완전 동차 대칭 다항식 (complete homogeneous symmetric polynomial)

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정의

• 변수의 개수 $n$
• $k$차의 완전 동차 다항식을 다음과 같이 정의

$$h_k(x_1,\cdots,x_n):=\sum_{1\leq i_1\leq i_2\cdot \leq i_k\leq n}x_{i_1}\cdots x_{i_k}$$

• $d$의 (0을 허용하며, 크기가 $n$인) 분할(partition) $$\lambda: \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\geq 0$$가 주어지면 $d$차 다항식 $$h_\lambda(x_1,\ldots,x_n)$$을 다음과 같이 정의

$$h_{\lambda}(x_1,\cdots,x_n):=h_{\lambda_1}(x_1,\cdots,x_n)\cdots h_{\lambda_n}(x_1,\cdots,x_n)$$

예

변수가 2개인 경우

$$\left( \begin{array}{cc} h_0\left(x_1,x_2\right) & 1 \\ h_1\left(x_1,x_2\right) & x_1+x_2 \\ h_2\left(x_1,x_2\right) & x_1^2+x_2 x_1+x_2^2 \\ h_3\left(x_1,x_2\right) & x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 \\ h_4\left(x_1,x_2\right) & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 \\ h_5\left(x_1,x_2\right) & x_1^5+x_2 x_1^4+x_2^2 x_1^3+x_2^3 x_1^2+x_2^4 x_1+x_2^5 \end{array} \right)$$

\begin{array}{c|c} \lambda & h_{\lambda } \\ \hline \{3\} & x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 \\ \{2,1\} & \left(x_1+x_2\right) \left(x_1^2+x_2 x_1+x_2^2\right) \\ \{1,1,1\} & \left(x_1+x_2\right){}^3 \\ \end{array}

\begin{array}{c|c} \lambda & h_{\lambda } \\ \hline \{4\} & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 \\ \{3,1\} & \left(x_1+x_2\right) \left(x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3\right) \\ \{2,2\} & \left(x_1^2+x_2 x_1+x_2^2\right){}^2 \\ \{2,1,1\} & \left(x_1+x_2\right){}^2 \left(x_1^2+x_2 x_1+x_2^2\right) \\ \{1,1,1,1\} & \left(x_1+x_2\right){}^4 \\ \end{array}

변수가 3개인 경우

$$\left( \begin{array}{cc} h_0\left(x_1,x_2,x_3\right) & 1 \\ h_1\left(x_1,x_2,x_3\right) & x_1+x_2+x_3 \\ h_2\left(x_1,x_2,x_3\right) & x_1^2+x_2 x_1+x_3 x_1+x_2^2+x_3^2+x_2 x_3 \\ h_3\left(x_1,x_2,x_3\right) & x_1^3+x_2 x_1^2+x_3 x_1^2+x_2^2 x_1+x_3^2 x_1+x_2 x_3 x_1+x_2^3+x_3^3+x_2 x_3^2+x_2^2 x_3 \\ h_4\left(x_1,x_2,x_3\right) & x_1^4+x_2 x_1^3+x_3 x_1^3+x_2^2 x_1^2+x_3^2 x_1^2+x_2 x_3 x_1^2+x_2^3 x_1+x_3^3 x_1+x_2 x_3^2 x_1+x_2^2 x_3 x_1+x_2^4+x_3^4+x_2 x_3^3+x_2^2 x_3^2+x_2^3 x_3 \end{array} \right)$$

\begin{array}{c|c} \lambda & h_{\lambda } \\ \hline \{2\} & x_1^2+x_2 x_1+x_3 x_1+x_2^2+x_3^2+x_2 x_3 \\ \{1,1\} & \left(x_1+x_2+x_3\right){}^2 \\ \end{array}

\begin{array}{c|c} \lambda & h_{\lambda } \\ \hline \{3\} & x_1^3+x_2 x_1^2+x_3 x_1^2+x_2^2 x_1+x_3^2 x_1+x_2 x_3 x_1+x_2^3+x_3^3+x_2 x_3^2+x_2^2 x_3 \\ \{2,1\} & \left(x_1+x_2+x_3\right) \left(x_1^2+x_2 x_1+x_3 x_1+x_2^2+x_3^2+x_2 x_3\right) \\ \{1,1,1\} & \left(x_1+x_2+x_3\right){}^3 \\ \end{array}

거듭제곱합 대칭다항식과의 관계

$\begin{array}{l} S_1-\Psi _1=0 \\ 2 S_2-S_1 \Psi _1-\Psi _2=0 \\ 3 S_3-S_2 \Psi _1-S_1 \Psi _2-\Psi _3=0\\ 4 S_4-S_3 \Psi _1-S_2 \Psi _2-S_1 \Psi _3-\Psi _4=0 \\ 5 S_5-S_4 \Psi _1-S_3 \Psi _2-S_2 \Psi _3-S_1 \Psi _4-\Psi _5=0\\ \cdots \end{array}$

거듭제곱합 대칭다항식을 이용한 표현

$$\begin{array}{l} S_1= \Psi _1 \\ S_2= \frac{1}{2} \left(\Psi _1^2+\Psi _2\right) \\ S_3= \frac{1}{6} \left(\Psi _1^3+3 \Psi _1 \Psi _2+2 \Psi _3\right) \\ S_4= \frac{1}{24} \left(\Psi _1^4+6 \Psi _1^2 \Psi _2+3 \Psi _2^2+8 \Psi _1 \Psi _3+6 \Psi _4\right) \\ S_5= \frac{1}{120} \left(\Psi _1^5+10 \Psi _1^3 \Psi _2+15 \Psi _1 \Psi _2^2+20 \Psi _1^2 \Psi _3+20 \Psi _2 \Psi _3+30 \Psi _1 \Psi _4+24 \Psi _5\right)\\ \cdots \end{array}$$

슈르 다항식

• 슈르 다항식(Schur polynomial) $s_{\lambda}$에 대하여 $s_{\lambda} = \operatorname{det}(h_{\lambda_{i}-i+j})$이 성립한다
• 예 $s_{(2,1,1)}(x_1,x_2,x_3)=h_1^2 h_2-h_2^2-h_1 h_3+h_4=x_1 x_2 x_3 \left(x_1+x_2+x_3\right)$

수학용어번역

• 완전, 완비, complete - 대한수학회 수학용어집