Kostant partition function

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.

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[{'LOWER': 'kostant'}, {'LOWER': 'partition'}, {'LEMMA': 'function'}]