Related papers: Relative Weyl Character formula, Relative Pieri fo…

We find the branching laws for the classical pairs $\mathrm{GL}(m, \mathbb{C}) \subset \mathrm{GL}(n, \mathbb{C})$, $\mathrm{Sp}(2m, \mathbb{C}) \subset \mathrm{Sp}(2n, \mathbb{C})$, $\mathrm{SO}(q, \mathbb{C}) \subset \mathrm{SO}(p,…

The decomposition of representations of compact classical Lie groups into representations of finite subgroups is discussed. A Mathematica package is presented that can be used to compute these branching rules using the Weyl character…

Let G be a split adjoint semisimple group over Q and K a maximal compact subgroup of the real points G(R). We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of G(R)/K.…

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…

Given a complex simply connected simple algebraic group $G$ of exceptional type and a maximal parabolic subgroup $P \subset G$, we classify all triples $(G,P,H)$ such that $H \subset G$ is a maximal reductive subgroup acting spherically on…

We derive group branching laws for formal characters of subgroups $H_\pi$ of GL(n) leaving invariant an arbitrary tensor $T^\pi$ of Young symmetry type $\pi$ where $\pi$ is an integer partition. The branchings $GL(n)\downarrow GL(n-1)$,…

Let B be a reductive Lie subalgebra of a semi-simple Lie algebra of the same rank both over the complex numbers. To each finite dimensional irreducible representation $V_\lambda$ of F we assign a multiplet of irreducible representations of…

Let G be a reductive algebraic group and V a G-module. We consider the question of when (GL(V), rho(G)) is a reductive pair of algebraic groups, where rho is the representation afforded by V. We first make some observations about general G…

For any representation of a complex simple Lie algebra $\mathfrak{sl}_n$, one problem of branching rules to $\mathfrak{sl}_2$-subalgebra is to determine the multiplicity of each irreducible component. In this paper, we derive a recursion…

The Pieri rule gives an explicit formula for the decomposition of the tensor product of irreducible representation of the complex general linear group GL(n,C) with a symmetric power of the standard representation on C^n. It is an important…

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,…

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…

The graded-fermion algebra and quasi-spin formalism are introduced and applied to obtain the $gl(m|n)\downarrow osp(m|n)$ branching rules for the "two-column" tensor irreducible representations of gl(m|n), for the case $m\leq n (n > 2)$. In…

We obtain explicit formulas for the product of a deformed Weyl denominator with the character of an irreducible representation of the spin group $\rm{Spin}_{2r+1}({\mathbb C})$, which is an analogue of the formulas of Tokuyama for Schur…

The traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important \qo{Pauli Exclusion Principle} but by the adoption of the complex standard relativistic quantum field…

In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(\lambda)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the…

We obtain the explicit direct integral decomposition of Stein's complementary series representations and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$ when restricted to the subgroup $\operatorname{GL}(2n-1, \mathbb{R})$. The…

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…

In this paper, we consider the branching law of the Speh representation $\mathrm{Sp}(\pi,n+l)$ of $\mathrm{GL}_{2n+2l}$ with respect to the block diagonal subgroup $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$ for any irreducible generic…

Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl…