대칭곱 (symmetric power)과 대칭텐서

개요

• 벡터공간 $V$에 대하여 대칭곱 $\operatorname{Sym}^n V$를 정의할 수 있다
• $V$에 작용하는 선형변환 $A$에 대하여 $\operatorname{Sym}^n A$를 정의할 수 있다

대칭곱의 기저

• 차원이 $n$인 벡터공간 $V$의 기저가 $\{v_1,\cdots,v_n\}$으로 주어진다고 하자
• $\operatorname{Sym}^d V$는 변수의 개수가 n이고, 차수가 d인 동차다항식(Homogeneous polynomial)이 이루는 벡터공간과 같다
• 차원은 $_n H_d$이며 기저는 다음과 같이 주어진다

$\dim V=1$인 경우

\begin{array}{c|c} d & \operatorname{Sym}^d V \\ \hline 0 & \{1\} \\ 1 & \left\{v_1\right\} \\ 2 & \left\{v_1^2\right\} \\ 3 & \left\{v_1^3\right\} \\ \end{array}

$\dim V=2$인 경우

\begin{array}{c|c} d & \operatorname{Sym}^d V \\ \hline 0 & \{1\} \\ 1 & \left\{v_1,v_2\right\} \\ 2 & \left\{v_1^2,v_1 v_2,v_2^2\right\} \\ 3 & \left\{v_1^3,v_1^2 v_2,v_1 v_2^2,v_2^3\right\} \\ \end{array}

$\dim V=3$인 경우

\begin{array}{c|c} d & \operatorname{Sym}^d V \\ \hline 0 & \{1\} \\ 1 & \left\{v_1,v_2,v_3\right\} \\ 2 & \left\{v_1^2,v_1 v_2,v_1 v_3,v_2^2,v_2 v_3,v_3^2\right\} \\ 3 & \left\{v_1^3,v_1^2 v_2,v_1^2 v_3,v_1 v_2^2,v_1 v_2 v_3,v_1 v_3^2,v_2^3,v_2^2 v_3,v_2 v_3^2,v_3^3\right\} \\ \end{array}

행렬의 대칭곱

• $V$에 작용하는 선형변환의 행렬표현 $A=(a_{ij})$를 생각하자

$\dim V=1$인 경우

$$ \begin{array}{c|c} d & \operatorname{Sym}^d A \\ \hline 0 & \left( \begin{array}{c} 1 \\ \end{array} \right) \\ 1 & \left( \begin{array}{c} a_{1,1} \\ \end{array} \right) \\ 2 & \left( \begin{array}{c} a_{1,1}^2 \\ \end{array} \right) \\ 3 & \left( \begin{array}{c} a_{1,1}^3 \\ \end{array} \right) \\ 4 & \left( \begin{array}{c} a_{1,1}^4 \\ \end{array} \right) \\ \end{array} $$

$\dim V=2$인 경우

$$ \begin{array}{c|c} d & \operatorname{Sym}^d A \\ \hline 0 & \left( \begin{array}{c} 1 \\ \end{array} \right) \\ 1 & \left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ \end{array} \right) \\ 2 & \left( \begin{array}{ccc} a_{1,1}^2 & a_{1,1} a_{1,2} & a_{1,2}^2 \\ 2 a_{1,1} a_{2,1} & a_{1,2} a_{2,1}+a_{1,1} a_{2,2} & 2 a_{1,2} a_{2,2} \\ a_{2,1}^2 & a_{2,1} a_{2,2} & a_{2,2}^2 \\ \end{array} \right) \\ 3 & \left( \begin{array}{cccc} a_{1,1}^3 & a_{1,1}^2 a_{1,2} & a_{1,1} a_{1,2}^2 & a_{1,2}^3 \\ 3 a_{1,1}^2 a_{2,1} & a_{2,2} a_{1,1}^2+2 a_{1,2} a_{2,1} a_{1,1} & a_{2,1} a_{1,2}^2+2 a_{1,1} a_{2,2} a_{1,2} & 3 a_{1,2}^2 a_{2,2} \\ 3 a_{1,1} a_{2,1}^2 & a_{1,2} a_{2,1}^2+2 a_{1,1} a_{2,2} a_{2,1} & a_{1,1} a_{2,2}^2+2 a_{1,2} a_{2,1} a_{2,2} & 3 a_{1,2} a_{2,2}^2 \\ a_{2,1}^3 & a_{2,1}^2 a_{2,2} & a_{2,1} a_{2,2}^2 & a_{2,2}^3 \\ \end{array} \right) \\ 4 & \left( \begin{array}{ccccc} a_{1,1}^4 & a_{1,1}^3 a_{1,2} & a_{1,1}^2 a_{1,2}^2 & a_{1,1} a_{1,2}^3 & a_{1,2}^4 \\ 4 a_{1,1}^3 a_{2,1} & a_{2,2} a_{1,1}^3+3 a_{1,2} a_{2,1} a_{1,1}^2 & 2 a_{1,2} a_{2,2} a_{1,1}^2+2 a_{1,2}^2 a_{2,1} a_{1,1} & a_{2,1} a_{1,2}^3+3 a_{1,1} a_{2,2} a_{1,2}^2 & 4 a_{1,2}^3 a_{2,2} \\ 6 a_{1,1}^2 a_{2,1}^2 & 3 a_{2,1} a_{2,2} a_{1,1}^2+3 a_{1,2} a_{2,1}^2 a_{1,1} & a_{1,2}^2 a_{2,1}^2+4 a_{1,1} a_{1,2} a_{2,2} a_{2,1}+a_{1,1}^2 a_{2,2}^2 & 3 a_{2,1} a_{2,2} a_{1,2}^2+3 a_{1,1} a_{2,2}^2 a_{1,2} & 6 a_{1,2}^2 a_{2,2}^2 \\ 4 a_{1,1} a_{2,1}^3 & a_{1,2} a_{2,1}^3+3 a_{1,1} a_{2,2} a_{2,1}^2 & 2 a_{1,2} a_{2,2} a_{2,1}^2+2 a_{1,1} a_{2,2}^2 a_{2,1} & a_{1,1} a_{2,2}^3+3 a_{1,2} a_{2,1} a_{2,2}^2 & 4 a_{1,2} a_{2,2}^3 \\ a_{2,1}^4 & a_{2,1}^3 a_{2,2} & a_{2,1}^2 a_{2,2}^2 & a_{2,1} a_{2,2}^3 & a_{2,2}^4 \\ \end{array} \right) \\ \end{array} $$

$\dim V=3$인 경우

$$ \begin{array}{c|c} d & \operatorname{Sym}^d A \\ \hline 0 & \left( \begin{array}{c} 1 \\ \end{array} \right) \\ 1 & \left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} \right) \\ 2 & \left( \begin{array}{cccccc} a_{1,1}^2 & a_{1,1} a_{1,2} & a_{1,1} a_{1,3} & a_{1,2}^2 & a_{1,2} a_{1,3} & a_{1,3}^2 \\ 2 a_{1,1} a_{2,1} & a_{1,2} a_{2,1}+a_{1,1} a_{2,2} & a_{1,3} a_{2,1}+a_{1,1} a_{2,3} & 2 a_{1,2} a_{2,2} & a_{1,3} a_{2,2}+a_{1,2} a_{2,3} & 2 a_{1,3} a_{2,3} \\ 2 a_{1,1} a_{3,1} & a_{1,2} a_{3,1}+a_{1,1} a_{3,2} & a_{1,3} a_{3,1}+a_{1,1} a_{3,3} & 2 a_{1,2} a_{3,2} & a_{1,3} a_{3,2}+a_{1,2} a_{3,3} & 2 a_{1,3} a_{3,3} \\ a_{2,1}^2 & a_{2,1} a_{2,2} & a_{2,1} a_{2,3} & a_{2,2}^2 & a_{2,2} a_{2,3} & a_{2,3}^2 \\ 2 a_{2,1} a_{3,1} & a_{2,2} a_{3,1}+a_{2,1} a_{3,2} & a_{2,3} a_{3,1}+a_{2,1} a_{3,3} & 2 a_{2,2} a_{3,2} & a_{2,3} a_{3,2}+a_{2,2} a_{3,3} & 2 a_{2,3} a_{3,3} \\ a_{3,1}^2 & a_{3,1} a_{3,2} & a_{3,1} a_{3,3} & a_{3,2}^2 & a_{3,2} a_{3,3} & a_{3,3}^2 \\ \end{array} \right) \\ \end{array} $$

관련된 항목들

• 동차다항식(Homogeneous polynomial)
• 중복조합의 공식 H(n,r)=C(n+r-1,r)
• 외대수(exterior algebra)와 다중선형대수(multilinear algebra)
• 행렬의 크로네커 곱 (Kronecker product)

사전 형태의 자료

• http://en.wikipedia.org/wiki/Symmetric_algebra#Distinction_with_symmetric_tensors
• http://en.wikipedia.org/wiki/Symmetric_tensor

리뷰, 에세이, 강의노트

• http://math.stanford.edu/~conrad/diffgeomPage/handouts/symmwedgebasis.pdf

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxWFFsckNSY2VhV28/edit