Related papers: The Gelfand-Tsetlin type base for the algebra $\ma…

In the paper we consider a realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $GL_n$. It is proved that functions corresponding to Gelfand-Tsetlin…

This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers…

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…

A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra o(2n) is constructed. The basis vectors are parametrized by the D-type Gelfand--Tsetlin patterns. Explicit formulas for the matrix elements…

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…

We consider the problem of constructing a Gelfand--Tsetlin basis in irreducible representations of an infinite-dimensional general linear group. For a finite-dimensional irreducible representation of a general linear group, all elements of…

We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for…

It is shown that the Gel'fand-Tsetlin realization of irreducible representations of the $A_n$ algebra is directly connected with a linear exactly integrable system in the n-dimensional space. General solution for this system is explicitly…

We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of…

Using the Zhelobenko's approach we investigate a branching of an irreducible representation of $g_n$ under the restriction of algebras $g_n\downarrow g_{n-1}$, where $g_n$ is a Lie algebra of type $B_n$, $C_n$, $D_n$ or a Lie algebra of…

A realization of representations of the Lie algebra $\mathfrak{o}_5$ in the space of functions on a group $Spin_5\simeq Sp_4$ is considered. In a representation we take a Gelfand-Tsetlin type base associated with a restriction…

We consider the problem of determination of the Gelfand-Tsetlin basis for unitary principal series representations of the Lie algebra $gl_n(\mathbb{C})$. The Gelfand-Tsetlin basis for an infinite-dimensional representation can be defined as…

In the paper a realization of representation of a Lie algebra $\mathfrak{sp}_4$ in the space of function on the Lie group $Sp_4$ is considered. We find a function corresponding to a Gelfand-Tsetlin type vector for $\mathfrak{sp}_4$…

Multidimensional contractions of irreducible representations of Cayley--Klein orthogonal algebras in Gel'fand--Zetlin basis are considered. Contracted over different parameters, algebras can turn out to be isomorphic. In this case method of…

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n,…

Multidimensional contractions of irreducible representations of the Cayley-Klein unitary algebras in the Gel'fand-Zetlin basis are considered. Contracted over different parameters, algebras can turn out to be isomorphic. In this case method…

A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m|n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore,…

We provide a classification and explicit bases of tableaux of all irreducible generic Gelfand-Tsetlin modules for the Lie algebra $\mathfrak{gl}(n)$.

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on…

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…