코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)
Pythagoras0 (토론 | 기여)님의 2020년 11월 16일 (월) 05:24 판
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, \(\mathfrak{M}_\lambda^\lambda(q)\equiv 1\);
- \(\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu\).
관련된 항목들