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| 4번째 줄: | 4번째 줄: | ||
** m | ** m | ||
** e | ** e | ||
| − | ** h | + | ** h : complete homogeneous symmetric polynomials |
| 38번째 줄: | 38번째 줄: | ||
<math>a_{\lambda+\delta}=\operatorname{det}(x_{i}^{\lambda_{j}+n-j})</math> | <math>a_{\lambda+\delta}=\operatorname{det}(x_{i}^{\lambda_{j}+n-j})</math> | ||
| − | + | <math>t_{\lambda} = a_{\lambda+\delta}/a_{\delta} =\sum_{w\in S_{n} } \epsilon(w) h_{\lambda+\delta - w.\lambda}</math> | |
| − | t_{\lambda} = det (h_{\lambda_{i}-i+j) | + | <math>t_{\lambda} = \operatorname{det}(h_{\lambda_{i}-i+j)</math> |
2011년 11월 18일 (금) 09:22 판
polynomial symmetric functions
- three bases
- m
- e
- h : complete homogeneous symmetric polynomials
algebraic independence result (Ruffini, around 1800)
- power sums
- A. Girard
- Waring
반데몬드 행렬과 행렬식 (Vandermonde matrix)
Jacobi-Trudi identity
sequence \delta : n-1,n-2,\cdots, 0
\lambda : partition \lambda_1\ geq \lambda_2,\cdots, \lambda_n\geq 0
\(a_{\lambda+\delta}=\operatorname{det}(x_{i}^{\lambda_{j}+n-j})\)
\(t_{\lambda} = a_{\lambda+\delta}/a_{\delta} =\sum_{w\in S_{n} } \epsilon(w) h_{\lambda+\delta - w.\lambda}\)
\(t_{\lambda} = \operatorname{det}(h_{\lambda_{i}-i+j)\)
Schur polynomials
J. Dieudonné, Schur functions and group representations , Young tableaux and Schur functors in algebra and geometry, Astéerisque, 87--88 , 7--19 (1981)