Related papers: Descriptions of the crystal $B(\infty)$ for $G_2$
We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of…
The crystal base of the modified quantized enveloping algebras of type $A_{+\infty}$ or $A_\infty$ is realized as a set of integral bimatrices. It is obtained by describing the decomposition of the tensor product of a highest weight crystal…
We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal…
We conjecture a deformation of the Weyl character formula for type G_2 in the spirit of Tokuyama's formula for type A. Using our conjecture we prove a combinatorial version of the Gindikin--Karpelevic formula for G_2, in the spirit of…
For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver…
The analysis of Coulomb crystallization is extended from one-component to two-component plasmas. Critical parameters for the existence of Coulomb crystals are derived for both classical and quantum crystals. In the latter case, a critical…
The quantum group structure of the Liouville theory is reviewd and shown to be an important tool for solving the theory.
Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the…
For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $\mathbb{C}[G^{e,c^2}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_-c^2B_-$ has a structure of cluster algebra of finite type,…
In this paper, we compute the derivations of the positive part of the two-parameter quantum group of type $G_2$ by embedding it into a quantum torus. We also show that the Hochschild cohomology group of degree $1$ of this algebra is a two…
We present a geometric construction of highest weight crystals for quantum generalized Kac-Moody algebras. It is given in terms of the irreducible components of certain Lagrangian subvarieties of Nakajima's quiver varieties associated to…
We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional $ \mathcal{N} =4 $ gauge theories. The result holds for simply-laced, non-simply laced and affine…
Some theoretical aspects of $B$ physics are reviewed. These include a brief recapitulation of information on weak quark transitions as described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix, a description of CP asymmetries in $B$ decays to…
A Lie theoretic interpretation is given to a pattern with five-fold symmetry occurring in aperiodic Penrose tiling based on isosceles triangles with length ratios equal to the Golden Section. Specifically a $B(\infty)$ crystal based on that…
We give an alternative proof of Naito--Sagaki's conjecture, which states that the restriction of $gl_{2n}(\mathbb{C})$-representations to $sp_{2n}(\mathbb{C})$ can be described in terms of crystals. Using the tableau model for crystals, we…
The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements…
We consider relativistic U(1) gauge theories in 2+1 dimensions, with N_b species of complex bosons and N_f species of Dirac fermions at finite temperature. The quantum phase transition between the Higgs and Coulomb phases is described by a…
A general theory of topological classification of defects is introduced. We illustrate the application of tools from algebraic topology, including homotopy and cohomology groups, to classify defects including several explicit calculations…
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…
We construct an integral PBW basis and an integral crystal basis of the quantum affine algebra of type A$_{2}^{(2)}$.