Related papers: Rationality properties of unipotent representation…

A representation $\Phi: G \to \mathrm{GL}_n(\mathbb{F})$ of a finite group $G$ is called unisingular if the matrix $\Phi(g)$ admits $1$ as an eigenvalue for any $g\in G$. In this paper, we determine all the complex irreducible unisingular…

We study the interpolation group whose elements are suitable pairs of formal power series. This group has a faithful representation into infinite lower triangular matrices and carries thus a natural structure as a Lie group. The matrix…

We define natural classes of rational and polynomial representations of the Yangian of the general linear Lie algebra. We also present the classification and explicit realizations of all irreducible rational representations of the Yangian.

This paper provides a construction of the unipotent representations for classical complex groups in terms of the Theta correspondence as introduced and studied by R. Howe. The K-type structure of unipotent representations is obtained as a…

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is…

This paper studies unitary representations with Dirac cohomology for complex groups, in particular relations to unipotent representations

We show that the rigid C*-tensor categories of finite dimensional type 1 unitary representations of the quantum groups $U_{q}(\mathfrak{g}_{2})$ corresponding to the exceptional Lie group $G_2$ for positive $q\ne 1$ have property (T).

In this paper we discuss some of Springer's work on unipotent elements in a reductive groups and on representations of Weyl groups. Among the topics considered are Springer's bijection from the unipotent variety to the nilpotent variety,…

The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for…

Stereotypical reasoning assumes that the situation at hand is one of a kind and that it enjoys the properties generally associated with that kind of situation. It is one of the most basic forms of nonmonotonic reasoning. A formal model for…

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that…

We give an apriori description of a set of irreducible representations of a Weyl group which parametrize the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre…

Our aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices…

We prove that there are infinitely many finite simple groups of symplectic Lie type, of any specified characteristic and rank, which appear as Galois groups over the field of rational numbers. This generalizes a result of Wiese, which…

We present an overview of characteristic identities for Lie algebras and superalgebras. We outline methods that employ these characteristic identities to deduce matrix elements of finite dimensional representations. To demonstrate the…

We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.

We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.

We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.

Hypercomplex numbers are unital algebras over the real numbers. We offer a short demonstration of the practical value of hypercomplex analytic functions in the field of partial differential equations.