코스트카 다항식 (Kostka polynomial)

introduction

• 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

• In 1978 Lascoux and Schützenberger proved the remarkable fact that $K_{\lambda,\mu}(q)$ is a polynomial in $q$ with non-negative integer coefficients.
• 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.

related items

• Fermionic formula and X=M=N conjecture
• Rigged configurations
• Macdonald polynomials

expositions

• 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.

articles

• 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.