코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)
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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 <math>\mathcal P:Q\to \mathbb{Z}</math> by
\[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-e^\alpha )}=:\sum_{\mu\in Q_+}{\mathcal P}(\mu)e^\mu\ . \]
- thm
Let <math>\lambda\in P_+</math>. For irreducible highest weight representation <math>V=L(\lambda)</math>, the weight multiplicity <math>m_{\mu}^{\lambda}:=\dim{V_{\mu}}</math> is given by
- <math>
m_{\mu}^{\lambda}=\sum_{w\in W}\ell(w){\mathcal P}(w(\lambda+\rho)-(\mu+\rho)) . </math>
Lusztig's q-analogue
- For a given weight the q-analog of Kostant’s partition function is a polynomial where the coefficient of <math>q^k</math> is the number of ways the weight can be written as a nonnegative integral sum of exactly <math>k</math> positive roots.
- Define functions <math>{\mathcal P}_q(\mu)</math> by the equation
\[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-qe^\alpha )}=:\sum_{\mu\in Q_+} {\mathcal P}_q(\mu)e^\mu\ . \]
- Then <math>\mathcal P_q(\mu)</math> is a polynomial in <math>q</math> with <math>\deg\mathcal P_q(\mu)=\mathsf{ht}(\mu)</math> and <math>\mu \mapsto {\mathcal P}(\mu):={\mathcal P}_q(\mu)\vert_{q=1}</math> is the usual Kostant's partition function.
- For <math>\lambda,\mu\in P</math>, Lusztig introduced a fundamental <math>q</math>-analogue of weight multipliciities <math>m_{\mu}^{\lambda}</math>:
- <math>
\mathfrak{M}_\lambda^\mu(q)=\sum_{w\in W}\ell(w){\mathcal P}_q(w(\lambda+\rho)-(\mu+\rho)) . </math>
properties
- <math>\mathfrak{M}_\lambda^\mu(q)\equiv 0</math> unless <math>\lambda \succcurlyeq \mu</math>;
- <math>\lambda\succcurlyeq\mu</math>, then <math>\mathfrak{M}_\lambda^\mu(q)</math> is a monic polynomial and <math>\deg\mathfrak{M}_\lambda^\mu(q)=\mathsf{ht}(\lambda-\mu)</math>; therefore, <math>\mathfrak{M}_\lambda^\lambda(q)\equiv 1</math>;
- <math>\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu</math>.
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