关于Grothendieck代数的特征标

On Character of Grothendieck Algebra

本文用群表示理论和复向量理论证明了Grothendieck代数的一些性质,特别是关于Grothendieck代数基的描述,即设B为K_G(X)的基,Θ={(θ,ρ)|θ是G在X上的轨道,ρ是G_x的一个不可约表示,x∈θ},其中G_x={g∈G|gx=x},则存在1—1对应f:B→Θ使得对于任意V∈B,f(v)=(θ,ρ):V_y≠0<==>y∈θ,且ρ是G_x在V_x上的一个不可约表示,x∈θ,利用这一性质,本文给出了一个求Grothendieck代数的特征标的方法,从而改进了由Luszrig,G.在文[1]中提出的方法,并且给出二面体群D_n关于其一些子群H的Grotheodieck代数的特征标表。

In this paper, we have obtained Some properties of the Grothendieck algebra algebra by using representation theory of group and vector bundles over , particularly about the description of the basts, that is let B be a basis of KG(X), (?)={(θ,p)|θ an orbit of G on X, p an irreducible representation of Gx, x∈θ}, where Gx={g∈G| gx=x}. Then,there exists an one to one map f: B→(?) such that for V∈B: f(V)=(θ,p) and Vy≠0 if and only if y∈θ, and p an irreducible representation of Gx on Vx, x∈θ. Using this property, the method of computing characters of the Grothendieck algebra have been given. We have, therefore, improved the method given in [1] by Lusztig G.. Moreover, some examples on this method are also presented.