"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.   
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* Define $\mathcal P:Q\to \mathbb{Z}$ by
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* 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
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Let $\lambda\in P_+$. For irreducible highest weight representation $V=L(\lambda)$, the weight multiplicity $m_{\mu}^{\lambda}:=\dim{V_{\mu}}$ is given by
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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
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$$
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:<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)) .  
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$$
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</math>
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==Lusztig's q-analogue==
 
==Lusztig's q-analogue==
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* 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.
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* 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.
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* Define functions ${\mathcal P}_q(\mu)$ by the equation
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* 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\ . \]
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* 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.  
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* 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.  
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* For $\lambda,\mu\in P$, Lusztig introduced a fundamental $q$-analogue of weight multipliciities $m_{\mu}^{\lambda}$:
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* For <math>\lambda,\mu\in P</math>, Lusztig introduced a fundamental <math>q</math>-analogue of weight multipliciities <math>m_{\mu}^{\lambda}</math>:
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$$
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:<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)) .  
−
$$
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</math>
 
===properties===
 
===properties===
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* $\mathfrak{M}_\lambda^\mu(q)\equiv 0$ unless $\lambda \succcurlyeq \mu$;
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* <math>\mathfrak{M}_\lambda^\mu(q)\equiv 0</math> unless <math>\lambda \succcurlyeq \mu</math>;
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* $\lambda\succcurlyeq\mu$, then $\mathfrak{M}_\lambda^\mu(q)$ is a monic polynomial and $\deg\mathfrak{M}_\lambda^\mu(q)=\mathsf{ht}(\lambda-\mu)$; therefore, $\mathfrak{M}_\lambda^\lambda(q)\equiv 1$;
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* <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>;
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* $\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu$.
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* <math>\mathfrak{M}_\lambda^\mu(1)=m_\lambda^\mu</math>.
  
  
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==매스매티카 파일==
 
==매스매티카 파일==
 
* https://drive.google.com/file/d/0B8XXo8Tve1cxSVN3eG1oc2VsLUk/view
 
* https://drive.google.com/file/d/0B8XXo8Tve1cxSVN3eG1oc2VsLUk/view
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 +
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2020년 11월 16일 (월) 05:24 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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