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.