Polyhedral Realizations Research Articles

Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type $$\textrm{A}^{(1)}_{n-1}$$, $$\textrm{C}^{(1)}_{n-1}$$, $$\textrm{A}^{(2)}_{2n-2}$$, $$\textrm{D}^{(2)}_{n}$$

In this paper, we consider polyhedral realizations for crystal bases $B(\lambda)$ of irreducible integrable highest weight modules of a quantized enveloping algebra $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a classical affine Lie algebra of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$ or ${\rm D}^{(2)}_{n}$. We will give explicit forms of polyhedral realizations in terms of extended Young diagrams or Young walls that appear in the representation theory of quantized enveloping algebras of classical affine type. As an application, a combinatorial description of $\varepsilon_k^*$ functions on $B(\infty)$ will be given.

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(∞).

On polyhedral realizations of Hurwitz's regular map {3, 7}_18 of genus 7 with geometric symmetries

In 2017 a first selfintersection-free polyhedral realization of Hurwitz’s regular map {3, 7}18 of genus 7 was found by Michael Cuntz and the first author. For any regular map which had previously been realized as a polyhedron without self-intersections in 3-space, it was also possible to find such a polyhedron with nontrivial geometric symmetries. So it is natural to ask of whether we can find for the above-mentioned regular map a corresponding version with some non-trivial geometric symmetry. The orientation-preserving combinatorial automorphism group of this Hurwitz map is the projective special linear group PSL(2,8) of order 504 = 23 ⋅ 32 ⋅ 7. All non-trivial subgroups of PSL(2,8) are candidates for such a geometric symmetry. Using the GAP software for exploring the subgroup structure, we found that it is sufficient to consider only four cyclic subgroups whose order is 9, 7, 3, and 2, respectively. We prove that there are obstructions for selfintersection-free polyhedral realizations of the Hurwitz map {3, 7}18 of genus 7 with geometric rotational symmetries of order 9 or 3. We provide new small integer coordinates within the realization space known from 2017, which are also suitable for making a 3D-printed model. We present Kepler–Poinsot type realizations, both with 7-fold and with 3-fold rotational symmetry, the latter with integer coordinates.

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).

Generalized Young Walls for Classical Lie Algebras

In this paper, we introduce an new combinatorial model, which we call generalized Young walls for classical Lie algebras, and we give two realizations of the crystal B(∞) over classical Lie algebras using generalized Young walls. Also, we construct natural crystal isomorphisms between generalized Young wall realizations and other realizations, for example, monomial realization, polyhedral realization and tableau realization. Moreover, as applications, we obtain a crystal isomorphism between two different polyhedral realizations of B(∞).

Folding procedure for Newton-Okounkov polytopes of Schubert varieties

ABSTRACTThe theory of Newton-Okounkov polytopes is a generalization of that of Newton polytopes for toric varieties, and gives a systematic method of constructing toric degenerations of projective varieties. In the case of Schubert varieties, their Newton-Okounkov polytopes are deeply connected with representation theory. Indeed, Littelmann’s string polytopes and Nakashima-Zelevinsky’s polyhedral realizations are obtained as Newton-Okounkov polytopes of Schubert varieties. In this paper, we apply the folding procedure to a Newton-Okounkov polytope of a Schubert variety, which relates Newton-Okounkov polytopes of Schubert varieties of different types. As an application, we obtain a new interpretation of Kashiwara’s similarity of crystal bases.

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.

On regular and equivelar Leonardo polyhedra

A Leonardo polyhedron is a 2-manifold without boundary, embedded in Euclidean 3-space E 3 , built up of convex polygons and with the geometric symmetry (or rotation) group of a Platonic solid and of genus g ≥ 2. The polyhedra are named in honour of Leonardo's famous illustrations in [1] (cf. also [2]). Only six combinatorially regular Leonardo polyhedra are known: Coxeter's four regular skew polyhedra, and the polyhedral realizations of the regular maps by Klein of genus 3 and by Fricke and Klein of genus 5. In this paper we construct infinite series of equivelar (i.e. locally regular) Leonardo polyhedra, which share some properties with the regular ones, namely the same Schläfli symbols and related topological structure. So the weaker condition of local regularity allows a much greater variety of symmetric polyhedra. [1] L. Pacioli, De Divina Proportione (Disegni di Leonardo da Vinci 1500-1503) , Faksimile Dominiani, Como, 1967. [2] D. Huylebrouk, Lost in triangulation: Leonardo da Vinci's mathematical slip-up, Scientific American , March 29, 2011.

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.

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.

Nested Complexes and their Polyhedral Realizations

This note which can be viewed as a complement to Alex Postnikov's paper math.CO/0507163, presents a self-contained overview of basic properties of nested complexes and their two dual polyhedral realizations: as complete simplicial fans, and as simple polytopes. Most of the results are not new; our aim is to bring into focus a striking similarity between nested complexes and associated fans and polytopes on one side, and cluster complexes and generalized associahedra introduced and studied in hep-th/0111053, math.CO/0202004, on the other side.

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 Integrable Highest Weight Modules

We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases of integrable highest weight modules for arbitrary rank 2 Kac-Moody algebra cases, the classical A_n-case and the affine A^{(1)}_{n-1}-case.

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)}.

A geometric realization without self-intersections does exist for Dyck's regular map

Even in our decade there is still an extensive search for analogues of the Platonic solids. In a recent paper Schulte and Wills [13] discussed properties of Dyck's regular map of genus 3 and gave polyhedral realizations for it allowing self-intersections. This paper disproves their conjecture in showing that there is a geometric polyhedral realization (without self-intersections) of Dyck's regular map {3, 8}6 already in Euclidean 3-space. We describe the shape of this new regular polyhedron.