코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)

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

• 유한바일군의 계산 강의노트
• 프로이덴탈 중복도 공식 (Freudenthal multiplicity formula)

매스매티카 파일

• https://drive.google.com/file/d/0B8XXo8Tve1cxSVN3eG1oc2VsLUk/view