"코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)"의 두 판 사이의 차이
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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 | ||
| + | $$ | ||
| + | m_{\mu}^{\lambda}=\sum_{w\in W}\ell(w){\mathcal P}(w(\lambda+\rho)-(\mu+\rho)) . | ||
| + | $$ | ||
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| + | ==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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==관련된 항목들== | ==관련된 항목들== | ||
* [[유한바일군의 계산 강의노트]] | * [[유한바일군의 계산 강의노트]] | ||
2016년 7월 18일 (월) 17:50 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