Related papers: Highest weight vectors in plethysms

Given a weight of sl(n), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module. Moreover, we completely solve the system in a…

Let $M$ be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra $\mathcal{G}$, where $\mathcal{G}$ is $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2m|2n)$ or $\mathfrak{spo}(2m|2n)$. We show, using super…

Fix any complex Kac-Moody Lie algebra $\mathfrak{g}$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. We study arbitrary highest weight $\mathfrak{g}$-modules $V$ (with any highest weight $\lambda\in \mathfrak{h}^*$, and let…

We consider the periplectic supergroup ${\bf P} (n)$ over a ground field $\Bbbk$ of characteristic $p>2$. We show that there are four blocks of ${\bf P} (n)$ of simple supermodules $L^{\epsilon}(\lambda)$ corresponding to dominant weights…

We prove that the weight multiplicities of the integrable irreducible highest weight module over the Kac-Moody algebra associated to a quiver are equal to the root multiplicities of the Kac-Moody algebra associated to some enlarged quiver.…

The present paper continues the work of [10] and [6]. For any symmetrizable generalized Cartan Matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider the associated quiver $Q$ with an admissible automorphism $a$. We…

In this paper we discuss the highest weight $\frak k_r$-finite representations of the pair $(\frak g_r,\frak k_r)$ consisting of $\frak g_r$, a real form of a complex basic Lie superalgebra of classical type $\frak g$ (${\frak g}\neq…

The general Lagrangian for maximal supergravity in five spacetime dimensions is presented with vector potentials in the \bar{27} and tensor fields in the 27 representation of E_6. This novel tensor-vector system is subject to an intricate…

We identify Whittaker vectors for $\mathcal{W}_k(\mathfrak{g})$-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable…

Let $\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension…

We study the vector space V_k^m(\lambda) of shifted polyharmonic Maass forms of weight k \in 2Z, depth m \geq 0, and shift \lambda \in C. This space is composed of real-analytic modular forms of weight k for PSL(2,Z) with moderate growth at…

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional…

Higher spin gravity is an interesting toy model of stringy geometry. Particularly intriguing is the presence of higher spin gauge transformations that redefine notions of invariance in gravity: the existence of event horizons and…

We completely determine the minimal polynomial of an arbitrary simple highest weight module $L(\lambda)$ over a complex classical Lie algebra $\mathfrak{g}\subseteq\mathfrak{gl}_N$ relative to its defining module $\pi=\mathbb{C}^{N}$. These…

Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator…

The purpose of this note is to define and construct highest weight modules for Felder's elliptic quantum groups. This is done by using exchange matrices for intertwining operators between (not necessarily finite-dimensional) modules over…

We revisit the classical Goddard-Kent-Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators…

Let $n \geq 2$ be an integer and let $K$ be a number field with ring of integers $\mathcal{O}_K$. We prove that the set of ternary $n$-ic forms with coefficients in $\mathcal{O}_K$ and fixed nonzero discriminant, breaks up into finitely…

We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the…

We describe a relationship between the Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercube graphs. Consider the $\mathbb C$-algebra $P$ of polynomials in four commuting variables. We turn $P$ into an $\mathfrak{sl}_4(\mathbb…