"대칭군의 지표(character)에 대한 프로베니우스 공식"의 두 판 사이의 차이

수학노트
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<h5>이 항목의 수학노트 원문주소</h5>
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==개요==
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* [[슈르 다항식(Schur polynomial)]]을 이용하여, [[대칭군 (symmetric group)]]의 지표를 계산하는 방법
 +
* 대칭군 <math>S_m</math>의 기약표현은 크기가 m인 영 다이어그램(또는 m의 분할)과 일대일대응된다
 +
* m의 분할 <math>\lambda</math>에 대응되는 <math>S_m</math>의 기약표현의 지표를 <math>\chi_{\lambda}</math> 로 나타내자
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* <math>C_{\mathbf{i}}=(1^{i_1},2^{i_2},\cdots,m^{i_m})</math>를 <math>i_1+2i_2+\cdots mi_m=m</math>를 만족시키는 대칭군 <math>S_m</math>의 공액류라 하면, <math>\chi_{\lambda}(C_{\mathbf{i}})</math>의 값은 다음 프로베니우스 공식으로 주어진다
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:<math>\left(\sum_{l=1}^{m} x_l\right)^{i_1}\left(\sum_{l=1}^{m} x_l^2\right)^{i_2}\cdots \left(\sum_{l=1}^{m} x_l^m\right)^{i_m}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}(x_1,\cdots, x_m)</math>
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여기서 <math>S_{\lambda}</math> 는 [[슈르 다항식(Schur polynomial)]]
 +
* 다음과 같이 표현하기도 한다
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:<math>\prod_{j=1}^{m}P_{j}(x)^{i_j}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}(x_1,\cdots, x_m)</math>
  
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==예==
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===<math>S_3</math>===
  
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* 대칭군 <math>S_3</math> 의 지표 테이블
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\begin{array}{c|ccc}
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  & \{1^3\} & \{1^1,2^1\} & \{3^1\} \\
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\hline
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\{3\} & 1 & 1 & 1 \\
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\{2,1\} & 2 & 0 & -1 \\
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\{1,1,1\} & 1 & -1 & 1
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\end{array}
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* 슈르 다항식
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:<math>
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\begin{align}
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S_{(3)} & = &x_1 x_2 x_3+\left(x_1+x_2+x_3\right){}^3-2 \left(x_1+x_2+x_3\right) \left(x_1 x_2+x_1 x_3+x_2 x_3\right) \\
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S_{(2,1)} & =&\left(x_1+x_2\right) \left(x_1+x_3\right) \left(x_2+x_3\right) \\
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S_{(1,1,1)} &= &x_1 x_2 x_3
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\end{align}
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</math>
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* 슈르 다항식과 거듭제곱의 합 (power sum) 대칭다항식
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:<math>
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\begin{align}
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\left(x_1+x_2+x_3\right){}^3&=S_{(3)}+2\cdot S_{(2,1)}+S_{(1,1,1)}\\
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\left(x_1+x_2+x_3\right) \left(x_1^2+x_2^2+x_3^2\right)& = S_{(3)}+0\cdot S_{(2,1)}-S_{(1,1,1)}\\
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x_1^3+x_2^3+x_3^3&=S_{(3)}-1\cdot S_{(2,1)}+S_{(1,1,1)}
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\end{align}
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</math>
  
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<h5>개요</h5>
 
  
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* <math>C_\mathbf{i}}=(1^{i_1},2^{i_2},\cdots,m^{i_m})</math> conjugacy class in <math>S_{m}</math> where <math>i_1+2i_2+\cdots mi_m=m</math>
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===<math>S_4</math>===
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* 프로베니우스 공식<br><math>\left(\sum_{l=1}^{m} x_l\right)^{i_1}\left(\sum_{l=1}^{m} x_l^2\right)^{i_2}\cdots \left(\sum_{l=1}^{m} x_l^m\right)^{i_m}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}</math><br><math>\lambda</math> 는 크기가 m인 영 다이어그램(또는 m의 분할)<br> 여기서 <math>\chi_{\lambda}</math> 는 character, <math>S_{\lambda}</math> 는 [[슈르 다항식(Schur polynomial)]]<br>
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* 지표 테이블
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\begin{array}{c|ccccc}
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  & \{1^4\} & \{1^2,2^1\} & \{1^1,3^1\} & \{2^2\} & \{4^1\} \\
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\hline
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\{4\} & 1 & 1 & 1 & 1 & 1 \\
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\{3,1\} & 3 & 1 & 0 & -1 & -1 \\
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\{2,2\} & 2 & 0 & -1 & 2 & 0 \\
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\{2,1,1\} & 3 & -1 & 0 & -1 & 1 \\
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\{1,1,1,1\} & 1 & -1 & 1 & 1 & -1
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\end{array}
  
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===<math>S_5</math>===
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* 지표 테이블
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\begin{array}{c|ccccccc}
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  & \{1^5\} & \{1^3,2^1\} & \{1^2,3^1\} & \{1^1,2^2\} & \{1^1,4^1\} & \{2^1,3^1\} & \{5^1\} \\
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\hline
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\{5\} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
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\{4,1\} & 4 & 2 & 1 & 0 & 0 & -1 & -1 \\
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\{3,2\} & 5 & 1 & -1 & 1 & -1 & 1 & 0 \\
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\{3,1,1\} & 6 & 0 & 0 & -2 & 0 & 0 & 1 \\
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\{2,2,1\} & 5 & -1 & -1 & 1 & 1 & -1 & 0 \\
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\{2,1,1,1\} & 4 & -2 & 1 & 0 & 0 & 1 & -1 \\
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\{1,1,1,1,1\} & 1 & -1 & 1 & 1 & -1 & -1 & 1
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\end{array}
  
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<h5>예</h5>
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===<math>S_6</math>===
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* 지표 테이블
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\begin{array}{c|ccccccccccc}
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  & \{1^6\} & \{1^4,2^1\} & \{1^3,3^1\} & \{1^2,2^2\} & \{1^2,4^1\} & \{1^1,2^1,3^1\} & \{1^1,5^1\} & \{2^3\} & \{2^1,4^1\} & \{3^2\} & \{6^1\} \\
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\hline
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\{6\} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
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\{5,1\} & 5 & 3 & 2 & 1 & 1 & 0 & 0 & -1 & -1 & -1 & -1 \\
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\{4,2\} & 9 & 3 & 0 & 1 & -1 & 0 & -1 & 3 & 1 & 0 & 0 \\
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\{4,1,1\} & 10 & 2 & 1 & -2 & 0 & -1 & 0 & -2 & 0 & 1 & 1 \\
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\{3,3\} & 5 & 1 & -1 & 1 & -1 & 1 & 0 & -3 & -1 & 2 & 0 \\
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\{3,2,1\} & 16 & 0 & -2 & 0 & 0 & 0 & 1 & 0 & 0 & -2 & 0 \\
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\{3,1,1,1\} & 10 & -2 & 1 & -2 & 0 & 1 & 0 & 2 & 0 & 1 & -1 \\
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\{2,2,2\} & 5 & -1 & -1 & 1 & 1 & -1 & 0 & 3 & -1 & 2 & 0 \\
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\{2,2,1,1\} & 9 & -3 & 0 & 1 & 1 & 0 & -1 & -3 & 1 & 0 & 0 \\
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\{2,1,1,1,1\} & 5 & -3 & 2 & 1 & -1 & 0 & 0 & 1 & -1 & -1 & 1 \\
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\{1,1,1,1,1,1\} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1
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\end{array}
  
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* 대칭군 <math>S_3</math> 의 character table
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===<math>S_7</math>===
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* 지표 테이블
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\begin{array}{c|ccccccccccccccc}
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  & \{1^7\} & \{1^52^1\} & \{1^43^1\} & \{1^32^2\} & \{1^34^1\} & \{1^22^13^1\} & \{1^25^1\} & \{1^12^3\} & \{1^12^14^1\} & \{1^13^2\} & \{1^16^1\} & \{2^23^1\} & \{2^15^1\} & \{3^14^1\} & \{7^1\} \\
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\hline
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\{7\} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
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\{6,1\} & 6 & 4 & 3 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 \\
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\{5,2\} & 14 & 6 & 2 & 2 & 0 & 0 & -1 & 2 & 0 & -1 & -1 & 2 & 1 & 0 & 0 \\
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\{5,1,1\} & 15 & 5 & 3 & -1 & 1 & -1 & 0 & -3 & -1 & 0 & 0 & -1 & 0 & 1 & 1 \\
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\{4,3\} & 14 & 4 & -1 & 2 & -2 & 1 & -1 & 0 & 0 & 2 & 0 & -1 & -1 & 1 & 0 \\
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\{4,2,1\} & 35 & 5 & -1 & -1 & -1 & -1 & 0 & 1 & 1 & -1 & 1 & -1 & 0 & -1 & 0 \\
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\{4,1,1,1\} & 20 & 0 & 2 & -4 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & -1 \\
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\{3,3,1\} & 21 & 1 & -3 & 1 & -1 & 1 & 1 & -3 & -1 & 0 & 0 & 1 & 1 & -1 & 0 \\
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\{3,2,2\} & 21 & -1 & -3 & 1 & 1 & -1 & 1 & 3 & -1 & 0 & 0 & 1 & -1 & 1 & 0 \\
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\{3,2,1,1\} & 35 & -5 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & -1 & -1 & -1 & 0 & 1 & 0 \\
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\{3,1,1,1,1\} & 15 & -5 & 3 & -1 & -1 & 1 & 0 & 3 & -1 & 0 & 0 & -1 & 0 & -1 & 1 \\
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\{2,2,2,1\} & 14 & -4 & -1 & 2 & 2 & -1 & -1 & 0 & 0 & 2 & 0 & -1 & 1 & -1 & 0 \\
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\{2,2,1,1,1\} & 14 & -6 & 2 & 2 & 0 & 0 & -1 & -2 & 0 & -1 & 1 & 2 & -1 & 0 & 0 \\
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\{2,1,1,1,1,1\} & 6 & -4 & 3 & 2 & -2 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1 \\
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\{1,1,1,1,1,1,1\} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1
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\end{array}
  
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==메모==
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* Math Overflow http://mathoverflow.net/search?q=
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{| class="dataTable2" style=""
 
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|-
 
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|  
 
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| (3)
 
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| (2,1)
 
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| (1,1,1)
 
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|-
 
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| <math>(1^3)</math>
 
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| 1
 
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| 2
 
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| 1
 
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|-
 
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| <math>(1^1,2^1)</math>
 
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| 1
 
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| 0
 
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| -1
 
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|-
 
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| <math>(3^1)</math>
 
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| 1
 
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| -1
 
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| 1
 
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|}
 
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<math>S_{(3)}=x_1 x_2 x_3+\left(x_1+x_2+x_3\right){}^3-2 \left(x_1+x_2+x_3\right) \left(x_1 x_2+x_1 x_3+x_2 x_3\right)</math>
 
  
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<math>S_{(2,1)}=\left(x_1+x_2\right) \left(x_1+x_3\right) \left(x_2+x_3\right)</math>
 
  
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<math>S_{(1,1,1)}=x_1 x_2 x_3</math>
 
  
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+
  
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<math>S_{(3)}+2S_{(2,1)}+S_{(1,1,1)}=\left(x_1+x_2+x_3\right){}^3</math>
+
  
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<math>S_{(3)}+0\cdot S_{(2,1)}-S_{(1,1,1)}=\left(x_1+x_2+x_3\right) \left(x_1^2+x_2^2+x_3^2\right)</math>
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==관련된 항목들==
 +
* [[대칭군의 표현론]]
  
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<math>S_{(3)}-1S_{(2,1)}+S_{(1,1,1)}=x_1^3+x_2^3+x_3^3</math>
 
  
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<h5>역사</h5>
 
  
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [[수학사연표 (역사)|수학사연표]]
 
  
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==수학용어번역==
 +
* 공액류, conjugacy class
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* 지표, {{학술용어집|url=character}}
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* 켤레변형, 공액연산자, {{학술용어집|url=conjugacy}}
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* 류, {{학술용어집|url=class}}
  
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<h5>메모</h5>
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<math>\prod_{j}P_{j}(x)^{i_j}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}</math>
+
  
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* Math Overflow http://mathoverflow.net/search?q=
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==매스매티카 파일 및 계산 리소스==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxYUZOWFFBZl9IMWM/edit
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<h5>관련된 항목들</h5>
 
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<h5>수학용어번역</h5>
 
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*  단어사전<br>
 
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** http://translate.google.com/#en|ko|
 
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** http://ko.wiktionary.org/wiki/
 
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* 발음사전 http://www.forvo.com/search/
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://www.kss.or.kr/pds/sec/dic.aspx 한국통계학회 통계학 용어 온라인 대조표]
 
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* [http://cgi.postech.ac.kr/cgi-bin/cgiwrap/sand/terms/terms.cgi 한국물리학회 물리학 용어집 검색기]
 
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* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
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<h5>매스매티카 파일 및 계산 리소스</h5>
 
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*  
 
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* http://www.wolframalpha.com/input/?i=
 
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* http://functions.wolfram.com/
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
 
 
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
 
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
 +
* http://mathoverflow.net/questions/96705/computer-package-for-representation-theory-of-the-symmetric-group
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* Stein, P.R., and C. Zemach. 1993. “Symmetric Function Algebra on a Computer.” Advances in Applied Mathematics 14 (4) (December): 430–454. doi:10.1006/aama.1993.1022.
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* http://mathoverflow.net/questions/162478/character-table-of-s-7
  
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<h5>사전 형태의 자료</h5>
 
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* [http://www.encyclopediaofmath.org/index.php/Main_Page Encyclopaedia of Mathematics]
 
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* [http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]
 
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* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]
 
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<h5>리뷰논문, 에세이, 강의노트</h5>
 
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<h5>관련논문</h5>
 
  
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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* http://www.ams.org/mathscinet
 
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* http://dx.doi.org/
 
  
−
 
+
==리뷰논문, 에세이, 강의노트==
 +
* http://mathematicalgemstones.wordpress.com/2012/05/21/characters-of-the-symmetric-group/
 +
  
−
 
+
==관련논문==
 +
* Rosa Orellana, Mike Zabrocki, Symmetric group characters as symmetric functions, arXiv:1605.06672 [math.CO], May 21 2016, http://arxiv.org/abs/1605.06672
 +
* Orellana, Rosa, and Mike Zabrocki. “Symmetric Group Characters as Symmetric Functions.” arXiv:1510.00438 [math], October 1, 2015. http://arxiv.org/abs/1510.00438.
 +
* Regev, Alon, Amitai Regev, and Doron Zeilberger. “Identities in Character Tables of <math>S_n</math>.” arXiv:1507.03499 [math], July 13, 2015. http://arxiv.org/abs/1507.03499.
  
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<h5>관련도서</h5>
 
  
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*  도서내검색<br>
+
[[분류:목록]]
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** http://books.google.com/books?q=
+
[[분류:대칭다항식]]
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** http://book.daum.net/search/contentSearch.do?query=
 

2020년 12월 28일 (월) 02:12 기준 최신판

개요

  • 슈르 다항식(Schur polynomial)을 이용하여, 대칭군 (symmetric group)의 지표를 계산하는 방법
  • 대칭군 \(S_m\)의 기약표현은 크기가 m인 영 다이어그램(또는 m의 분할)과 일대일대응된다
  • m의 분할 \(\lambda\)에 대응되는 \(S_m\)의 기약표현의 지표를 \(\chi_{\lambda}\) 로 나타내자
  • \(C_{\mathbf{i}}=(1^{i_1},2^{i_2},\cdots,m^{i_m})\)를 \(i_1+2i_2+\cdots mi_m=m\)를 만족시키는 대칭군 \(S_m\)의 공액류라 하면, \(\chi_{\lambda}(C_{\mathbf{i}})\)의 값은 다음 프로베니우스 공식으로 주어진다

\[\left(\sum_{l=1}^{m} x_l\right)^{i_1}\left(\sum_{l=1}^{m} x_l^2\right)^{i_2}\cdots \left(\sum_{l=1}^{m} x_l^m\right)^{i_m}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}(x_1,\cdots, x_m)\] 여기서 \(S_{\lambda}\) 는 슈르 다항식(Schur polynomial)

  • 다음과 같이 표현하기도 한다

\[\prod_{j=1}^{m}P_{j}(x)^{i_j}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}(x_1,\cdots, x_m)\]

예

\(S_3\)

  • 대칭군 \(S_3\) 의 지표 테이블

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

  • 슈르 다항식

\[ \begin{align} S_{(3)} & = &x_1 x_2 x_3+\left(x_1+x_2+x_3\right){}^3-2 \left(x_1+x_2+x_3\right) \left(x_1 x_2+x_1 x_3+x_2 x_3\right) \\ S_{(2,1)} & =&\left(x_1+x_2\right) \left(x_1+x_3\right) \left(x_2+x_3\right) \\ S_{(1,1,1)} &= &x_1 x_2 x_3 \end{align} \]

  • 슈르 다항식과 거듭제곱의 합 (power sum) 대칭다항식

\[ \begin{align} \left(x_1+x_2+x_3\right){}^3&=S_{(3)}+2\cdot S_{(2,1)}+S_{(1,1,1)}\\ \left(x_1+x_2+x_3\right) \left(x_1^2+x_2^2+x_3^2\right)& = S_{(3)}+0\cdot S_{(2,1)}-S_{(1,1,1)}\\ x_1^3+x_2^3+x_3^3&=S_{(3)}-1\cdot S_{(2,1)}+S_{(1,1,1)} \end{align} \]


\(S_4\)

  • 지표 테이블

\begin{array}{c|ccccc} & \{1^4\} & \{1^2,2^1\} & \{1^1,3^1\} & \{2^2\} & \{4^1\} \\ \hline \{4\} & 1 & 1 & 1 & 1 & 1 \\ \{3,1\} & 3 & 1 & 0 & -1 & -1 \\ \{2,2\} & 2 & 0 & -1 & 2 & 0 \\ \{2,1,1\} & 3 & -1 & 0 & -1 & 1 \\ \{1,1,1,1\} & 1 & -1 & 1 & 1 & -1 \end{array}


\(S_5\)

  • 지표 테이블

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

\(S_6\)

  • 지표 테이블

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

\(S_7\)

  • 지표 테이블

\begin{array}{c|ccccccccccccccc} & \{1^7\} & \{1^52^1\} & \{1^43^1\} & \{1^32^2\} & \{1^34^1\} & \{1^22^13^1\} & \{1^25^1\} & \{1^12^3\} & \{1^12^14^1\} & \{1^13^2\} & \{1^16^1\} & \{2^23^1\} & \{2^15^1\} & \{3^14^1\} & \{7^1\} \\ \hline \{7\} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \{6,1\} & 6 & 4 & 3 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 \\ \{5,2\} & 14 & 6 & 2 & 2 & 0 & 0 & -1 & 2 & 0 & -1 & -1 & 2 & 1 & 0 & 0 \\ \{5,1,1\} & 15 & 5 & 3 & -1 & 1 & -1 & 0 & -3 & -1 & 0 & 0 & -1 & 0 & 1 & 1 \\ \{4,3\} & 14 & 4 & -1 & 2 & -2 & 1 & -1 & 0 & 0 & 2 & 0 & -1 & -1 & 1 & 0 \\ \{4,2,1\} & 35 & 5 & -1 & -1 & -1 & -1 & 0 & 1 & 1 & -1 & 1 & -1 & 0 & -1 & 0 \\ \{4,1,1,1\} & 20 & 0 & 2 & -4 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & -1 \\ \{3,3,1\} & 21 & 1 & -3 & 1 & -1 & 1 & 1 & -3 & -1 & 0 & 0 & 1 & 1 & -1 & 0 \\ \{3,2,2\} & 21 & -1 & -3 & 1 & 1 & -1 & 1 & 3 & -1 & 0 & 0 & 1 & -1 & 1 & 0 \\ \{3,2,1,1\} & 35 & -5 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & -1 & -1 & -1 & 0 & 1 & 0 \\ \{3,1,1,1,1\} & 15 & -5 & 3 & -1 & -1 & 1 & 0 & 3 & -1 & 0 & 0 & -1 & 0 & -1 & 1 \\ \{2,2,2,1\} & 14 & -4 & -1 & 2 & 2 & -1 & -1 & 0 & 0 & 2 & 0 & -1 & 1 & -1 & 0 \\ \{2,2,1,1,1\} & 14 & -6 & 2 & 2 & 0 & 0 & -1 & -2 & 0 & -1 & 1 & 2 & -1 & 0 & 0 \\ \{2,1,1,1,1,1\} & 6 & -4 & 3 & 2 & -2 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1 \\ \{1,1,1,1,1,1,1\} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 \end{array}

메모





관련된 항목들





수학용어번역

  • 공액류, conjugacy class
  • 지표, character - 대한수학회 수학용어집
  • 켤레변형, 공액연산자, conjugacy - 대한수학회 수학용어집
  • 류, class - 대한수학회 수학용어집




매스매티카 파일 및 계산 리소스


리뷰논문, 에세이, 강의노트


관련논문

  • Rosa Orellana, Mike Zabrocki, Symmetric group characters as symmetric functions, arXiv:1605.06672 [math.CO], May 21 2016, http://arxiv.org/abs/1605.06672
  • Orellana, Rosa, and Mike Zabrocki. “Symmetric Group Characters as Symmetric Functions.” arXiv:1510.00438 [math], October 1, 2015. http://arxiv.org/abs/1510.00438.
  • Regev, Alon, Amitai Regev, and Doron Zeilberger. “Identities in Character Tables of \(S_n\).” arXiv:1507.03499 [math], July 13, 2015. http://arxiv.org/abs/1507.03499.
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