영 태블로(Young tableau)

개요

• 영 다이어그램 또는 Ferrers Diagram

영 다이어그램

• 자연수 $d$의 분할 $$\lambda: \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\geq 0\,; \lambda_1+\cdots+\lambda_n=d$$

에 대응되는 다이어그램

• 7의 분할 $(4,2,1)$의 경우, 영 다이어그램은 다음과 같다

$$ \begin{array}{cccc} \square & \square & \square & \square \\ \square & \square & \text{} & \text{} \\ \square & \text{} & \text{} & \text{} \end{array} $$

영 태블로

• 주어진 자연수 $n$의 분할에 대응되는 영 다이어그램에 있는 $n$개의 상자에, $\{1,2,\cdots,n\}$의 원소을 채워넣어 얻어진다

표준 영 태블로

• 영 태블로의 각 행과 열을 따라 수가 강하게 증가(strictly increasing)할 때 이를 표준 영 태블로(standard Young tableau)라 한다
• 주어진 분할에 대한 표준 영 태블로의 개수는 갈고리 길이 공식 (hook length formula)으로 주어진다

준표준 영 태블로

• 영 태블로의 각 열을 따라 수가 강하게 증가(strictly increasing)하고 각 행을 따라 수가 약하게 증가(weakly increasing)할 때 이를 준표준 영 태블로(semistandard Young tableau)라 한다
  • 분할 λ 형태의 준표준 영 태블로의 집합으로부터 슈르 다항식(Schur polynomial)을 기술할 수 있다

표준 영 태블로

• 7의 분할 $(4,2,1)$의 영 다이어그램에 다음과 같은 수를 채워넣어 얻어진다

\begin{array}{ccc} \{1,4,6,7\} & \{2,5\} & \{3\} \\ \{1,3,6,7\} & \{2,5\} & \{4\} \\ \{1,2,6,7\} & \{3,5\} & \{4\} \\ \{1,3,6,7\} & \{2,4\} & \{5\} \\ \{1,2,6,7\} & \{3,4\} & \{5\} \\ \{1,4,5,7\} & \{2,6\} & \{3\} \\ \{1,3,5,7\} & \{2,6\} & \{4\} \\ \{1,2,5,7\} & \{3,6\} & \{4\} \\ \{1,3,4,7\} & \{2,6\} & \{5\} \\ \{1,2,4,7\} & \{3,6\} & \{5\} \\ \{1,2,3,7\} & \{4,6\} & \{5\} \\ \{1,3,5,7\} & \{2,4\} & \{6\} \\ \{1,2,5,7\} & \{3,4\} & \{6\} \\ \{1,3,4,7\} & \{2,5\} & \{6\} \\ \{1,2,4,7\} & \{3,5\} & \{6\} \\ \{1,2,3,7\} & \{4,5\} & \{6\} \\ \{1,4,5,6\} & \{2,7\} & \{3\} \\ \{1,3,5,6\} & \{2,7\} & \{4\} \\ \{1,2,5,6\} & \{3,7\} & \{4\} \\ \{1,3,4,6\} & \{2,7\} & \{5\} \\ \{1,2,4,6\} & \{3,7\} & \{5\} \\ \{1,2,3,6\} & \{4,7\} & \{5\} \\ \{1,3,4,5\} & \{2,7\} & \{6\} \\ \{1,2,4,5\} & \{3,7\} & \{6\} \\ \{1,2,3,5\} & \{4,7\} & \{6\} \\ \{1,2,3,4\} & \{5,7\} & \{6\} \\ \{1,3,5,6\} & \{2,4\} & \{7\} \\ \{1,2,5,6\} & \{3,4\} & \{7\} \\ \{1,3,4,6\} & \{2,5\} & \{7\} \\ \{1,2,4,6\} & \{3,5\} & \{7\} \\ \{1,2,3,6\} & \{4,5\} & \{7\} \\ \{1,3,4,5\} & \{2,6\} & \{7\} \\ \{1,2,4,5\} & \{3,6\} & \{7\} \\ \{1,2,3,5\} & \{4,6\} & \{7\} \\ \{1,2,3,4\} & \{5,6\} & \{7\} \end{array}

• 이렇게 얻어진 35개의 표준 영 태블로는 다음과 같다

\begin{array}{cccc} \boxed{1} & \boxed{4} & \boxed{6} & \boxed{7} \\ \boxed{2} & \boxed{5} & \text{} & \text{} \\ \boxed{3} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{6} & \boxed{7} \\ \boxed{2} & \boxed{5} & \text{} & \text{} \\ \boxed{4} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{6} & \boxed{7} \\ \boxed{3} & \boxed{5} & \text{} & \text{} \\ \boxed{4} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{6} & \boxed{7} \\ \boxed{2} & \boxed{4} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{6} & \boxed{7} \\ \boxed{3} & \boxed{4} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{4} & \boxed{5} & \boxed{7} \\ \boxed{2} & \boxed{6} & \text{} & \text{} \\ \boxed{3} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{5} & \boxed{7} \\ \boxed{2} & \boxed{6} & \text{} & \text{} \\ \boxed{4} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{5} & \boxed{7} \\ \boxed{3} & \boxed{6} & \text{} & \text{} \\ \boxed{4} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{4} & \boxed{7} \\ \boxed{2} & \boxed{6} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{4} & \boxed{7} \\ \boxed{3} & \boxed{6} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{7} \\ \boxed{4} & \boxed{6} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{5} & \boxed{7} \\ \boxed{2} & \boxed{4} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{5} & \boxed{7} \\ \boxed{3} & \boxed{4} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{4} & \boxed{7} \\ \boxed{2} & \boxed{5} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{4} & \boxed{7} \\ \boxed{3} & \boxed{5} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{7} \\ \boxed{4} & \boxed{5} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{4} & \boxed{5} & \boxed{6} \\ \boxed{2} & \boxed{7} & \text{} & \text{} \\ \boxed{3} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{5} & \boxed{6} \\ \boxed{2} & \boxed{7} & \text{} & \text{} \\ \boxed{4} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{5} & \boxed{6} \\ \boxed{3} & \boxed{7} & \text{} & \text{} \\ \boxed{4} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{4} & \boxed{6} \\ \boxed{2} & \boxed{7} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{4} & \boxed{6} \\ \boxed{3} & \boxed{7} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{6} \\ \boxed{4} & \boxed{7} & \text{} & \text{} \\ \boxed{5} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{4} & \boxed{5} \\ \boxed{2} & \boxed{7} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{4} & \boxed{5} \\ \boxed{3} & \boxed{7} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{5} \\ \boxed{4} & \boxed{7} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} \\ \boxed{5} & \boxed{7} & \text{} & \text{} \\ \boxed{6} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{5} & \boxed{6} \\ \boxed{2} & \boxed{4} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{5} & \boxed{6} \\ \boxed{3} & \boxed{4} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{4} & \boxed{6} \\ \boxed{2} & \boxed{5} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{4} & \boxed{6} \\ \boxed{3} & \boxed{5} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{6} \\ \boxed{4} & \boxed{5} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{3} & \boxed{4} & \boxed{5} \\ \boxed{2} & \boxed{6} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{4} & \boxed{5} \\ \boxed{3} & \boxed{6} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{5} \\ \boxed{4} & \boxed{6} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array},\begin{array}{cccc} \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} \\ \boxed{5} & \boxed{6} & \text{} & \text{} \\ \boxed{7} & \text{} & \text{} & \text{} \end{array}

관련된 항목들

• 대칭군의 표현론

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxeGoxMzFlUTRnMUE/edit
• http://mathworld.wolfram.com/YoungTableau.html
• http://www.sagemath.org/doc/reference/sage/combinat/tableau.html

사전 형태의 참고자료

• http://en.wikipedia.org/wiki/Young_tableau

리뷰, 에세이, 강의노트

• http://www.thehcmr.org/issue2_2/tableaux.pdf
• A. Yong. What is a Young tableau? Notices of the American MathematicalSociety 54(2) (2007), 240-241. http://www.ams.org/notices/200702/whatis-yong.pdf