﻿ 点群胚的Orbifold K-理论

# 点群胚的Orbifold K-理论Orbifold K-Theory of Point Orbifolds

Abstract: In this paper we study the point orbifold, and calculate some kinds of ring structure of point orr-bifolds. Then we compare the difference among these ring structures.

1. 引言

Orbifold K-理论是Adem和Ruan [1] 首先提出来的。他们把orbifold X上的orbifold K-理论 ${K}_{orb}\left(X\right)$ 定义为orbifold X上所有orbifold丛的等价类构成的Grothendieck群。若X是一个global quotient $\left[X/G\right]$ ，则 ${K}_{orb}\left(X\right)$ 刚好是等变K-理论 ${K}_{G}\left(X\right)$ 。给定一个twisting $\alpha :G×G\to C$ ，Adem和Ruan [1] 还定义了twisted orbifold K-理论，但是他们并没有给出两类orbifold K-理论的环结构。

${e}^{*}:{K}_{orb}\left(X\right)\to {K}_{orb}\left(\Lambda X\right)$ ,

${\alpha }_{1},{\alpha }_{2}\in {K}_{orb}\left(X\right)$ 限制在 $\Lambda X$ 的每个分支上。在 ${K}_{orb}\left(\Lambda X\right)$ 上做弦积，然后利用 ${e}^{*}$ 的左逆 ${e}_{#}$ 拉回到 ${K}_{orb}\left(X\right)$ 上。即

${\alpha }_{1}{·}_{HW}{\alpha }_{2}={e}_{#}\left({e}^{*}{\alpha }_{1}{·}_{ARZ}{e}^{*}{\alpha }_{2}\right)$ .

$\left({V}_{1},{\pi }_{1}\right){·}_{L}\left({V}_{2},{\pi }_{2}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}\ne {\pi }_{2},\\ \frac{|G|}{n}\left({V}_{1},{\pi }_{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}={\pi }_{2}.\end{array}$

2. 准备知识

Orbifold X的每一个不动子集 ${X}_{\left(g\right)}$ 上都有自然的orbifold结构。因此X可对应一个更大的orbifold $\Lambda X={\coprod }_{\left(g\right)}{X}_{\left(g\right)}$ 。称之为X的inertia orbifold。Adem，Ruan和Zhang [2] 定义了 ${}^{\alpha }{K}_{orb}\left(\Lambda X\right)$ 上的弦积 ${·}_{ARZ}$ 。对任意 $\stackrel{˜}{\beta },\stackrel{˜}{\gamma }\in {}^{\alpha }K{}_{orb}\left(\Lambda X\right)$

$\stackrel{˜}{\beta }{·}_{ARZ}\stackrel{˜}{\gamma }={\left({e}_{12}\right)}_{\ast }\left({e}_{1}^{\ast }\stackrel{˜}{\beta }\otimes {e}_{2}^{\ast }\stackrel{˜}{\gamma }\otimes {\lambda }_{-1}\left({E}^{\left(2\right)}\right)\right)$ ,

$\begin{array}{l}{e}_{1}:{\left(\Lambda X\right)}^{\left(2\right)}\to \Lambda X,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,\left(g,h\right)\right)↦\left(x,g\right);\\ {e}_{2}:{\left(\Lambda X\right)}^{\left(2\right)}\to \Lambda X,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,\left(g,h\right)\right)↦\left(x,h\right);\\ {e}_{12}:{\left(\Lambda X\right)}^{\left(2\right)}\to \Lambda X,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,\left(g,h\right)\right)↦\left(x,gh\right).\end{array}$

${E}^{\left(2\right)}$${\left(\Lambda X\right)}^{\left(2\right)}$ 上的阻碍丛。

$e={\coprod }_{\left(g\right)}{e}_{\left(g\right)}:\Lambda X\to X.$

${e}^{\ast }:{K}_{orb}\left(X\right)\to {K}_{orb}\left(\Lambda X\right)$

${e}_{#}:{K}_{orb}\left(\Lambda X\right)\to {K}_{orb}\left(X\right)$ .

${K}_{orb}\left(X\right)$ 上的环结构 ${·}_{HW}$ 如下定义，对任意 $\beta ,\gamma \in {K}_{orb}\left(X\right)$

$\beta {·}_{HW}\gamma ={e}_{#}\left({e}^{\ast }\beta {·}_{ARZ}{e}^{\ast }\gamma \right)$ .

$\begin{array}{l}\phi {:}^{\alpha }{K}_{orb}\left(X\right)\to {\oplus }_{g\in G}K\left({X}^{g}\right)\otimes C,\\ E↦{\oplus }_{g\in G}{E}_{g}.\end{array}$

${}^{\alpha }{K}_{orb}\left(X\right)\cong {\left({\oplus }_{g\in G}K\left({X}^{g}\right)\otimes C\right)}^{{G}_{\alpha }}$ .

$\stackrel{˜}{\beta }·\stackrel{˜}{\gamma }={\left({e}_{12}\right)}_{\ast }\left({e}_{1}^{\ast }\stackrel{˜}{\beta }\otimes {e}_{2}^{\ast }\stackrel{˜}{\gamma }\otimes {\lambda }_{-1}\left({E}^{\left(2\right)}\right)\right)$ ,

$\phi$ 把这个乘法拉回到 ${}^{\alpha }{K}_{orb}\left(X\right)$ 上，便得到 ${}^{\alpha }{K}_{orb}\left(X\right)$ 的乘法 ${·}_{L}$ 。即对于任意 $\beta ,\gamma \in {}^{\alpha }K{}_{orb}\left(X\right)$

$\beta {·}_{L}\gamma ={\phi }^{-1}\left(\phi \left(\beta \right)·\phi \left(\gamma \right)\right)$ .

3. 几类Orbifold K-理论的环结构

3.1. 一般Orbifold K-理论的环结构 ${·}_{HW}$

${·}^{G}$ 的inertia orbifold 为 $\Lambda {·}^{G}=\left(G,G×G\right)$ 。其源映射和靶映射分别为

$\begin{array}{l}s:G×G\to G,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(g,h\right)↦g;\\ t:G×G\to G,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(g,h\right)↦h.\end{array}$

$\Lambda {·}^{G}=\left(G,G×G\right)$ 等价于global quotient $\left[G/G\right]$ 。其中G在自身的作用为 $g·h={g}^{-1}hg$ 。因此，我们有

${K}_{orb}\left(\Lambda {·}^{G}\right)\cong {K}_{G}\left(G\right)$ .

$\stackrel{˜}{V}{·}_{ARZ}\stackrel{˜}{W}={\oplus }_{g\in G}\left({\oplus }_{h\in G}{\stackrel{˜}{V}}_{h}\otimes {\stackrel{˜}{W}}_{{h}^{-1}g}\right)$ .

$\begin{array}{c}V{·}_{HW}W={e}_{#}\left({e}^{\ast }V{·}_{ARZ}{e}^{\ast }W\right)\\ ={e}_{#}\left(\left({\oplus }_{g\in G}{V}_{g}\right){·}_{ARZ}\left({\oplus }_{g\in G}{W}_{g}\right)\right)\\ ={e}_{#}\left({\oplus }_{g,h\in G}{V}_{h}\otimes {W}_{{h}^{-1}g}\right)\\ =|G|V\otimes W\end{array}$

3.2. 一般Orbifold K-理论的环结构 ${·}_{L}$

$\left({V}_{1},{\pi }_{1}\right),\left({V}_{2},{\pi }_{2}\right)$ 为G的两个不可约表示， ${\chi }_{1},{\chi }_{2}$ 分别为它们的特征标，则有正交性定理

$\left({\chi }_{1},{\chi }_{2}\right)=\frac{1}{|G|}\underset{g\in G}{\sum }\stackrel{\to }{{\chi }_{1}\left(g\right)}{\chi }_{2}\left(g\right)=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\chi }_{1}={\chi }_{2},\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\chi }_{1}\ne {\chi }_{2}.\end{array}$

${\pi }_{1},{\pi }_{2}$ 可以构造线性变换

${\varphi }_{{\pi }_{1},{\pi }_{2}}=\frac{1}{|G|}\underset{g}{\sum }{\pi }_{1}^{\ast }\left(g\right){\pi }_{2}\left(g\right):{V}_{1}^{\ast }\otimes {V}_{2}\to {V}_{1}^{\ast }\otimes {V}_{2}$ .

${\varphi }_{{\pi }_{1},{\pi }_{2}}\circ \left({\pi }_{1}^{\ast }\left(h\right){\pi }_{2}\left(h\right)\right)={\varphi }_{{\pi }_{1},{\pi }_{2}}$ .

${\varphi }_{{\pi }_{1},{\pi }_{2}}\circ {\varphi }_{{\pi }_{1},{\pi }_{2}}={\varphi }_{{\pi }_{1},{\pi }_{2}}$ .

${\pi }_{1}\ne {\pi }_{2}$ ，则由特征标的正交性，有

$tr{\varphi }_{{\pi }_{1},{\pi }_{2}}=\frac{1}{|G|}\underset{g\in G}{\sum }{\chi }_{{V}_{1}^{\ast }\otimes {V}_{2}}\left(g\right)=\frac{1}{|G|}\underset{g\in G}{\sum }\stackrel{\to }{{\chi }_{1}\left(g\right)}{\chi }_{2}\left(g\right)=0$ .

${\pi }_{1}={\pi }_{2}$ 。记 $\pi \doteq {\pi }_{1}={\pi }_{2}$$V\doteq {V}_{1}={V}_{2}$$\varphi \doteq {\varphi }_{{\pi }_{1},{\pi }_{2}}$ 。取V中的一个规范正交基 ${v}_{1},{v}_{2},\cdots ,{v}_{n}$ 使得对任意 $g\in G$$\pi \left(g\right)$ 都为酉矩阵。取 ${v}_{1}^{\ast },{v}_{2}^{\ast },\cdots ,{v}_{n}^{\ast }$ 为其对偶基，则 ${\pi }^{\ast }\left(g\right)={\left(\pi {\left(g\right)}^{-1}\right)}^{\text{T}}=\pi \left(g\right)$

$\begin{array}{c}tr\varphi =\frac{1}{|G|}\underset{g\in G}{\sum }{\chi }_{{V}_{1}^{\ast }\otimes {V}_{2}}\left(g\right)=\frac{1}{|G|}{\chi }_{{V}^{\ast }}\left(g\right){\chi }_{V}\left(g\right)\\ =\frac{1}{|G|}\stackrel{¯}{{\chi }_{V}\left(g\right)}{\chi }_{V}\left(g\right)=1\end{array}$

$\varphi \left({v}_{1}^{\ast }\otimes {v}_{1}+{v}_{2}^{\ast }\otimes {v}_{2}+\cdots +{v}_{n}^{\ast }\otimes {v}_{n}\right)={v}_{1}^{\ast }\otimes {v}_{1}+{v}_{2}^{\ast }\otimes {v}_{2}+\cdots +{v}_{n}^{\ast }\otimes {v}_{n}$ .

$\varphi \left({v}_{1}^{\ast }\otimes {v}_{1}\right)={a}_{1}{v}_{1}^{\ast }\otimes {v}_{1}+{a}_{2}{v}_{2}^{\ast }\otimes {v}_{2}+\cdots +{a}_{n}{v}_{n}^{\ast }\otimes {v}_{n}$ ,

${a}_{1}+{a}_{2}+\cdots +{a}_{n}=1$ .

$\varphi \left({v}_{i}^{\ast }\otimes {v}_{j}\right)=\left\{\begin{array}{l}\frac{1}{n}\left({v}_{1}^{\ast }\otimes {v}_{1}+{v}_{2}^{\ast }\otimes {v}_{2}+\cdots +{v}_{n}^{\ast }\otimes {v}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=j,\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ne j.\end{array}$

$\left({V}_{1},{\pi }_{1}\right),\left({V}_{2},{\pi }_{2}\right)$ 为两个不可约的G表示，则

$\begin{array}{l}\phi \left({V}_{1},{\pi }_{1}\right)=\underset{g\in G}{\sum }{\chi }_{1}\left(g\right){V}_{g},\\ \phi \left({V}_{2},{\pi }_{2}\right)=\underset{g\in G}{\sum }{\chi }_{2}\left(g\right){V}_{g}\end{array}$

$\phi \left({V}_{1},{\pi }_{1}\right)\phi \left({V}_{2},{\pi }_{2}\right)=\left(\underset{g\in G}{\sum }{\chi }_{1}\left(g\right){V}_{g}\right)\left(\underset{g\in G}{\sum }{\chi }_{2}\left(g\right){V}_{g}\right)=\underset{h\in G}{\sum }\left(\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right){V}_{h}\right)$

$\begin{array}{l}\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right)\\ =\underset{g\in G}{\sum }{\chi }_{1}^{\ast }\left(g\right){\chi }_{2}\left(gh\right)\\ =\underset{g\in G}{\sum }tr\left({\pi }_{1}^{\ast }\left(g\right)\right)·tr\left({\pi }_{2}\left(g\right){\pi }_{2}\left(h\right)\right)\\ =\underset{g\in G}{\sum }tr\left[\left({\pi }_{1}^{\ast }\left(g\right)\otimes {\pi }_{2}\left(g\right)\right)·\left({\pi }_{1}\left(1\right)\otimes {\pi }_{2}\left(h\right)\right)\right]\\ =|G|tr\left[{\varphi }_{{\pi }_{1},{\pi }_{2}}\circ \left({\pi }_{1}\left(1\right)\otimes {\pi }_{2}\left(h\right)\right)\right]\end{array}$

${\pi }_{1}\ne {\pi }_{2}$ ，则 ${\varphi }_{{\pi }_{1},{\pi }_{2}}=0$ ，因此 $\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right)=0$

${\pi }_{1}={\pi }_{2}$ ，取规范正交基 ${v}_{1},{v}_{2},\cdots ,{v}_{n}$ 。对于任意 $i,j$

$\begin{array}{l}\varphi \circ \left({\pi }^{\ast }\left(1\right)\otimes \pi \left(h\right)\right)\left({v}_{i}^{\ast }\otimes {v}_{j}\right)\\ =\varphi \left({v}_{i}^{\ast }\otimes \underset{k=1}{\overset{n}{\sum }}{h}_{kj}{v}_{k}\right)=\underset{k=1}{\overset{n}{\sum }}{h}_{kj}\varphi \left({v}_{i}^{\ast }\otimes {v}_{k}\right)\\ ={h}_{ij}\frac{1}{n}\left({v}_{1}^{\ast }\otimes {v}_{1}+{v}_{2}^{\ast }\otimes {v}_{2}+\cdots +{v}_{n}^{\ast }\otimes {v}_{n}\right)\end{array}$

$\begin{array}{l}\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right)\\ =|G|tr\left[{\varphi }_{{\pi }_{1},{\pi }_{2}}\circ \left({\pi }_{1}\left(1\right)\otimes {\pi }_{2}\left(h\right)\right)\right]\\ =\frac{|G|}{n}\left({h}_{11}+{h}_{22}+\cdots +{h}_{nn}\right)\\ =\frac{|G|}{n}\chi \left(h\right)\end{array}$

$\left({V}_{1},{\pi }_{1}\right){·}_{L}\left({V}_{2},{\pi }_{2}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}\ne {\pi }_{2},\\ \frac{|G|}{n}\left({V}_{1},{\pi }_{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}{\pi }_{1}={\pi }_{2}.\end{array}$

$\left({V}_{1},{\pi }_{1}\right){·}_{L}\left({V}_{2},{\pi }_{2}\right)={\phi }^{-1}\left(\phi \left({V}_{1},{\pi }_{1}\right)·\phi \left({V}_{2},{\pi }_{2}\right)\right)=0$ .

${\pi }_{1}={\pi }_{2}$ ，由于 $\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right)=\frac{|G|}{n}\chi \left(h\right)$ ，则 $\left({V}_{1},{\pi }_{1}\right){·}_{L}\left({V}_{2},{\pi }_{2}\right)={\phi }^{-1}\left(\frac{|G|}{n}\chi \left(h\right)\right){V}_{h}=\frac{|G|}{n}\left({V}_{1},{\pi }_{1}\right)$ 。证毕

3.3. α-Twisted Orbifold K-理论的环结构 ${·}_{L}$

$\phi {:}^{\alpha }{K}_{orb}\left({·}^{G}\right)\to {\left(K\left(G\right)\right)}^{{G}_{\alpha }}$ ,

${\left(K\left(G\right)\right)}^{{G}_{\alpha }}$ 上的乘法为：对任意 $\underset{g\in G}{\sum }{a}_{g}{V}_{g}$$\underset{h\in G}{\sum }{a}_{h}{V}_{h}\in {\left(K\left(G\right)\right)}^{{G}_{\alpha }}$$\left(\underset{g\in G}{\sum }{a}_{g}{V}_{g}\right)·\left(\underset{h\in G}{\sum }{a}_{h}{V}_{h}\right)=\underset{g,h\in G}{\sum }{a}_{g}{b}_{h}\frac{1}{\alpha \left(g,h\right)}{V}_{gh}$

$\begin{array}{l}\left({V}_{1},{\pi }_{1}\right){·}_{L}\left({V}_{2},{\pi }_{2}\right)\\ ={\phi }^{-1}\left(\phi \left({V}_{1},{\pi }_{1}\right)·\phi \left({V}_{2},{\pi }_{2}\right)\right)\\ ={\phi }^{-1}\left[\left(\underset{g\in G}{\sum }{\chi }_{1}\left(g\right){V}_{g}\right)\otimes \left(\underset{g\in G}{\sum }{\chi }_{2}\left(g\right){V}_{g}\right)\right]\\ ={\phi }^{-1}\left[\underset{h\in G}{\sum }\left(\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right)\frac{1}{\alpha \left({g}^{-1},gh\right)}\right){V}_{h}\right]\end{array}$

$\begin{array}{l}\underset{g\in G}{\sum }{\chi }_{1}\left({g}^{-1}\right){\chi }_{2}\left(gh\right)\frac{1}{\alpha \left({g}^{-1},gh\right)}\\ =\underset{g\in G}{\sum }tr{\pi }_{1}\left({g}^{-1}\right)tr{\pi }_{2}\left(gh\right)\frac{1}{\alpha \left({g}^{-1},gh\right)}\\ =\underset{g\in G}{\sum }tr\left(\alpha \left(g,{g}^{-1}\right){\pi }_{1}^{\ast }\left(g\right)\right)tr\left(\frac{1}{\alpha \left(g,h\right)}{\pi }_{2}\left(g\right){\pi }_{2}\left(h\right)\right)\frac{1}{\alpha \left({g}^{-1},gh\right)}\\ =\underset{g\in G}{\sum }tr\left({\pi }_{1}^{\ast }\left(g\right)\right)tr\left({\pi }_{2}\left(g\right){\pi }_{2}\left(h\right)\right)\frac{\alpha \left(g,{g}^{-1}\right)}{\alpha \left(g,h\right)\alpha \left({g}^{-1},gh\right)}\end{array}$

$\begin{array}{l}=\underset{g\in G}{\sum }tr\left({\pi }_{1}^{\ast }\left(g\right)\right)tr\left({\pi }_{2}\left(g\right){\pi }_{2}\left(h\right)\right)\\ =\underset{g\in G}{\sum }tr\left[\left({\pi }_{1}^{\ast }\left(g\right)\otimes {\pi }_{2}\left(g\right)\right)\left({\pi }_{1}^{\ast }\left(1\right){\pi }_{2}\left(h\right)\right)\right]\\ =|G|tr\left[{\varphi }_{{\pi }_{1},{\pi }_{2}}\circ \left({\pi }_{1}^{\ast }\otimes {\pi }_{2}\left(h\right)\right)\right]\end{array}$

$|G|tr\left[{\varphi }_{{\pi }_{1},{\pi }_{2}}\circ \left({\pi }_{1}^{\ast }\otimes {\pi }_{2}\left(h\right)\right)\right]=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}\ne {\pi }_{2},\\ \frac{|G|}{n}{\chi }_{1}\left(h\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}={\pi }_{2}.\end{array}$

$\left({V}_{1},{\pi }_{1}\right){·}_{L}\left({V}_{2},{\pi }_{2}\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}\ne {\pi }_{2},\\ \frac{|G|}{n}\left({V}_{1},{\pi }_{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\pi }_{1}={\pi }_{2}.\end{array}$

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