Abstract We explicitly construct, in terms of Gelfand–Tsetlin tableaux, a new family simple positive energy representations for the affine vertex algebra $V_k(\mathfrak{s}\mathfrak{l}_{n+1})$ minimal nilpotent orbit $\mathfrak{s}\mathfrak{l}_{n+1}$. These are quotients induced modules over Kac–Moody $\widehat{\mathfrak{s}\mathfrak{l}}_{n+1} $ and include particular all admissible highest weight from $\mathfrak{s}\mathfrak{l}_2$. Any such module has bounded multiplicities.

Abstract We prove a uniqueness theorem for irreducible non-critical Gelfand–Tsetlin modules. The result leads to complete classification of the modules with $1$-singularity. An explicit construction such was given in Futorny et al. [7]. In particular, we show that constructed [7] exhaust all To result, introduce new category (called Drinfeld category) related generators Yangian $Y({\mathfrak{gl}}_n)$ and define functor from category.

We propose a new effective method of constructing explicitly Gelfand -Tsetlin modules for $\mathfrak{gl}_n$. obtain large family simple that have basis consisting Gelfand-Tsetlin tableaux, the action Lie algebra is given by formulas and with all multiplicities equal $1$. As an application our construction we prove necessary sufficient condition Graev's continuation to define module which was conjectured Lemire Patera.

Consider a Hölder continuous potential ϕ defined on the full shift , where A is finite alphabet. Let be specified sofic subshift. It well known that there unique Gibbs measure μϕ X associated with ϕ. In addition, natural nested sequence of subshifts type (Xm) converging to subshift X. To this we can associate measures . paper, prove these converge weakly at exponential speed (in classical distance metrizing weak topology). We also establish mixing property implies Bernoulli. Finally,...

We prove a conjecture for the irreducibility of singular Gelfand-Tsetlin modules. describe explicitly irreducible subquotients certain classes

The classical Gelfand-Tsetlin formulas provide a basis in terms of tableaux and an explicit action the generators $\mathfrak{gl} (n)$ for every irreducible finite-dimensional (n)$-module. These can be used to define (n)$-module structure on some infinite-dimensional modules - so-called generic modules. are convenient work with since tableau there exists unique module containing this as element. In paper we initiate systematic study large class non-generic $1$-singular An realization these is...

We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlin modules. The result leads to complete classification of the modules with 1-singularity. An explicit construction such was given in \cite{FGR2}. In particular, we show that constructed \cite{FGR2} exhaust all To introduce new category (called Drinfeld category) related generators Yangian Y(gl_n) and define functor from category.

We provide an explicit combinatorial realization of all simple and injective (hence, projective) modules in the category bounded $\mathfrak{sp}(2n)$-modules. This is defined via a natural tableaux correspondence between spinor-type $\mathfrak{so}(2n)$ oscillator-type $\mathfrak{sp}(2n)$. In particular, we show that, contrast with $A$-type case, generic $\mathfrak{sp}(2n)$-modules admit analog Gelfand-Graev continuation from finite-dimensional representations.

In this paper we study realizations of highest weight modules for the complex Lie algebra $\mathfrak{gl}_n$ with respect to non-standard Gelfand-Tsetlin subalgebras. We also provide sufficient conditions such subalgebras have a diagonalizable action on these realizations.