Related papers: On the Ext groups between Weyl modules for GL_n

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given representation is symplectic or…

An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative g in a semisimple algebraic group G of an elliptic element w in the…

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where…

By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral…

In this paper we study Weyl modules for a toroidal Lie algebra $\CT$ with arbitrary $n$ variables. Using the work of Rao \cite{1995}, we prove that the level one global Weyl modules of $\CT$ are isomorphic to suitable submodules of a Fock…

Let $G$ be a connected complex algebraic group and $A$ a connected abelian algebraic group endowed with an algebraic action of $G$ by group automorphisms. In the present note we describe the abelian group $\Ext_{alg}(G,A)$ of algebraic…

The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite…

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…

In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra.…

We show that certain characteristic varieties of a finitely generated module over a given Weyl algebra arising from weighted degree filtrations are equal to the critical cone of some other characteristic varieties. This behaviour of the…

Let F be a local non-archimedean field and let G be the group of F-valued points of a reductive algebraic group over F. In this paper we compute the Ext-groups of generalized Steinberg representations in the category of smooth…

Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities $\leq 2$, where for nullity 2 there is only one known such presentation. We…

We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we…

We investigate which Weyl groups have a Coxeter presentation and which of them at least have the presentation by conjugation with respect to their root system. For most concepts of root systems the Weyl group has both. In the context of…

We study a class of algebras B(n,l) associated with integrable models with boundaries. These algebras can be identified with coideal subalgebras in the Yangian for gl(n). We construct an analog of the quantum determinant and show that its…

Locally affine Lie algebras are generalizations of affine Kac--Moody algebras with Cartan subalgebras of infinite rank whose root system is locally affine. In this note we study a class of representations of locally affine algebras…

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…

Using the \texttt{WeylModules} \textsf{GAP} Package, we compute structural information about certain Weyl modules for type $G_2$ in characteristic $2$. This gives counterexamples to two conjectures stated by S.~Donkin in 1990. It also…

A unified splitting result for Ext calculated in the category of modules over a Leibniz algebra is given for the case where coefficients are either both symmetric modules or both antisymmetric modules. This is a generalization of results of…

We develop a general framework for studying relative weight representations for certain pairs consisting of an associative algebra and a commutative subalgebra. Using these tools we describe projective and simple weight modules for quantum…