# Young 對稱化子

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Young對稱化子(Young symmetriser)係表示論入面嘅一種架生，用來整對稱羣${\displaystyle {\mathfrak {S}}_{d}}$唔約得表示

## 設

• ${\displaystyle {\mathfrak {S}}_{d}}$係 對稱羣
• ${\displaystyle \mathbb {C} {\mathfrak {S}}_{d}}$${\displaystyle {\mathfrak {S}}_{d}}$羣環
• ${\displaystyle e_{g}}$係渠嘅基
• ${\displaystyle \lambda =(\lambda _{1},\lambda _{2},\cdot \cdot \cdot ,\lambda _{k})}$
• 其中${\displaystyle \lambda _{1}+\cdot \cdot \cdot +\lambda _{k}=d}$係整數d 嘅分解(partition)，啲${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3},......}$越來越細。
• Y係相應嘅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}
• ${\displaystyle a_{\lambda }=\sum _{g\in P}e_{g}\in \mathbb {C} {\mathfrak {S}}_{d}}$
• ${\displaystyle b_{\lambda }=\sum _{g\in Q}(-1)^{g}e_{g}\in \mathbb {C} {\mathfrak {S}}_{d}}$

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

## 參攷

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