Related papers: Clebsh-Gordan coefficients for the algebra $\mathf…

Let X be a smooth affine variety over a field k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in k. The answer is given in topological terms…

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…

This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra.…

We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method…

We use a (pre)-Kuznetsov type formula to prove a density result for the Borel-type congruence subgroup of GLn. This has some arithmetic applications to optimal lifting and counting considered earlier by A. Kamber and H. Lavner for $GL_3$.

Two descriptions of the dual $-1$ Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra $\mathfrak{o}(4)$ and the Lie superalgebra $\mathfrak{osp}(1|2)$.

In third paper of the series we construct a large family of representations of the quantum toroidal $\gl_1$ algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the…

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces $\mathbb{A}^{m,n}$ with trivial coefficients. In this paper we extend the methodology developed in the previous paper (arXiv:2210.16585) to perform the…

Consider an extension of finite dimensional nilpotent Lie algebras $0 \to \mathfrak{h} \to \tilde{\mathfrak{g}} \to \mathfrak{g} \to 0$ (over a field $k$ of characteristic zero) corresponding to an extension of unipotent algebraic groups $1…

If R, S, T are irreducible SL_3-representations, we give an easy and explicit description of a basis of the space of equivariant maps from R tensor S to T. We apply this method to the rationality problem for invariant function fields. In…

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of the quadratic space $\mathbb{R}^{3,3}$. We show that this algebra describes in a unified way…

Let $S$ be a discrete abelian semigroup with unit and concellations and $\widehat {S}$ the semigroup of semicharacters of $S$. We shaw that the strong boundary and the Shilov boundary of the algebra of generalized analytic functions defined…

Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $\Delta$. Let $\Gamma$ be a finite set of integral weights of $\mathfrak{g}$ containing…

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…

In this note, we formulate and prove branching rules of simple polynomial modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. Our branching rules depend on the conjugacy class of the Borel subalgebra. A Gelfand-Tsetlin basis of a…

The irreducible $*$-representations of the Lie algebra $su(1,1)$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist…

A type of closed exterior algebra in R3 under the cross product is revealed to hold between differential forms from the three Whittaker scalar potentials, associated with the fields of a moving electron. A special algebraic structure is…

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of the Bessel function can be realized as the…

We study the structure of the Gelfand-Tsetlin subalgebras of a family of idempotent truncations of the generalized blob algebras. We obtain optimal presentations in terms of generators and relations, monomial basis and dimension formula.