Related papers: Faces of polyhedra associated with relation module…
Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his…
Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating…
We present a simplified way to construct the Gelfand-Tsetlin modules over $\mathfrak{gl}(n,\mathbb C)$ related to a $1$-singular GT-tableau defined by Futorny, Grantcharov and Ramirez. We begin by reframing the classical construction of…
Recently, a class of interaction round the face (IRF) solvable lattice models were introduced, based on any rational conformal field theory (RCFT). We investigate here the connection between the general solvable IRF models and the fusion…
A Gelfand-Tsetlin tableau $T(v)$ induces a character $\chi_v$ of the Gelfand-Tsetlin subalgebra $\Gamma$ of $U = U(\mathfrak{gl}(n,\mathbb C))$. By a theorem due to Ovsienko, for each tableau $T(v)$ there exists a finite number of…
Singular Gelfand-Tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row. In this paper we construct singular Gelfand-Tsetlin modules for arbitrary singular…
We study a class of mechanisms known as Kokotsakis polyhedra with a quadrangular base. These are $3\times3$ quadrilateral meshes whose faces are rigid bodies and joined by hinges at the common edges. In contrast to existing work, the…
Remarkable subalgebras of the Yangian for gl_n called the shifted Yangians were introduced in a recent work by Brundan and Kleshchev in relation to their study of finite W-algebras. In particular, in that work a classification of…
In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over $\mathfrak{gl}_n$ and diagrammatic KLRW algebras, and develop some of its consequences. In addition to a new proof of this…
In this paper we provide an explicit tableaux realization for all simple subquotients of a relation Gelfand-Tsetlin $\mathfrak{gl}(n)$-module.
The most famous simple Lie algebra is $sl_n$ (the $n \times n$ matrices with trace equals $0$). The representation theory for $sl_n$ has been one of the most important research areas for the past hundred years and within their the simple…
We construct explicitly a large family of Gelfand-Tsetlin modules for an arbitrary finite W-algebra of type A and establish their irreducibility. A basis of these modules is formed by the Gelfand-Tsetlin tableaux whose entries satisfy…
We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the…
We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and…
In this paper, we study the Gelfand--Cetlin systems and polytopes of the co-adjoint $\mathrm{SO}(n)$-orbits. We describe the face structure of Gelfand--Cetlin polytopes and iterated bundle structure of Gelfand--Cetlin fibers in terms of…
We investigate the permutation modules associated to the set of $k$-dimensional faces of the hyperoctahedron in dimension $n$, denoted $H^{n}.$ For any $k\leq n$ such a module can be defined over an arbitrary field $F$, it is called a face…
The purpose of this paper is to construct new families of irreducible Gelfand-Tsetlin modules for U_q(gl_n). These modules have arbitrary singularity and Gelfand-Tsetlin multiplicities bounded by 2. Most previously known irreducible modules…
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular…
This is the third in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin…
We give a crystal structure on the set of Gelfand-Tsetlin patterns which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form, expressed using…