Related papers: Crystal $\Bla$ in $B(\infty)$ for $G_2$ type Lie A…

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing…

Thin Lie algebras are infinite-dimensional graded Lie algebras $L=\bigoplus_{i=1}^{\infty}$, with $\dim(L_1)=2$ and satisfying a covering property: for each $i$, each nonzero $z\in L_i$ satisfies $[zL_1]=L_{i+1}$. It follows that each…

Let $G$ be a graph with adjacency matrix $A(G)$ and Laplacian matrix $L(G)$. In 2024, Samanta \textit{et} \textit{al.} defined the convex linear combination of $A(G)$ and $L(G)$ as $B_\alpha(G) = \alpha A(G) + (1-\alpha)L(G)$, for $\alpha…

Let $L$ be a Lie algebra of Block type over $\C$ with basis $\{L_{\alpha,i}\,|\,\alpha,i\in\Z\}$ and brackets $[L_{\alpha,i},L_{\beta,j}]=(\beta(i+1)-\alpha(j+1))L_{\alpha+\beta,i+j}$. In this paper, we shall construct a formal distribution…

We develop a theory of bicrystalline ideals, synthesizing Gr\"obner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and…

A ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebra $\mathfrak g$ is a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded algebra $\mathfrak g$ with a bracket $[|. , . |]$ that satisfies certain graded versions of the symmetry and Jacobi…

Lusztig's theory of PBW bases gives a way to realize the infinity crystal for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced…

We study the subgroup B_0(G) of H^2(G,Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B_0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic…

$\nabla$-algebra is a natural generalization of Heyting algebra, unifying many algebraic structures including bounded lattices, Heyting algebras, temporal Heyting algebras and the algebraic presentation of the dynamic topological systems.…

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals…

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.

We strengthen the revised GCH theorem by showing, e.g., that for lambda=cf(lambda)>beth_omega, for all but finitely many regular kappa<beth_omega, lambda is accessible on cofinality kappa in a weak version of it holds. In particular,…

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…

We construct an explicit basis for the coordinate ring of the Bott-Samelson variety Z_i associated to G = GL(n) and an arbitrary sequence of simple reflections i. Our basis is parametrized by certain standard tableaux and generalizes the…

The tableau model for Kirillov-Reshetikhin (KR) crystals, which are finite dimensional crystals corresponding to certain affine Lie algebras, is commonly used for its ease of crystal operator calculations. However, its simplicity makes…

For a complex simple Lie algebra of type A_l,B_l,C_l or D_l, given a family of elements f_\lambda\ in commutative ring Z[\Lambda], we show that f_\lambda\ is just the formal character of the Weyl module V(\lambda) if f_\lambda\ satisfy…

Let $\Lambda$ be a lattice of rank $n$. A Lie algebra on the lattice $\Lambda$ is a Lie algebra ${\cal L}=\oplus_{\lambda\in\Lambda}\,{\cal L}_{\lambda}$ such that $\dim\,{\cal L}_\lambda=1$ for all $\lambda$. In this article, we classify…

The crystals for finite dimensional representations of sl(n+1) can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of the embedding of…

We determine the singularity category of an arbitrary finite dimensional gentle algebra $\Lambda$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of…

We use Kang-Misra's combinatorial description of the crystal graphs for $U_{q}(G_{2})$ to introduce the plactic monoid for type $G_{2}$. Then we describe the corresponding insertion algorithm which yields a Schensted type correspondence.…