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 dealt with C and D types). A weight basis for each representation of the Lie algebra o(2n+1) is constructed. The basis vectors are parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the matrix elements of generators of o(2n+1) in this basis are given. The construction is based on the representation theory of the Yangians. A similar approach is applied to the A type case where the well-known formulas due to Gelfand and Tsetlin are reproduced.
We construct functors categorifying the branching rules for $U_q(\mathfrak{g})$ for $\mathfrak{g}$ of type $B_n$, $C_n$, and $D_n$ for the embeddings $so_{2n+1}\supset so_{2n-1}$, $sp_{2n}\supset sp_{2n-2}$, and $so_{2n}\supset so_{2n-2}$. We give the corresponding categorical Gelfand-Tsetlin basis.
The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For $\mathfrak{g}=\mathfrak{gl}_n$ the Gelfand-Tsetlin commutative subalgebra in $U(\mathfrak{g})$ arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{so}_{2n+1}$. The limit subalgebra is described explicitly in terms of Bet…
The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For $\mathfrak{g}=\mathfrak{gl}_n$ the Gelfand-Tsetlin commutative subalgebra in $U(\mathfrak{g})$ arises as some limit of subalgebras from this family. In our previous work (arXiv:1807.11126) we studied the analogous limit of shift of argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{o}_{2n+1}$. We described the li…
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 finite-dimensional irreducible representations of the shifted Yangians and the associated finite W-algebras was given. We construct a class of these representations in an explicit form via bases of Gelfand-Tsetlin type.
In this note we compute the leading term with respect to the De Concini-Kac filtration of $U_q(\mathfrak{gl}_n)$ of a generating set for the quantum Gelfand-Tsetlin subalgebra.
We define a quotient of the category of finitely generated modules over the cyclotomic Khovanov-Lauda-Rouquier algebra for type An and show it has a module category structure over a direct sum of certain cyclotomic Khovanov-Lauda-Rouquier algebras of type An-1, this way categorifying the branching rules for the inclusion of sl(n) in sl(n+1). Using this we give a new, elementary proof of Khovanov-Lauda cyclotomic conjecture. We show that continuing recursively gives the Gelfand-Tsetlin basis for type An. As an application we prove a conjecture of Mackaay, Stosic and Vaz concerning categorical W…
Representations of Super Yangians with Gelfand-Tsetlin bases
Shenzhen International Center for Mathematics, Southern University of Science and Technology（深圳数学中心，南方科技大学）；School of Artificial Intelligence, Jianghan University（江汉大学人工智能学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
研究超杨代数的表示，通过评价同态将其协变张量模视为超杨代数模，并给出简单商模块的 tame 条件，扩展了纳扎罗夫和塔拉索夫的工作。
This is a review paper on the Gelfand-Tsetlin type bases for representations of the classical Lie algebras. Different approaches to construct the original Gelfand-Tsetlin bases for representations of the general linear Lie algebra are discussed. Weight basis constructions for representations of the orthogonal and symplectic Lie algebras are reviewed. These rely on the representation theory of the B,C,D type twisted Yangians
We construct generic Gelfand-Tsetlin representations of the $\imath$quantum groups $U_q^{\text{tw}}(\mathfrak{so}_3)$ and $U_q^{\text{tw}}(\mathfrak{so}_4)$. These representations are infinite-dimensional analogs to the finite-dimensional irreducible representations provided by Gavrilik and Klimyk. They are quantum analogs of generic Gelfand-Tsetlin representations constructed by Mazorchuk. We give sufficient conditions for irreducibility and provide an upper bound for the length with the help of Casimir elements found by Molev, Ragoucy, and Sorba.
In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonneg…
We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052-3132, arXiv:1301.0634], and are expected to provide tools for the study of statistical mechanical models, representation theory and random matrices. As first application of our formulas, we characterize the boundary of the $(q,t)$-deformation of the Gelfand-Tsetlin graph when $t=q^θ$ and $θ$ is a positive integer.
For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over $\…
Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in $U(\g)^{\otimes n}$ as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.
We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard $q$-deformed enveloping algebra $U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra $U(\mathfrak{so}_n)$. In this paper we only consider the case when $q$ is not a root of unity, and $q\to 1$ for the classical case. Extending work by Mazorchuk on $\mathfrak{so}_n$, we provide rational matrix coefficients for these infinite-dimensional modules of both $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$. We use these modules with r…
The problem of the description of finite factor representations of the infinite-dimensional unitary group, investigated by Voiculescu in 1976, is equivalent to the description of all totally positive Toeplitz matrices. Vershik-Kerov showed that this problem is also equivalent to the description of the simplex of central (i.e. possessing a certain Gibbs property) measures on paths in the Gelfand-Tsetlin graph. We study a quantum version of the latter problem. We introduce a notion of a q-centrality and describe the simplex of all q-central measures on paths in the Gelfand-Tsetlin graph. Conject…
We provide Harish-Chandra type formulas for the multivariate Bessel functions and Heckman-Opdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof-Kirillov Jr. Integration over the Liouville tori of the Gelfand-Tsetlin integrable system and adjunction for higher Calogero-Moser Hamiltonians recovers and gives a new proof of the integral realization over Gelfand-Tsetlin polytopes which appeared in the recent work of Bo…
We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we…
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 of generators of o(2n) in this basis are given. The construction is based on the representation theory of the Yangians and extends our previous results for the symplectic Lie algebras.
We consider $U_{q}(\mathfrak{gl}_{n})$, the quantum group of type $A$ for $|q| = 1$, $q$ generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary $q$: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit $q \rightarrow 1$ that were obtained by different means…
In [8], the affine vertex algebra $L_k(\mathfrak{sl}_2)$ is realized as a subalgebra of the vertex algebra $Vir_c \otimes Π(0)$, where $Vir_c$ is a simple Virasoro vertex algebra and $Π(0)$ is a half-lattice vertex algebra. Moreover, all $L_k(\mathfrak{sl}_2)$--modules (including, modules in the category $KL_k$, relaxed highest weight modules and logarithmic modules) are realized as $Vir_c \otimes Π(0)$--modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case $\mathfrak{g}= \mathfrak{sl}_3$ and present realization of the VO…
