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Pythagoras0 (토론 | 기여) |
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| + | ==introduction== | ||
| + | * Kostant’s partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra. | ||
| + | * For a given weight the q-analog of Kostant’s partition function is a polynomial where the coefficient of <math>q^k</math> is the number of ways the weight can be written as a nonnegative integral sum of exactly <math>k</math> positive roots. | ||
| + | * Define functions <math>{\mathcal P}_q(\mu)</math> by the equation | ||
| + | \[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-qe^\alpha )}=:\sum_{\mu\in Q_+} | ||
| + | {\mathcal P}_q(\mu)e^\mu\ . \] | ||
| + | * Then <math>\mathcal P_q(\mu)</math> is a polynomial in <math>q</math> with <math>\deg\mathcal P_q(\mu)=\mathsf{ht}(\mu)</math> and | ||
| + | <math>\mu \mapsto {\mathcal P}(\mu):={\mathcal P}_q(\mu)\vert_{q=1}</math> is the usual Kostant's partition | ||
| + | function. For <math>\lambda,\mu\in P</math>, | ||
| + | Lusztig \cite[(9.4)]{Lus} (see also \cite[(1.2)]{kato}) introduced a fundamental <math>q</math>-analogue of weight multipliciities <math>m_{\mu}^{\lambda}</math>: | ||
| + | :<math> | ||
| + | \mathfrak{M}_\lambda^\mu(q)=\sum_{w\in W}\ell(w){\mathcal P}_q(w(\lambda+\rho)-(\mu+\rho)) . | ||
| + | </math> | ||
| + | ===properties=== | ||
| + | * <math>\mathfrak{M}_\lambda^\mu(q)\equiv 0</math> unless <math>\lambda \succcurlyeq \mu</math>; | ||
| + | * <math>\lambda\succcurlyeq\mu</math>, then <math>\mathfrak{M}_\lambda^\mu(q)</math> is a monic polynomial and <math>\deg\mathfrak{M}_\lambda^\mu(q)=\mathsf{ht}(\lambda-\mu)</math>; therefore, <math>\mathfrak{M}_\lambda^\lambda(q)\equiv 1</math>; | ||
| + | * <math>\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu</math>. | ||
| + | |||
| + | |||
| + | ==history== | ||
| + | * Kostant’s partition function was introduced and studied by F.A. Berezin and I.M. Gelfand (Proc. Moscow Math. Soc. 5 (1956), 311-351) for the case <math>g=sl(n)</math>, and by B. Kostant (Trans. Amer. Math. Soc., 93 (1959), 53-73) for arbitrary semi–simple finite dimensional Lie algebra <math>g</math> | ||
| + | |||
| + | |||
| + | |||
| + | ==computational resource== | ||
| + | * https://drive.google.com/file/d/0B8XXo8Tve1cxR1VOVW5CMG5DV2c/view | ||
| + | * Kolman, B., and R. Beck. “Computers in Lie Algebras. I: Calculation of Inner Multiplicities.” SIAM Journal on Applied Mathematics 25, no. 2 (September 1, 1973): 300–312. doi:10.1137/0125032. | ||
| + | |||
| + | ==articles== | ||
| + | * [http://www-math.mit.edu/~karola/ Flow polytopes and the Kostant partition function] | ||
| + | * Harris, Pamela E., Erik Insko, and Mohamed Omar. “The <math>q</math>-Analog of Kostant’s Partition Function and the Highest Root of the Classical Lie Algebras.” arXiv:1508.07934 [math], August 31, 2015. http://arxiv.org/abs/1508.07934. | ||
| + | * Panyushev, Dmitri I. “On Lusztig’s <math>q</math>-Analogues of All Weight Multiplicities of a Representation.” arXiv:1406.1453 [Math], June 5, 2014. http://arxiv.org/abs/1406.1453. | ||
| + | * lecouvey, Cedric. “Kostka-Foulkes Polynomials Cyclage Graphs and Charge Statistic for the Root System <math>C_{n}</math>.” arXiv:math/0310370, October 23, 2003. http://arxiv.org/abs/math/0310370. | ||
| + | * Lansky, Joshua M. “A q-Analog of Freudenthal’s Weight Multiplicity Formula.” Indagationes Mathematicae 11, no. 1 (January 1, 2000): 87–94. doi:10.1016/S0019-3577(00)88576-6. | ||
| + | * Broer, Bram. “Line Bundles on the Cotangent Bundle of the Flag Variety.” Inventiones Mathematicae 113, no. 1 (n.d.): 1–20. doi:10.1007/BF01244299. | ||
| + | * Gupta, R. K. “Characters and the q-Analog of Weight Multiplicity.” Journal of the London Mathematical Society s2-36, no. 1 (August 1, 1987): 68–76. doi:10.1112/jlms/s2-36.1.68. | ||
| + | * Kato, Shin-ichi. “Spherical Functions and A q-analogue of Kostant's weight multiplicity formula.” Inventiones Mathematicae 66, no. 3 (n.d.): 461–68. doi:10.1007/BF01389223. | ||
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| + | |||
| + | [[분류:개인노트]] | ||
| + | [[분류:cluster algebra]] | ||
| + | [[분류:math and physics]] | ||
| + | [[분류:math]] | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q6433478 Q6433478] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'kostant'}, {'LOWER': 'partition'}, {'LEMMA': 'function'}] | ||
2021년 2월 17일 (수) 02:41 기준 최신판
introduction
- Kostant’s partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra.
- For a given weight the q-analog of Kostant’s partition function is a polynomial where the coefficient of \(q^k\) is the number of ways the weight can be written as a nonnegative integral sum of exactly \(k\) positive roots.
- Define functions \({\mathcal P}_q(\mu)\) by the equation
\[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-qe^\alpha )}=:\sum_{\mu\in Q_+} {\mathcal P}_q(\mu)e^\mu\ . \]
- Then \(\mathcal P_q(\mu)\) is a polynomial in \(q\) with \(\deg\mathcal P_q(\mu)=\mathsf{ht}(\mu)\) and
\(\mu \mapsto {\mathcal P}(\mu):={\mathcal P}_q(\mu)\vert_{q=1}\) is the usual Kostant's partition function. For \(\lambda,\mu\in P\), Lusztig \cite[(9.4)]{Lus} (see also \cite[(1.2)]{kato}) introduced a fundamental \(q\)-analogue of weight multipliciities \(m_{\mu}^{\lambda}\): \[ \mathfrak{M}_\lambda^\mu(q)=\sum_{w\in W}\ell(w){\mathcal P}_q(w(\lambda+\rho)-(\mu+\rho)) . \]
properties
- \(\mathfrak{M}_\lambda^\mu(q)\equiv 0\) unless \(\lambda \succcurlyeq \mu\);
- \(\lambda\succcurlyeq\mu\), then \(\mathfrak{M}_\lambda^\mu(q)\) is a monic polynomial and \(\deg\mathfrak{M}_\lambda^\mu(q)=\mathsf{ht}(\lambda-\mu)\); therefore, \(\mathfrak{M}_\lambda^\lambda(q)\equiv 1\);
- \(\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu\).
history
- Kostant’s partition function was introduced and studied by F.A. Berezin and I.M. Gelfand (Proc. Moscow Math. Soc. 5 (1956), 311-351) for the case \(g=sl(n)\), and by B. Kostant (Trans. Amer. Math. Soc., 93 (1959), 53-73) for arbitrary semi–simple finite dimensional Lie algebra \(g\)
computational resource
- https://drive.google.com/file/d/0B8XXo8Tve1cxR1VOVW5CMG5DV2c/view
- Kolman, B., and R. Beck. “Computers in Lie Algebras. I: Calculation of Inner Multiplicities.” SIAM Journal on Applied Mathematics 25, no. 2 (September 1, 1973): 300–312. doi:10.1137/0125032.
articles
- Flow polytopes and the Kostant partition function
- Harris, Pamela E., Erik Insko, and Mohamed Omar. “The \(q\)-Analog of Kostant’s Partition Function and the Highest Root of the Classical Lie Algebras.” arXiv:1508.07934 [math], August 31, 2015. http://arxiv.org/abs/1508.07934.
- Panyushev, Dmitri I. “On Lusztig’s \(q\)-Analogues of All Weight Multiplicities of a Representation.” arXiv:1406.1453 [Math], June 5, 2014. http://arxiv.org/abs/1406.1453.
- lecouvey, Cedric. “Kostka-Foulkes Polynomials Cyclage Graphs and Charge Statistic for the Root System \(C_{n}\).” arXiv:math/0310370, October 23, 2003. http://arxiv.org/abs/math/0310370.
- Lansky, Joshua M. “A q-Analog of Freudenthal’s Weight Multiplicity Formula.” Indagationes Mathematicae 11, no. 1 (January 1, 2000): 87–94. doi:10.1016/S0019-3577(00)88576-6.
- Broer, Bram. “Line Bundles on the Cotangent Bundle of the Flag Variety.” Inventiones Mathematicae 113, no. 1 (n.d.): 1–20. doi:10.1007/BF01244299.
- Gupta, R. K. “Characters and the q-Analog of Weight Multiplicity.” Journal of the London Mathematical Society s2-36, no. 1 (August 1, 1987): 68–76. doi:10.1112/jlms/s2-36.1.68.
- Kato, Shin-ichi. “Spherical Functions and A q-analogue of Kostant's weight multiplicity formula.” Inventiones Mathematicae 66, no. 3 (n.d.): 461–68. doi:10.1007/BF01389223.
메타데이터
위키데이터
- ID : Q6433478
Spacy 패턴 목록
- [{'LOWER': 'kostant'}, {'LOWER': 'partition'}, {'LEMMA': 'function'}]