코스트카 다항식 (Kostka polynomial)

개요

• 코스트카 수 (Kostka number)의 q-버전
• 갈고리 공식
• 라스꾸-슈첸베르제 (Lascoux-Schützenberger) 공식
• 대수적 조합수학의 중요한 연구대상

코스트카 수

• 코스트카 수 (Kostka number) \(K_{\lambda,\mu}\)
• 군 \(\mathrm{GL}_n(\mathbb{C})\)의 기약표현 \(V_{\lambda}\)에서 \(\mu\)를 무게(weight)로 갖는 무게 공간(weight space)의 차원
• 코스트카 수 \(K_{\lambda,\mu}\)는 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

\[ s_\lambda(\mathbb{x})= \sum_\mu K_{\lambda\mu}(q)P_\mu(\mathbb{x};q) \]

• \(K_{\lambda\mu}(1)=K_{\lambda\mu}\)이 성립

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.

테이블

• \(n\geq d\)를 가정하면, 코스트카 다항식 \(K_{\lambda,\mu}(\mathbb{x})\)는 \(n\)에 의존하지 않고, \(d\)에만 의존

\(d=1\)

\begin{array}{c|c} \text{} & \{1\} \\ \hline \{1\} & 1 \\ \end{array}

\(d=2\)

\begin{array}{c|cc} \text{} & \{2\} & \{1,1\} \\ \hline \{2\} & 1 & q \\ \{1,1\} & 0 & 1 \\ \end{array}

\(d=3\)

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

\(d=4\)

\begin{array}{c|cccc} \text{} & \{4\} & \{3,1\} & \{2,2\} & \{2,1,1\} & \{1,1,1,1\} \\ \hline \{4\} & 1 & q & q^2 & q^3 & q^6 \\ \{3,1\} & 0 & 1 & q & q^2+q & q^5+q^4+q^3 \\ \{2,2\} & 0 & 0 & 1 & q & q^4+q^2 \\ \{2,1,1\} & 0 & 0 & 0 & 1 & q^3+q^2+q \\ \{1,1,1,1\} & 0 & 0 & 0 & 0 & 1 \\ \end{array}

\(d=5\)

\begin{array}{c|ccccccc} \text{} & \{5\} & \{4,1\} & \{3,2\} & \{3,1,1\} & \{2,2,1\} & \{2,1,1,1\} & \{1,1,1,1,1\} \\ \hline \{5\} & 1 & q & q^2 & q^3 & q^4 & q^6 & q^{10} \\ \{4,1\} & 0 & 1 & q & q^2+q & q^3+q^2 & q^5+q^4+q^3 & q^9+q^8+q^7+q^6 \\ \{3,2\} & 0 & 0 & 1 & q & q^2+q & q^4+q^3+q^2 & q^8+q^7+q^6+q^5+q^4 \\ \{3,1,1\} & 0 & 0 & 0 & 1 & q & q^3+q^2+q & q^7+q^6+2 q^5+q^4+q^3 \\ \{2,2,1\} & 0 & 0 & 0 & 0 & 1 & q^2+q & q^6+q^5+q^4+q^3+q^2 \\ \{2,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 1 & q^4+q^3+q^2+q \\ \{1,1,1,1,1\} & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}

관련된 항목들

• 코스트카 수 (Kostka number)
• 홀-리틀우드(Hall-Littlewood) 대칭함수

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

• Shoji, Toshiaki. ‘Enhanced Variety of Higher Level and Kostka Functions Associated to Complex Reflection Groups’. arXiv:1507.01240 [math], 5 July 2015. http://arxiv.org/abs/1507.01240.
• 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.