相关论文: Combinatorics of $A_2$-crystals

A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that $3$-regular graphs are $2$-reconstructible.

We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated.…

We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

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 Ammann A2 tiling is a simple aperiodically ordered tiling of the plane. We consider the graph derived from this tiling, by treating each corner of each tile as a vertex and each side of each tile as an edge. We present a closed-form…

We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.

Following an idea of Gon\c{c}alvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.

To each 2-dimensional irreducible p-adic representation of Gal(Qpbar/Qp) which becomes crystalline over an abelian extension of Q_p, we associate a Banach space B(V) endowed with a linear continuous unitary action of GL_2(Q_p). When V is…

In this note, we compute the split Grothendieck ring of a generalized category of Soergel bimodules of type $A_2$, where we take one generator for each reflection. We give a presentation by generators and relations of it and a…

We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible…

Stembridge characterized regular crystals associated with a simply-laced generalized Cartan matrix (GCM) in terms of local graph-theoretic quantities. We give a similar axiomatization for $B_2$ regular crystals and thus for regular crystals…

Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of g are described: ``tensor product'' and ``multiplicity'' varieties. These varieties are closely related to Nakajima's…

We provide a procedure for generating the irreducible representations of crystallography groups in any dimension. We also furnish a strategy to investigate the topology of the unitary dual of a crystallography group using sequences of…

A new kind of aperiodic tiling is introduced. It is shown to underlie a structure obtained as a superposition of waves with incommensurate periods. Its connections to other other tilings and quasicrystals are discussed.

We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of…

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…

Several combinatorial problems of (quasi-)crystallography are reviewed with special emphasis on a unified approach, valid for both crystals and quasicrystals. In particular, we consider planar sublattices, similarity sublattices,…

We give a recursion formula to generate all equivalence classes of biconnected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We give a linear map to produce a connected graph with say, u,…

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One con show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a…

We introduce a recursive procedure for computing the number of realizations of a minimally rigid graph on the sphere up to rotations. We accomplish this by combining two ingredients. The first is a framework that allows us to think of such…