Polyhedral Realizations Of Crystal Bases Research Articles

Adapted sequences and polyhedral realizations of crystal bases for highest weight modules

The polyhedral realizations for crystal bases of the integrable highest weight modules of Uq(g) have been introduced in Nakashima (1999) [13], which describe the crystal bases as sets of lattice points in the infinite Z-lattice Z∞ given by some system of linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. If the pair (ι,λ) (λ: a dominant integral weight) satisfies the ‘ample’ condition then there are some procedure to calculate the sets of linear inequalities.In this article, we show that if ι is an adapted sequence (defined in our paper [5]) then the pair (ι, λ) satisfies the ample condition for any dominant integral weight λ in the case g is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases B(λ) associated with arbitrary adapted sequences ι in terms of column tableaux. As an application, we will give a combinatorial description of the function εi⁎ on the crystal base B(∞).

Adapted sequence for polyhedral realization of crystal bases

The polyhedral realization of crystal base has been introduced by Zelevinsky and Nakashima, which describe the crystal base as a polyhedral convex cone in the infinite -lattice To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. According to this one has certain set of linear functions defining a polyhedral convex cone and under the “positivity condition” on ι, and it has been shown that the polyhedral convex cone is isomorphic to the crystal base To confirm the positivity condition for a given ι, we need to obtain the whole feature of the set of linear functions, which requires, in general, a bunch of explicit calculations. In this article, we introduce the notion of the adapted sequence and show that if ι is an adapted sequence, then the positivity condition holds for classical Lie algebras. Furthermore, we reveal the explicit forms of the polyhedral realizations associated with arbitrary adapted sequences ι in terms of column tableaux.

Polyhedral realizations of crystal bases and convex-geometric Demazure operators

The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.

Polyhedral realizations of crystal bases B(λ) for quantum algebras of nonexceptional affine types

We give explicit forms of the crystal bases B(λ) for the integrable highest weight modules of the quantum affine algebras for A2n(2), A2n−1(2), Bn(1), Cn(1), and Dn+1(2).

Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara's involution (*-operation) corresponds to a change of valuations on the rational function field.

Polyhedral Realizations of Crystal Bases B(λ) for Dn1

We give the explicit forms of the crystal bases B(λ) (λ ∈P+) for the integrable highest weight modules of the quantum affine algebras for .

Polyhedral realizations of crystal bases for quantum algebras of classical affine types

We give the explicit forms of the crystal bases B(∞) for the quantum affine algebras of types \documentclass[12pt]{minimal}\begin{document}$A^{(2)}_{2n-1}$\end{document}A2n−1(2), \documentclass[12pt]{minimal}\begin{document}$A^{(2)}_{2n}$\end{document}A2n(2), \documentclass[12pt]{minimal}\begin{document}$B^{(1)}_{n}$\end{document}Bn(1), \documentclass[12pt]{minimal}\begin{document}$C^{(1)}_{n}$\end{document}Cn(1), \documentclass[12pt]{minimal}\begin{document}$D^{(1)}_{n}$\end{document}Dn(1), and \documentclass[12pt]{minimal}\begin{document}$D^{(2)}_{n+1}$\end{document}Dn+1(2) by using the method of polyhedral realizations of crystal bases.

Polyhedral realization of crystal bases for generalized Kac-Moody algebras

In this paper, we give a polyhedral realization of the crystal B(∞) of Uq−(𝔤) for the generalized Kac–Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac–Moody algebras of rank 2 and 3, and Monster Lie algebras.

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Rank 2

We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component B 0(λ) containing the unit element for arbitrary rank 2 cases. We also present the explicit form of B 0(λ) containing the highest (or lowest) weight vector by the polyhedral realization method.

Polyhedral realizations of crystal bases for quantum algebras of finite types

Polyhedral realization of crystal bases is one of the methods for describing the crystal base B(∞) of a quantized enveloping algebra explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases B(λ) of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases B(∞) and B(λ) are only given in the case of arbitrary rank 2, of An and of An(1). So, we will give the polyhedral realizations of crystal bases B(∞) and B(λ) for all simple Lie algebras in this paper.

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Type A #

ABSTRACT 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 . We also present the explicit form of the unique highest-weight vector in the connected component.

Polyhedral Realizations of Crystal Bases for Quantized Kac–Moody Algebras

Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.