研究僧的学习小站

\(SU(l)\)群的直积分解

\(SU(2)\)群的直积分解

例1：\(SU(2)\)群两个不可约表示的直积\(D^{\left( 2 \right)}\left( 1 \right) \otimes D^{\left( 2 \right)}\left( 1 \right)\)的杨图分解如图所示

图1 \(D^{\left( 2 \right)}\left( 1 \right) \otimes D^{\left( 2 \right)}\left( 1 \right)\)的杨图分解

则

\( D^{\left( 2 \right)}\left( 1 \right) \otimes D^{\left( 2 \right)}\left( 1 \right) =D^{\left( 2 \right)}\left( 2 \right) +D^{\left( 2 \right)}\left( 0 \right) \)

或者可用表示的维度简记为

\( 2 \otimes 2 = 3 \oplus 1 \)

例2：\(SU(2)\)群两个不可约表示的直积\(D^{\left( 2 \right)}\left( 2 \right) \otimes D^{\left( 2 \right)}\left( 1 \right)\)的杨图分解如图所示

图2 \(D^{\left( 2 \right)}\left( 2 \right) \otimes D^{\left( 2 \right)}\left( 1 \right)\)群的杨图分解

则

\( D^{\left( 2 \right)}\left( 2 \right) \otimes D^{\left( 2 \right)}\left( 1 \right) =D^{\left( 2 \right)}\left( 3 \right) +D^{\left( 2 \right)}\left( 1 \right) \)

或者可用表示的维度简记为

\( 3 \otimes 2 = 4 \oplus 2 \)

\(SU(3)\)群的直积分解