"코스트카 다항식 (Kostka polynomial)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) 잔글 (Pythagoras0 사용자가 Kostka polynomial and its generalizations 문서를 코스트카 다항식 (Kostka polynomial) 문서로 옮겼습니다.) |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
| − | == | + | ==개요== |
| + | * [[코스트카 수 (Kostka number)]]의 q-버전 | ||
| + | |||
* 갈고리 공식 | * 갈고리 공식 | ||
* 라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식 | * 라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식 | ||
| 9번째 줄: | 11번째 줄: | ||
* [[슈르 다항식(Schur polynomial)]] $s_{\lambda}(\mathbb{x})$ 을 [[단항 대칭 다항식 (monomial symmetric polynomial)]] $m_{\mu}(\mathbb{x})$의 선형결합으로 표현할 때 다음을 얻는다 | * [[슈르 다항식(Schur polynomial)]] $s_{\lambda}(\mathbb{x})$ 을 [[단항 대칭 다항식 (monomial symmetric polynomial)]] $m_{\mu}(\mathbb{x})$의 선형결합으로 표현할 때 다음을 얻는다 | ||
$$s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}m_\mu(\mathbb{x})$$ | $$s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}m_\mu(\mathbb{x})$$ | ||
| + | |||
==코스트카 다항식== | ==코스트카 다항식== | ||
| 15번째 줄: | 18번째 줄: | ||
s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}P_\mu(\mathbb{x},q) | s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}P_\mu(\mathbb{x},q) | ||
$$ | $$ | ||
| + | |||
==라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식== | ==라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식== | ||
| − | * | + | * 1978년에 라스꾸와 슈첸베르제는 $K_{\lambda,\mu}(q)$가 음이 아닌 정수 계수 다항식임을 증명 |
* They proved this by showing that $K_{\lambda,\mu}(q)=\sum q^{c(T)}$, where $T$ varies over all semi-standard tableaux of shape $\lambda$ and weight $\mu$ and $c(T)$ is an integer-valued function, called the charge of the tableau $T$, which is still a mysterious object in combinatorics. | * They proved this by showing that $K_{\lambda,\mu}(q)=\sum q^{c(T)}$, where $T$ varies over all semi-standard tableaux of shape $\lambda$ and weight $\mu$ and $c(T)$ is an integer-valued function, called the charge of the tableau $T$, which is still a mysterious object in combinatorics. | ||
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| − | |||
| − | |||
| − | |||
| + | ==관련된 항목들== | ||
| − | == | + | |
| + | ==리뷰, 에세이, 강의노트== | ||
* [http://www.aimath.org/WWN/kostka/ Generalized Kostka Polynomials] | * [http://www.aimath.org/WWN/kostka/ Generalized Kostka Polynomials] | ||
* Yamada, Yasuhiko. 1996. “Kostka Polynomials and Crystals.” Sūrikaisekikenkyūsho Kōkyūroku (962): 86–96. | * Yamada, Yasuhiko. 1996. “Kostka Polynomials and Crystals.” Sūrikaisekikenkyūsho Kōkyūroku (962): 86–96. | ||
| 34번째 줄: | 36번째 줄: | ||
| − | + | ==관련논문== | |
| − | == | ||
* Takeyama, Yoshihiro. “A Deformation of Affine Hecke Algebra and Integrable Stochastic Particle System.” arXiv:1407.1960 [cond-Mat, Physics:math-Ph], July 8, 2014. http://arxiv.org/abs/1407.1960. | * Takeyama, Yoshihiro. “A Deformation of Affine Hecke Algebra and Integrable Stochastic Particle System.” arXiv:1407.1960 [cond-Mat, Physics:math-Ph], July 8, 2014. http://arxiv.org/abs/1407.1960. | ||
* Okado, Masato, Anne Schilling, and Mark Shimozono. “A Crystal to Rigged Configuration Bijection for Nonexceptional Affine Algebras.” arXiv:math/0203163, March 15, 2002. http://arxiv.org/abs/math/0203163. | * Okado, Masato, Anne Schilling, and Mark Shimozono. “A Crystal to Rigged Configuration Bijection for Nonexceptional Affine Algebras.” arXiv:math/0203163, March 15, 2002. http://arxiv.org/abs/math/0203163. | ||
| 44번째 줄: | 45번째 줄: | ||
* Nakayashiki, Atsushi, and Yasuhiko Yamada. 1997. “Kostka Polynomials and Energy Functions in Solvable Lattice Models.” Selecta Mathematica. New Series 3 (4): 547–599. doi:10.1007/s000290050020. | * Nakayashiki, Atsushi, and Yasuhiko Yamada. 1997. “Kostka Polynomials and Energy Functions in Solvable Lattice Models.” Selecta Mathematica. New Series 3 (4): 547–599. doi:10.1007/s000290050020. | ||
* Lascoux, Alain, and Marcel-Paul Schützenberger. 1978. “Sur Une Conjecture de H. O. Foulkes.” C. R. Acad. Sci. Paris Sér. A-B 286 (7): A323–A324. | * Lascoux, Alain, and Marcel-Paul Schützenberger. 1978. “Sur Une Conjecture de H. O. Foulkes.” C. R. Acad. Sci. Paris Sér. A-B 286 (7): A323–A324. | ||
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| + | [[분류:대칭다항식]] | ||
2015년 3월 31일 (화) 21:19 판
개요
- 코스트카 수 (Kostka number)의 q-버전
- 갈고리 공식
- 라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식
코스트카 수
- 코스트카 수 (Kostka number)
- 군 \(\mathrm{GL}_n(\mathbb{C})\)의 기약표현 $V_{\lambda}$에서 $\mu$를 무게(weight)로 갖는 무게 공간(weight space)의 차원
- 슈르 다항식(Schur polynomial) $s_{\lambda}(\mathbb{x})$ 을 단항 대칭 다항식 (monomial symmetric polynomial) $m_{\mu}(\mathbb{x})$의 선형결합으로 표현할 때 다음을 얻는다
$$s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}m_\mu(\mathbb{x})$$
코스트카 다항식
- 코스트카 다항식 $K_{\lambda,\mu}(q)$은 슈르 다항식 $s_{\lambda}(x)$과 홀-리틀우드(Hall-Littlewood) 대칭함수 $P_{\mu}(x,q)$의 연결계수로서 나타난다
$$ s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}P_\mu(\mathbb{x},q) $$
라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식
- 1978년에 라스꾸와 슈첸베르제는 $K_{\lambda,\mu}(q)$가 음이 아닌 정수 계수 다항식임을 증명
- They proved this by showing that $K_{\lambda,\mu}(q)=\sum q^{c(T)}$, where $T$ varies over all semi-standard tableaux of shape $\lambda$ and weight $\mu$ and $c(T)$ is an integer-valued function, called the charge of the tableau $T$, which is still a mysterious object in combinatorics.
관련된 항목들
리뷰, 에세이, 강의노트
- Generalized Kostka Polynomials
- Yamada, Yasuhiko. 1996. “Kostka Polynomials and Crystals.” Sūrikaisekikenkyūsho Kōkyūroku (962): 86–96.
- Foulkes, H. O. 1974. “A Survey of Some Combinatorial Aspects of Symmetric Functions.” In Permutations (Actes Colloq., Univ. René-Descartes, Paris, 1972), 79–92. Gauthier-Villars, Paris. http://www.ams.org/mathscinet-getitem?mr=0416934.
관련논문
- Takeyama, Yoshihiro. “A Deformation of Affine Hecke Algebra and Integrable Stochastic Particle System.” arXiv:1407.1960 [cond-Mat, Physics:math-Ph], July 8, 2014. http://arxiv.org/abs/1407.1960.
- Okado, Masato, Anne Schilling, and Mark Shimozono. “A Crystal to Rigged Configuration Bijection for Nonexceptional Affine Algebras.” arXiv:math/0203163, March 15, 2002. http://arxiv.org/abs/math/0203163.
- Schilling, Anne, and Mark Shimozono. 2001. “Fermionic Formulas for Level-Restricted Generalized Kostka Polynomials and Coset Branching Functions.” Communications in Mathematical Physics 220 (1): 105–164. doi:10.1007/s002200100443.
- Kirillov, Anatol N., Anne Schilling, and Mark Shimozono. 1999. “Various Representations of the Generalized Kostka Polynomials.” Séminaire Lotharingien de Combinatoire 42: Art. B42j, 19 pp. (electronic). http://www.emis.de/journals/SLC/wpapers/s42schil.pdf
- Feigin, B., and S. Loktev. 1999. “On Generalized Kostka Polynomials and the Quantum Verlinde Rule.” In Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, 194:61–79. Amer. Math. Soc. Transl. Ser. 2. Providence, RI: Amer. Math. Soc.
- Kirillov, A. N. 1988. “On the Kostka-Green-Foulkes Polynomials and Clebsch-Gordan Numbers.” Journal of Geometry and Physics 5 (3): 365–389. doi:10.1016/0393-0440(88)90030-7.
- Nakayashiki, Atsushi, and Yasuhiko Yamada. 1997. “Kostka Polynomials and Energy Functions in Solvable Lattice Models.” Selecta Mathematica. New Series 3 (4): 547–599. doi:10.1007/s000290050020.
- Lascoux, Alain, and Marcel-Paul Schützenberger. 1978. “Sur Une Conjecture de H. O. Foulkes.” C. R. Acad. Sci. Paris Sér. A-B 286 (7): A323–A324.