Components of V(ρ)\otimes V(ρ)

$V(ρ) \otimes V(ρ)$ 的分量与仿射 Kac-Moody 李代数的支配权多面体

Components of $V(ρ) \otimes V(ρ)$ and dominant weight polyhedra for affine Kac-Moody Lie algebras

本文将 Kostant 关于单李代数张量积分量的结果推广至非扭曲仿射 Kac-Moody 李代数（相差饱和因子），并对仿射 $sl_n$ 给出了无饱和因子的证明。

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g be an affine Kac-Moody Lie algebra and let $λ, μ$ be two dominant integral weights for g. We prove that under some mild restriction, for any positive root $β$, $V(λ)\otimes V(μ)$ contains $V(λ+μ-β)$ as a component, where $V(λ)$ denotes the integrable highest weight (irreducible) g-module with highest weight $λ$. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product $V(…

Schur-Weyl duality for orthogonal groups

We prove Schur--Weyl duality between the Brauer algebra $\mathfrak{B}_n(m)$ and the orthogonal group $O_{m}(K)$ over an arbitrary infinite field $K$ of odd characteristic. If $m$ is even, we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of $n$-tensor space $V^{\otimes n}$ in the Brauer algebra $mathfrak{B}_n(m)$ is also given.

Exterior Algebra Structure for Relative Invariants of Reflection Groups

Let $G$ be a reflection group acting on a vector space $V$ (over a field with zero characteristic). We denote by $S(V^*)$ the coordinate ring of $V$, by $M$ a finite dimensional $G$-module and by $χ$ a one-dimensional character of $G$. In this article, we define an algebra structure on the isotypic component associated to $χ$ of the algebra $S(V^*) \otimes Λ(M^*)$. This structure is then used to obtain various generalizations of usual criterions on regularity of integers.