코스트카 수 (Kostka number)

수학노트
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개요[편집]

  • 코스트카 수(Kostka number) $K_{\lambda\mu}$ : 형태가 $\lambda$이고 weight이 $\mu$인 준표준 영 태블로의 수
  • 군 \(\mathrm{GL}_n(\mathbb{C})\)의 기약표현 $V_{\lambda}$에서 $\mu$를 무게(weight)로 갖는 무게 공간(weight space)의 차원
  • 여러 대칭 다항식 사이의 연결 계수로서 나타난다


연결 계수[편집]

\[s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}m_\mu(\mathbb{x}).\ \]

\[h_\mu(\mathbb{x})= \sum_\lambda K_{\lambda\mu}s_\mu(\mathbb{x}).\ \]


$n=3,d=4$의 예[편집]

슈르 다항식과 단항 대칭 다항식[편집]

  • 슈르 다항식 $s_{\lambda}$와 단항 대칭 다항식 $m_{\lambda}$는 다음과 같은 표로 주어진다

\begin{array}{c|c|c} \lambda & s_{\lambda } & m_{\lambda } \\ \hline (4) & x_1^4+x_2 x_1^3+x_3 x_1^3+x_2^2 x_1^2+x_3^2 x_1^2+x_2 x_3 x_1^2+x_2^3 x_1+x_3^3 x_1+x_2 x_3^2 x_1+x_2^2 x_3 x_1+x_2^4+x_3^4+x_2 x_3^3+x_2^2 x_3^2+x_2^3 x_3 & x_1^4+x_2^4+x_3^4 \\ (3,1) & x_2 x_1^3+x_3 x_1^3+x_2^2 x_1^2+x_3^2 x_1^2+2 x_2 x_3 x_1^2+x_2^3 x_1+x_3^3 x_1+2 x_2 x_3^2 x_1+2 x_2^2 x_3 x_1+x_2 x_3^3+x_2^2 x_3^2+x_2^3 x_3 & x_2 x_1^3+x_3 x_1^3+x_2^3 x_1+x_3^3 x_1+x_2 x_3^3+x_2^3 x_3 \\ (2,2) & x_2^2 x_1^2+x_3^2 x_1^2+x_2 x_3 x_1^2+x_2 x_3^2 x_1+x_2^2 x_3 x_1+x_2^2 x_3^2 & x_1^2 x_2^2+x_3^2 x_2^2+x_1^2 x_3^2 \\ (2,1,1) & x_2 x_3 x_1^2+x_2 x_3^2 x_1+x_2^2 x_3 x_1 & x_2 x_3 x_1^2+x_2 x_3^2 x_1+x_2^2 x_3 x_1 \\ (1,1,1,1) & 0 & 0 \\ \end{array}


코스트카 수의 계산[편집]

$$ \begin{align} s_{(4)}(x_1,x_2,x_3) & =x_1^4+x_2 x_1^3+x_3 x_1^3+x_2^2 x_1^2+x_3^2 x_1^2+x_2 x_3 x_1^2+x_2^3 x_1+x_3^3 x_1+x_2 x_3^2 x_1+x_2^2 x_3 x_1+x_2^4+x_3^4+x_2 x_3^3+x_2^2 x_3^2+x_2^3 x_3 \\ & = (x_1^4+x_2^4+x_3^4)+(x_2 x_1^3+x_3 x_1^3+x_2^3 x_1+x_3^3 x_1+x_2 x_3^3+x_2^3 x_3)+(x_1^2 x_2^2+x_3^2 x_2^2+x_1^2 x_3^2)+(x_2 x_3 x_1^2+x_2 x_3^2 x_1+x_2^2 x_3 x_1)\\ & = m_{(4)}(x_1,x_2,x_3)+m_{(3,1)}(x_1,x_2,x_3)+m_{(2,2)}(x_1,x_2,x_3)+m_{(2,1,1)}(x_1,x_2,x_3) \\ s_{(3,1)}(x_1,x_2,x_3) & =x_1^3 x_2+x_1^2 x_2^2+x_1 x_2^3+x_1^3 x_3+2 x_1^2 x_2 x_3+2 x_1 x_2^2 x_3+x_2^3 x_3+x_1^2 x_3^2+2 x_1 x_2 x_3^2+x_2^2 x_3^2+x_1 x_3^3+x_2 x_3^3 \\ & = (x_1^3 x_2+x_1 x_2^3+x_1^3 x_3+x_2^3 x_3+x_1 x_3^3+x_2 x_3^3)+(x_1^2 x_2^2+x_1^2 x_3^2+x_2^2 x_3^2)+2(x_1^2 x_2 x_3+x_1 x_2^2 x_3+x_1 x_2 x_3^2)\\ & = m_{(3,1)}(x_1,x_2,x_3)+m_{(2,2)}(x_1,x_2,x_3)+2m_{(2,1,1)}(x_1,x_2,x_3) \\ s_{(2,2)}(x_1,x_2,x_3) & = x_2^2 x_1^2+x_3^2 x_1^2+x_2 x_3 x_1^2+x_2 x_3^2 x_1+x_2^2 x_3 x_1+x_2^2 x_3^2 \\ & =(x_1^2 x_2^2+x_1^2 x_3^2+x_2^2 x_3^2)+(x_1^2 x_2 x_3+x_1 x_2^2 x_3+x_1 x_2 x_3^2)\\ & = m_{(2,2,0)}(x_1,x_2,x_3)+m_{(2,1,1)}(x_1,x_2,x_3) \\ s_{(2,1,1)}(x_1,x_2,x_3) & = x_2 x_3 x_1^2+x_2 x_3^2 x_1+x_2^2 x_3 x_1 \\ & =(x_1^2 x_2 x_3+x_1 x_2^2 x_3+x_1 x_2 x_3^2)\\ & = m_{(2,1,1)}(x_1,x_2,x_3) \end{align} $$


\begin{array}{c|cccc} \lambda\backslash \mu & (4) & (3,1) & (2,2) & (2,1,1) \\ \hline (4) & 1 & 1 & 1 & 1 \\ (3,1) & 0 & 1 & 1 & 2 \\ (2,2) & 0 & 0 & 1 & 1 \\ (2,1,1) & 0 & 0 & 0 & 1 \\ \end{array}

테이블[편집]

  • $n\geq d$를 가정하면, 코스트카 수 $K_{\lambda,\mu}$는 $n$에 의존하지 않고, $d$에만 의존


$d=1$[편집]

\begin{array}{c|c} \text{} & \{1\} \\ \hline \{1\} & 1 \\ \end{array}


$d=2$[편집]

\begin{array}{c|cc} \text{} & \{2\} & \{1,1\} \\ \hline \{2\} & 1 & 1 \\ \{1,1\} & 0 & 1 \\ \end{array}

$d=3$[편집]

\begin{array}{c|ccc} \text{} & \{3\} & \{2,1\} & \{1,1,1\} \\ \hline \{3\} & 1 & 1 & 1 \\ \{2,1\} & 0 & 1 & 2 \\ \{1,1,1\} & 0 & 0 & 1 \\ \end{array}


$d=4$[편집]

\begin{array}{c|cccc} \text{} & \{4\} & \{3,1\} & \{2,2\} & \{2,1,1\} & \{1,1,1,1\} \\ \hline \{4\} & 1 & 1 & 1 & 1 & 1 \\ \{3,1\} & 0 & 1 & 1 & 2 & 3 \\ \{2,2\} & 0 & 0 & 1 & 1 & 2 \\ \{2,1,1\} & 0 & 0 & 0 & 1 & 3 \\ \{1,1,1,1\} & 0 & 0 & 0 & 0 & 1 \\ \end{array}


$d=5$[편집]

\begin{array}{c|ccccccc} \text{} & \{5\} & \{4,1\} & \{3,2\} & \{3,1,1\} & \{2,2,1\} & \{2,1,1,1\} & \{1,1,1,1,1\} \\ \hline \{5\} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \{4,1\} & 0 & 1 & 1 & 2 & 2 & 3 & 4 \\ \{3,2\} & 0 & 0 & 1 & 1 & 2 & 3 & 5 \\ \{3,1,1\} & 0 & 0 & 0 & 1 & 1 & 3 & 6 \\ \{2,2,1\} & 0 & 0 & 0 & 0 & 1 & 2 & 5 \\ \{2,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ \{1,1,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}


$d=6$[편집]

\begin{array}{c|ccccccccccc} \text{} & \{6\} & \{5,1\} & \{4,2\} & \{4,1,1\} & \{3,3\} & \{3,2,1\} & \{3,1,1,1\} & \{2,2,2\} & \{2,2,1,1\} & \{2,1,1,1,1\} & \{1,1,1,1,1,1\} \\ \hline \{6\} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \{5,1\} & 0 & 1 & 1 & 2 & 1 & 2 & 3 & 2 & 3 & 4 & 5 \\ \{4,2\} & 0 & 0 & 1 & 1 & 1 & 2 & 3 & 3 & 4 & 6 & 9 \\ \{4,1,1\} & 0 & 0 & 0 & 1 & 0 & 1 & 3 & 1 & 3 & 6 & 10 \\ \{3,3\} & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 3 & 5 \\ \{3,2,1\} & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 & 8 & 16 \\ \{3,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 4 & 10 \\ \{2,2,2\} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 & 5 \\ \{2,2,1,1\} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 9 \\ \{2,1,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 5 \\ \{1,1,1,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}

메모[편집]


관련된 항목들[편집]


매스매티카 파일 및 계산 리소스[편집]


관련논문[편집]