Related papers: On characters of $L_{\frak{sl}_n}(-\Lambda_0)$-mod…
In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all…
Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\hgltwo$ and the finite-dimensional group $GL_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\hgltwo$ in…
In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether $\operatorname{St}_r…
The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of…
Basic modules of McLain groups $M=M(\Lambda,\leq, R)$ are defined and investigated. These are (possibly infinite dimensional) analogues of Andr\'e's supercharacters of $U_n(q)$. The ring $R$ need not be finite or commutative and the field…
In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group $\GL_n(\mathbb F_q)$. Green's theory of $\GL_n(\mathbb F_q)$ is recovered and enhanced under the…
Let $\mathfrak{g}$ be a simple finite dimensional complex Lie algebra and let $\widehat{\mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a…
In this paper we give a new formula for characters of finite dimensional irreducible $\frak{gl}(m,n)$ modules. We use two main ingredients: Su-Zhang formula and Brion's theorem.
In this paper we compute the characters of certain non-irreducible N=4 superconformal modules which are different from the ones treated in our previous paper, and study their relation with characters of N=2 superconformal modules. Also, for…
The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many…
The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are…
In this paper we study the representations of loop Affine-Virasoro Algebras. As they have canonical triangular decomposition, we define Verma modules and its irreducible quotients. We give necessary and sufficient condition for an…
We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The \textit{level-$\ell$ Appell functions} $K_\ell$ satisfy open quasiperiodicity relations…
The level-$1$ integrable highest weight modules of $U_q(\widehat{sl}_2)$ admit a level-$0$ action of the same algebra. This action is defined using the affine Hecke algebra and the basis of the level-$1$ module generated by components of…
Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k + 2)-Mod. In this paper, we use the functor successively to obtain an universal first-order differential operator realization for any highest-weight representation…
We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness…
We study the representation theory of the N=1 super Heisenberg-Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg-Virasoro vertex algebra arXiv:math/0201314, arXiv:1405.1707 and arXiv:1703.00531 to the…
We begin a systematic study of unitary representations of minimal $W$-algebras. In particular, we classify unitary minimal $W$-algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We…
We develop a theory of weights for a quantum analogue of the symmetric pair (gl4,gl2 x gl2) realised as a quantum symmetric pair subalgebra. Based on Letzter's triangular decomposition we define Verma modules. Using magical operators that…
For a field $F$ of characteristic zero and an additive subgroup $G$ of $F$, a Lie algebra $B(G)$ of lock type is defined with basis $\{L_{a,i},c|a \in G, i>-2\}$ and relations…