"코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)"의 두 판 사이의 차이
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==개요== | ==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. | * 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 | + | * Define <math>\mathcal P:Q\to \mathbb{Z}</math> by |
\[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-e^\alpha )}=:\sum_{\mu\in Q_+}{\mathcal P}(\mu)e^\mu\ . \] | \[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-e^\alpha )}=:\sum_{\mu\in Q_+}{\mathcal P}(\mu)e^\mu\ . \] | ||
;thm | ;thm | ||
| − | Let | + | 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)) . | m_{\mu}^{\lambda}=\sum_{w\in W}\ell(w){\mathcal P}(w(\lambda+\rho)-(\mu+\rho)) . | ||
| − | + | </math> | |
==Lusztig's q-analogue== | ==Lusztig's q-analogue== | ||
| − | * For a given weight the q-analog of Kostant’s partition function is a polynomial where the coefficient of | + | * 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 | + | * Define functions <math>{\mathcal P}_q(\mu)</math> by the equation |
\[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-qe^\alpha )}=:\sum_{\mu\in Q_+} | \[ \frac{1}{\prod_{\alpha\in \Delta_+}(1-qe^\alpha )}=:\sum_{\mu\in Q_+} | ||
{\mathcal P}_q(\mu)e^\mu\ . \] | {\mathcal P}_q(\mu)e^\mu\ . \] | ||
| − | * Then | + | * 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 | + | * 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)) . | \mathfrak{M}_\lambda^\mu(q)=\sum_{w\in W}\ell(w){\mathcal P}_q(w(\lambda+\rho)-(\mu+\rho)) . | ||
| − | + | </math> | |
===properties=== | ===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>. |
2020년 11월 16일 (월) 04:24 기준 최신판
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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