Borel subgroups, and W the associated Weyl group. For u, v W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v = BuB BvB is isomorphic to an upper cluster algebra A(i)C and the generalized minors (k i) are the cluster variables of C[Gu,v][5]. In the case v = e, we shall describe the generalized minor (k i) explicitly.

Let C be a simply connected simple algebraic group over C, B and B_ be two opposite Borel subgroups in C and W be the Weyl group. For u, v is an element of W, it is known that the coordinate ring C[C-u,C- v] of the double Bruhat cell C-u,C- v = BuB boolean AND B_vB_ is isomorphic to an upper cluster algebra (A) over bar( i)(C) and the generalized minors {Delta(k; i)} are the cluster variables belonging to a given initial seed in C[C-u,C- v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case C = SLr+1 (C), v = e and some special u is an element of W, we shall describe the generalized minors {Delta(k; i)} as summations of monomial realizations of certain Demazure crystals.

We shall describe explicitly the decoration functions for certain decorated geometric crystals of classical groups and we shall show that they are represented in terms of monomial realizations of crystal bases. (C) 2013 Elsevier Inc. All rights reserved.

We present a Robinson-Schensted-Knuth type one-to-one correspondence between the set of pictures and the set of pairs of Littlewood-Richardson crystals.

Let g be an affine Lie algebra with index set I = {0, 1, 2,..., n} and gL be its Langlands dual. It is conjectured in Kashiwara et al. (Trans. Am. Math. Soc. 360(7):3645-3686, 2008) that for each i ∈ I \\ {0} the affine Lie algebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for gL.We prove this conjecture for i = 2 and g = A(1)n. © Springer-Verlag London 2013.

Let g be an affine Lie algebra and g L its Langlands dual. It was conjectured by Kashiwara, Nakashima, and Okado that g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for g L. We prove that the ultradiscretization of the positive geometric crystal for g=D 4 (3) given by Igarashi and Nakashima is isomorphic to the limit of the coherent family of perfect crystals for g L= G 2 (1) constructed by Misra, Mohamad, and Okado. © 2012 by Pacific Journal of Mathematics.

We shall describe the one-to-one correspondence between the set of pictures and the set of Littlewood-Richardson crystals.

For a classical simple algebraic group G, we obtain the affirmative answer for the conjecture in Nakashima (2005)[3] that there exists an isomorphism between the geometric crystal on the flag variety and the one on the maximal uni potent subgroup U(-). (C) 2011 Elsevier B.V. All rights reserved.

We present a one-to-one correspondence between the set of admissible pictures and the Littlewood-Richardson crystals. As a simple consequence, we shall show that the set of pictures does not depend on the choice of admissible orders.

By modifying an earlier method of the authors (2008), certain affine geometric crystals are realized in affinization of the fundamental representation W(formula presented)l, and the tropical R maps for the affine geometric crystals are described explicitly. We also define prehomogeneous geometric crystals and show that for a positive geometric crystal, the connectedness of the corresponding ultra-discretized crystal is the sufficient condition for prehomogeneity of the positive geometric crystal. Moreover, the uniqueness of tropical R maps is discussed. © 2010 American Mathematical Society.

We shall realize certain affine geometric crystal of type D(4)((3)) associated with the fundamental representation W((w) over bar1) explicitly. By its explicit form, we see that it has a positive structure.

We introduce an epsilon system on a geometric crystal of type A(n), which is a certain set of rational functions with some nice properties. We shall show that it is equipped with a product structure and that it is invariant under the action of tropical R maps.

We obtain the affirmative answer to the conjecture in [14]. More precisely, let chi := (V, {e(i)}, {gamma(i)}, {epsilon(i})) be the affine geometric crystal of type G(2)((1)) in [14] and U D (chi, T, theta) a ultra-discretization of chi with respect to a certain positive structure theta. Then we show that U D (chi, T, theta) is isomorphic to the limit of coherent family of perfect crystals of type D(4)((3)) in [9].

For every non- exceptional a. ne Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra- discretization is isomorphic to the limit of certain perfect crystals of the Langlands dual affine Lie algebra.

We shall realize certain affine geometric crystal of type G(2)((1)) ex plicitly in the fundamental representation W(pi(1)). Its explicit form is rather complicated but still keeps positive structure.

We define geometric/unipotent crystal structure on maximal unipotent subgroups of semi-simple algebraic groups. We shall show that in An-case, their ultra-discretizations coincide with crystals obtained by generalizing Young tableaux. (c) 2005 Elsevier Inc. All rights reserved.

We define geometric crystals and unipotent crystals for arbitrary Kac-Moody groups and describe geometric and unipotent crystal structures on the Schubert varieties. We give some examples in affine sl(2)-case. (C) 2004 Elsevier BX All rights reserved.

We describe the crystal bases of modified quantum algebras and their connected component containing "zero vector" by the polyhedral realization method for the type A(n) and A(1)((1)). We also present the explicit form of the unique highest-weight vector in the connected component.

We define the extremal projector of the q-boson Kashiwara algebra and study their basic properties. Applying their properties to the representation theory of the category , whose objects are ''upper bounded'' -modules, we obtain its semi-simplicity and the classification of simple modules.

By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type A may be transformed into modules over Lusztig's infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we find that they are never isomorphic to infinitesimal Verma modules unless they are irreducible. The same carry over to the modular setup for the special linear groups in positive characteristic.

Properly specializing the parameters contained in the maximal cyclic representation of the nonrestricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. In this case, the representation is no longer irreducible. We show that the submodule generated by the primitive vector is the unique irreducible submodule and can be identified with an irreducible highest weight module of the finite dimensional A-type quantum algebra, which is defined as the subalgebra of the restricted quantum algebra at roots of unity. (C) 2002 American Institute of Physics.

We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.

Crystal base of the level 0 part of the modified quantum affine algebra (U) over tilde(q)(<(sI(2))over cap>)(0) is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.