Related papers: Breaking classical Lie groups to finite subgroups …

The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra…

We consider the finite Weyl groups of classical type -- $W(A_{r})$ for $r \geq 1$, $W(B_{r}) = W(C_{r})$ for $r \geq 2$, and $W(D_{r})$ for $r \geq 4$ -- as supergroups in which the reflections are of odd superdegree. Viewing the…

It is known that characters of irreducible representations of finite Lie algebras can be obtained using theWeyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras…

This note presents a procedure to determine the reduction of the irreducible and the induced characters of the symmetric group in terms of the irreducible and induced characters of the hyperoctahedral group Key Words: Symmetric Group,…

I calculate characters of certain representations of loop groups based on non simply connected Lie groups. This gives a generalization of the Kac-Weyl character formula.

We survey some aspects of the pseudo-differential Weyl calculus for irreducible unitary representations of nilpotent Lie groups, ranging from the classical ideas to recently obtained results. The classical Weyl-H\"ormander calculus is…

We compute the fake degrees of representations of classical Weyl groups in terms of domino tableaux.

The orbits of Weyl groups W(B(n)), W(C(n)) and W(D(n)) of the simple Lie algebras B(n), C(n) and D(n) are reduced to the union of the orbits of Weyl groups of the maximal reductive subalgebras of B(n), C(n) and D(n). Matrices transforming…

Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.

By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction…

In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights…

We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of…

For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…

Super Weyl group plays an important role in the study of representations of basic classical Lie superalgebras. The Coxeter graphs for super Weyl groups of basis classical Lie superalgebras have been given in \cite{CLS}, where the authors…

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of…

We give alternate proofs of the classical branching rules for highest weight representations of a complex reductive group $G$ restricted to a closed regular reductive subgroup $H$, where $(G,H)$ consist of the pairs $(GL(n+1),GL(n))$, $…

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…

Let $\gg$ be the Lie algebra of a compact Lie group and let $\theta$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^\theta$. In this paper we present LiE programs that, for any finite dimensional complex…

Given a grading on a nonassociative algebra by an abelian group, we have two subgroups of automorphisms attached to it: the automorphisms that stabilize each homogeneous component (as a subspace) and the automorphisms that permute the…