Related papers: Geometric crystals and Cluster ensembles in Kac-Mo…

A duality property for star products is exhibited. In view of it, known star-product schemes, like the Weyl-Wigner-Moyal formalism, the Husimi and the Glauber-Sudarshan maps are revisited and their dual partners elucidated. The tomographic…

Let G be a split real Kac-Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan-Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a…

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…

For simply-laced Kac-Moody algebras $\frak g$, Stembridge (2003) proposed a `local' axiomatization of crystal graphs of representations of $U_q(\frak g)$. In this paper we propose axioms for edge-2-colored graphs which characterize the…

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK…

Buan, Iyama, Reiten and Smith proved that the superpotential of a quiver corresponding to an element of Coxeter group is rigid. In this paper, we extend this result to the Berenstein-Fomin-Zelevinsky quivers corresponding to double Bruhat…

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of…

We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with…

To accelerate development of innovative materials, their modelings and predictions with useful functionalities are of vital importance. Here, based on a recently developed crystal structure prediction method, we find a new family of stable…

The mass distribution and chemical composition of globular cluster (GC) systems preserve fossil record of the early stages of galaxy formation. The observed distribution of GC colors within massive early-type galaxies in the ACS Virgo…

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…

When flat or on a firm mechanical substrate, the atomic composition and atomistic structure of two-dimensional crystals dictate their chemical, electronic, optical, and mechanical properties. These properties change when the two-dimensional…

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…

For an arbitrary compact Lie group G, we describe a model for rational G-spectra with toral geometric isotropy and show that there is a convergent Adams spectral sequence based on it. The contribution from geometric isotropy at a subgroup K…

We generalise the notion of cluster structures from the work of Buan-Iyama-Reiten-Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi-Yau category, the set of…

We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter…

We establish a Bruhat decomposition indexed by the wreath product $\Sigma_m\wr \Sigma_d$ between two symmetric groups -- note that $\Sigma_m\wr \Sigma_d$ is not a Coxeter group in general. We show that such a decomposition affords a…

We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties $G/P$ associated to an arbitrary…

A cluster algebra is an algebraic structure generated by operations of a quiver (a directed graph) called the mutations and their associated simple birational mappings. By using a graph-combinatorial approach, we present a systematic way to…

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra…