杨对称化子

• ${\displaystyle {\mathfrak {S}}_{d}}$ 是 对称群
• ${\displaystyle \mathbb {C} {\mathfrak {S}}_{d}}$ 是${\displaystyle {\mathfrak {S}}_{d}}$ 的群环
  • ${\displaystyle e_{g}}$ 是它的基
• ${\displaystyle \lambda =(\lambda _{1},\lambda _{2},\cdot \cdot \cdot ,\lambda _{k})}$ 
  • 其中${\displaystyle \lambda _{1}+\cdot \cdot \cdot +\lambda _{k}=d}$ 是整数d 的分解(partition)，这些${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3},......}$ 越来越小。
• Y是相应的Young diagram
• P={${\displaystyle g\in {\mathfrak {S}}_{d}}$ | g 保存每一行}
• Q={${\displaystyle g\in {\mathfrak {S}}_{d}}$ | g 保存每一列}
• ${\displaystyle a_{\lambda }=\sum _{g\in P}e_{g}\in \mathbb {C} {\mathfrak {S}}_{d}}$ 
• ${\displaystyle b_{\lambda }=\sum _{g\in Q}(-1)^{g}e_{g}\in \mathbb {C} {\mathfrak {S}}_{d}}$ 

这样

• ${\displaystyle c_{\lambda }:=a_{\lambda }b_{\lambda }\in \mathbb {C} {\mathfrak {S}}_{d}}$ 

就叫杨对称化子(Young symmetriser)

• William Fulton / Joe Harris (1991): "Representation Theory", ISBN 0-387-97495-4 , pp.46,45