코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)

수학노트
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개요

  • 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.
  • Define $\mathcal P:Q\to \mathbb{Z}$ by

\[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-e^\alpha )}=:\sum_{\mu\in Q_+}{\mathcal P}(\mu)e^\mu\ . \]

thm

Let $\lambda\in P_+$. For irreducible highest weight representation $V=L(\lambda)$, the weight multiplicity $m_{\mu}^{\lambda}:=\dim{V_{\mu}}$ is given by $$ m_{\mu}^{\lambda}=\sum_{w\in W}\ell(w){\mathcal P}(w(\lambda+\rho)-(\mu+\rho)) . $$

Lusztig's q-analogue

  • 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 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, $\mathfrak{M}_\lambda^\lambda(q)\equiv 1$;
  • $\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu$.



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