Related papers: A local characterization of $B_2$ regular crystals

Generalized splines are a simultaneous generalization of GKM theory -- which studies equivariant cohomology -- and classical splines, which provide piecewise approximations of functions. Generalized splines can also be understood via…

We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications…

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer…

The (reduced) characteristic group of a locally conformally product manifold is obtained by restricting the action of its fundamental group to the non-flat factor of the universal cover, and taking the connected component of the identity in…

Let $C$ be a simply laced generalized Cartan matrix. Given an element $b$ of the generalized braid semigroup related to $C$, we construct a collection of mutation-equivalent quivers with potentials. A quiver with potential in such a…

In this paper we describe all group gradings by a finite abelian group $\Gamma$ of a simple Lie algebra of type $G_2$ over an algebraically closed field $F$ of characteristic 0.

We define geometric/unipotent crystal structure on unipotent subgroups of semi-simple algebraic groups. We shall show that in $A_n$-case, their ultra-discretizations coincide with crystals obtained by generalizing Young tableaux.

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of…

We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel…

Periodic frameworks with crystallographic symmetry are investigated from the perspective of a general deformation theory of periodic bar-and-joint structures in $R^d$. It is shown that natural parametrizations provide affine section…

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…

Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible…

On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the…

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental…

For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with…

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.

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise a comparison theorem between the rational crystalline cohomology of the special fibre and the rational $p$-adic \'etale…

There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and…