"Talk on String functions and quantum affine algebras"의 두 판 사이의 차이
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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(다른 사용자 한 명의 중간 판 27개는 보이지 않습니다) | |||
3번째 줄: | 3번째 줄: | ||
===key message=== | ===key message=== | ||
* string functions know about Kirillov-Reshetikhin modules | * string functions know about Kirillov-Reshetikhin modules | ||
+ | * infinite vs. finite | ||
+ | <math> | ||
+ | \newcommand{\g}{\mathfrak{g}} | ||
+ | \newcommand{\h}{\mathfrak{h}} | ||
+ | \newcommand{\res}{\operatorname{res}} | ||
+ | \newcommand{\uqg}{U_{q}(\g)} | ||
+ | \newcommand{\ghat}{\widehat{\g}} | ||
+ | \newcommand{\uqghat}{U_{q}(\ghat)} | ||
+ | </math> | ||
==review of affine Lie algebras and their integrable representations== | ==review of affine Lie algebras and their integrable representations== | ||
===affine Lie algebras=== | ===affine Lie algebras=== | ||
* [[Affine Kac-Moody algebra]] | * [[Affine Kac-Moody algebra]] | ||
− | * | + | * <math>\overline{\mathfrak{g}}</math> : complex simple Lie algebra of rank <math>r</math> assoc. to Cartan matrix <math>(a_{ij})_{i,j\in \overline{I}}</math>, <math>\overline{I}=\{1,\cdots, r\}</math> |
− | * untwisted affine Kac-Moody algebra | + | * untwisted affine Kac-Moody algebra <math>\mathfrak{g}</math> |
− | + | :<math>\mathfrak{g}=\overline{\mathfrak{g}}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d</math> | |
− | * <math>(a_{ij})_{i,j\in I}</math> : extended Cartan matrix | + | * <math>(a_{ij})_{i,j\in I}</math> : extended Cartan matrix <math>I=\{0\}\cup \overline{I}</math> |
* can be also defined as a Lie algebra with generators <math>e_i,h_i,f_i , (i=0,1,2,\cdots, r)</math> and relations, for example, | * can be also defined as a Lie algebra with generators <math>e_i,h_i,f_i , (i=0,1,2,\cdots, r)</math> and relations, for example, | ||
** <math>\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0</math> (<math>i\neq j</math>) | ** <math>\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0</math> (<math>i\neq j</math>) | ||
− | * basis of the Cartan subalgebra | + | * basis of the Cartan subalgebra <math>\mathfrak{h}</math>; <math>h_0,h_ 1,\cdots,h_r,d</math> |
− | * dual basis for | + | * dual basis for <math>\mathfrak{h}^{*}</math>; <math>\Lambda_0,\Lambda_1,\cdots,\Lambda_r,\delta</math> |
* we call <math>\Lambda_0,\Lambda_1,\cdots,\Lambda_r</math> the fundamental weights and <math>\delta</math> the imaginary root | * we call <math>\Lambda_0,\Lambda_1,\cdots,\Lambda_r</math> the fundamental weights and <math>\delta</math> the imaginary root | ||
* simple roots <math>\alpha_0,\alpha_1,\cdots,\alpha_r</math> | * simple roots <math>\alpha_0,\alpha_1,\cdots,\alpha_r</math> | ||
− | * | + | * <math>a_i,\, i=0,1,\dots, r</math> : marks |
− | * | + | * <math>a_i^{\vee},\, i=0,1,\dots, r</math> : comarks |
* distinguished elements | * distinguished elements | ||
− | ** longest root of | + | ** longest root of <math>\overline{\mathfrak{g}}</math> : <math>\theta = \sum_{i=1}^{r}a_i\alpha_i</math> |
** central element <math>c=\sum_{i=0}^{r}a_i^{\vee}h _i</math> | ** central element <math>c=\sum_{i=0}^{r}a_i^{\vee}h _i</math> | ||
** imaginary root <math>\delta=\sum_{i=0}^{r}a_i\alpha_i</math> | ** imaginary root <math>\delta=\sum_{i=0}^{r}a_i\alpha_i</math> | ||
26번째 줄: | 35번째 줄: | ||
===remarks on affine weights=== | ===remarks on affine weights=== | ||
− | * call | + | * call <math>k=\lambda(c)</math> the level of <math>\lambda\in \mathfrak{h}^{*}</math> |
− | * sometimes convenient to write | + | * sometimes convenient to write <math>\lambda\in \mathfrak{h}^{*}</math> as <math>\lambda=(k;\overline{\lambda};\xi)\in \mathbb{C}\times \overline{\mathfrak{h}}^{*}\times \mathbb{C}</math> where <math>k=\lambda(c)</math>, <math>\overline{\lambda}</math> is the restriction of <math>\lambda</math> on <math>\overline{\mathfrak{h}}</math>, <math>\xi=\lambda(\delta)</math> |
− | ** | + | ** <math>\Lambda_0 = (a_0^{\vee};0;0)</math> |
− | ** | + | ** <math>\Lambda_i = (a_i^{\vee};\omega_i;0)</math>, for <math>i=1,\dots, r</math> (<math>\omega_i</math> is fundamental weight for <math>\overline{\mathfrak{g}}</math>) |
− | ** | + | ** <math>\delta = (0;0;0)</math>, for <math>i=1,\dots, r</math> |
− | ** | + | ** <math>\alpha_0 = (0;-\theta;1)</math> |
− | ** | + | ** <math>\alpha_i = (0;\alpha_i;0)</math>, for <math>i=1,\dots, r</math> (<math>\alpha_i</math> simple root for <math>\overline{\mathfrak{g}}</math>) |
− | * bilinear form | + | * bilinear form <math>(\cdot|\cdot)</math> on <math>\mathfrak{h}^{*}</math> |
− | ** | + | ** <math>\left((k_1;\overline{\lambda}_1;\xi_1)|(k_2;\overline{\lambda}_2;\xi_2)\right) = k_1\xi_2+k_2\xi_1+(\overline{\lambda}_1|\overline{\lambda}_2)_{\overline{\mathfrak{h}}^{*}}</math> |
− | * normalize | + | * normalize <math>(\cdot|\cdot)</math> so that <math>(\theta|\theta)_{\overline{\mathfrak{h}}^{*}}=2</math> |
− | * sometimes write | + | * sometimes write <math>\overline{\lambda} = (0;\overline{\lambda};0)</math> by abusing notation |
− | * let | + | * let <math>Q=\sum_{i=1}^{r}\Z\, \alpha_i\subseteq \mathfrak{h}^{*}</math> (root lattice of <math>\overline{\mathfrak{g}}</math>) |
− | * define | + | * define <math>M\subseteq Q</math> by <math>M=\{\sum_{i=1}^{r}\Z\, \alpha_i^{\vee}\}</math> where <math>\alpha_i^{\vee}=t_i\alpha_i</math> where <math>t_i=\frac{2}{(\alpha_i|\alpha_i)}</math> |
===affine Weyl group=== | ===affine Weyl group=== | ||
* [[Affine Weyl group]] | * [[Affine Weyl group]] | ||
− | * The affine Weyl group | + | * The affine Weyl group <math>W</math> is generated by <math>s_0,s_1,\cdots, s_r\in \operatorname{Aut}\,\mathfrak{h}^{*}</math> defined by |
− | + | :<math>s_{i}\lambda = \lambda -\lambda(h_i)\alpha_i</math> | |
− | for | + | for <math>i=0,1, \cdots, r</math>. |
− | * for | + | * for <math>\gamma\in \mathfrak{h}^{*}</math>, define <math>t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}</math> by |
− | + | :<math>t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma|\lambda)\right)\delta </math> | |
;thm | ;thm | ||
− | Let | + | Let <math>T=\{t_{\gamma}|\gamma\in M\}</math>. Then <math>W=\overline{W} \ltimes T</math> |
===integrable representations and characters=== | ===integrable representations and characters=== | ||
* [[Unitary representations of affine Kac-Moody algebras]] | * [[Unitary representations of affine Kac-Moody algebras]] | ||
− | * for each | + | * for each <math>\lambda\in \mathfrak{h}^{*}</math>, <math>\exists</math> irreducible <math>\mathfrak{g}</math>-module <math>L(\lambda)</math> (quotient of Verma module) |
− | * A | + | * A <math>\mathfrak{g}</math>-module <math>V</math> is ''integrable'' if <math>V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}</math> and if <math>e_i : V\to V</math> and <math>f_i : V\to V</math> are locally nilpotent for all <math>i=0,1,\cdots, r</math> |
− | * | + | * <math>\Lambda\in \mathfrak{h}^{*}</math> is dominant integral if <math>\Lambda(\mathfrak{h}_i)\in \mathbb{Z}_{\geq 0},\, i=0,1,\cdots,r</math> |
− | * let | + | * let <math>P_{+}</math> be the set of dominant integral weights, i.e. <math>\{\Lambda\in \mathfrak{h}^{*}|\Lambda=\sum_{i=0}^{l}\lambda_{i}\Lambda_i+\xi \delta, \lambda_i \in\mathbb{Z}_{\geq 0},\xi \in \mathbb{C}\} |
− | + | </math> | |
;thm | ;thm | ||
− | Let | + | Let <math>V</math> be an irreducible <math>\mathfrak{g}</math>-module in a certain category <math>\mathcal{O}</math>. Then <math>V=L(\Lambda)</math> for some <math>\Lambda\in \mathfrak{h}^{*}</math> and |
− | + | <math>L(\Lambda)</math> is integrable if and only if <math>\Lambda\in P_{+}</math> | |
* why care irreducible and integrable representation? Weyl's character formula holds | * why care irreducible and integrable representation? Weyl's character formula holds | ||
* character of <math>L(\Lambda)</math> | * character of <math>L(\Lambda)</math> | ||
− | + | :<math>\operatorname{ch} L(\Lambda):=\sum_{\lambda\in \mathfrak{h}^{*}}\operatorname{mult}_{\Lambda}(\lambda) e^{\lambda}</math> | |
;thm (Weyl-Kac formula) | ;thm (Weyl-Kac formula) | ||
− | Let | + | Let <math>\Lambda\in P_{+}</math>. Then |
− | + | :<math> | |
\operatorname{ch} L(\Lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\Lambda+\rho})}{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})} | \operatorname{ch} L(\Lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\Lambda+\rho})}{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})} | ||
− | + | </math> | |
;remark | ;remark | ||
− | For actual computation of | + | For actual computation of <math>m_{\lambda} = \operatorname{mult}_{\lambda}(\lambda)</math>, more practical to use Freudenthal multiplicity formula |
− | + | :<math> | |
(|\Lambda+\rho|^2-|\lambda+\rho|^2)m_{\lambda}=2\sum_{\alpha\in \Delta_{+}}\sum_{j\geq 1}(\operatorname{mult} \alpha)(\lambda+j\alpha|\alpha)m_{\lambda+j\alpha} | (|\Lambda+\rho|^2-|\lambda+\rho|^2)m_{\lambda}=2\sum_{\alpha\in \Delta_{+}}\sum_{j\geq 1}(\operatorname{mult} \alpha)(\lambda+j\alpha|\alpha)m_{\lambda+j\alpha} | ||
− | + | </math> | |
==string functions== | ==string functions== | ||
* [[String functions and branching functions]] | * [[String functions and branching functions]] | ||
− | * Fix | + | * Fix <math>\Lambda\in P_{+}^{k}</math> |
;def | ;def | ||
− | For each | + | For each <math>\lambda\in \mathfrak{h}^{*}</math>, the string function <math>c_{\lambda }^{\Lambda}</math> is |
− | + | :<math> | |
c_{\lambda }^{\Lambda}=e^{-m_{\Lambda,\lambda}\delta}\sum_{n=-\infty}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n \delta} | c_{\lambda }^{\Lambda}=e^{-m_{\Lambda,\lambda}\delta}\sum_{n=-\infty}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n \delta} | ||
− | + | </math> | |
− | where | + | where <math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}</math> and <math>m_{\Lambda,\lambda}=m_{\Lambda}-\frac{\lambda^2}{2k}</math> |
− | * note that | + | * note that <math>m_{\Lambda}=h_{\Lambda}-\frac{c(k)}{24}+\xi</math> where <math>h_{\Lambda}=\frac{(\bar{\Lambda}+2\bar{\rho}|\bar{\Lambda})}{2(k+h^{\vee})}</math> and <math>c(k)=\frac{k}{k+h^{\vee}}\dim \mathfrak{\overline{g}}</math> (these number frequently appear in rep. theory of Virasoro algebra) |
;remarks | ;remarks | ||
− | * modular form of weight | + | * modular form of weight <math>-r/2</math> after setting <math>q:=e^{-\delta}</math> |
* an explicit expression for the string functions is not known in general | * an explicit expression for the string functions is not known in general | ||
* the few that are known were guessed using the modular transformations | * the few that are known were guessed using the modular transformations | ||
− | * | + | * <math>c_{\lambda }^{\Lambda}=c_{w\lambda }^{\Lambda}</math> for <math>w\in W</math> |
* [[Theta functions in Kac-Moody algebras]] | * [[Theta functions in Kac-Moody algebras]] | ||
− | * for each | + | * for each <math>\lambda\in P^k</math>, define the theta function as |
− | + | :<math> | |
\Theta_{\lambda}= | \Theta_{\lambda}= | ||
e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)}=e^{k\Lambda_0}\sum_{\gamma\in M+\overline{\lambda}/k}e^{-\frac{1}{2}k|\gamma|^2 \delta + k \gamma} | e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)}=e^{k\Lambda_0}\sum_{\gamma\in M+\overline{\lambda}/k}e^{-\frac{1}{2}k|\gamma|^2 \delta + k \gamma} | ||
− | + | </math> | |
− | * A weight | + | * A weight <math>\lambda</math> of <math>L(\Lambda)</math> is ''maximal'' if <math>\lambda+\delta</math> is not a weight |
− | * the set | + | * the set <math>\max(\Lambda)</math> of maximal weights is stable under <math>W</math> |
;thm | ;thm | ||
− | + | :<math> | |
e^{-m_{\Lambda}\delta}\operatorname{ch} L(\Lambda)=\sum_{\lambda}c^{\Lambda}_{\lambda }\Theta_{\lambda} | e^{-m_{\Lambda}\delta}\operatorname{ch} L(\Lambda)=\sum_{\lambda}c^{\Lambda}_{\lambda }\Theta_{\lambda} | ||
− | + | </math> | |
;proof | ;proof | ||
− | + | :<math> | |
\begin{aligned} | \begin{aligned} | ||
\operatorname{ch} L(\Lambda)&=\sum_{\lambda\in \max{L(\Lambda)}}\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{\lambda-n \delta} | \operatorname{ch} L(\Lambda)&=\sum_{\lambda\in \max{L(\Lambda)}}\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{\lambda-n \delta} | ||
&& | && | ||
− | \text{(any weight | + | \text{(any weight <math>\mu</math> is of the form <math>\lambda-n \delta</math> for some unique <math>\lambda, n</math>)} |
\\ | \\ | ||
&=\sum_{\lambda\in \max{L(\Lambda)}/T} \left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\left(\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ | &=\sum_{\lambda\in \max{L(\Lambda)}/T} \left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\left(\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ | ||
&=\sum_{\lambda\in \max{L(\Lambda)}/T} e^{m_{\Lambda}\delta}c^{\Lambda}_{\lambda}\Theta_{\lambda} | &=\sum_{\lambda\in \max{L(\Lambda)}/T} e^{m_{\Lambda}\delta}c^{\Lambda}_{\lambda}\Theta_{\lambda} | ||
\end{aligned} | \end{aligned} | ||
− | + | </math> | |
■ | ■ | ||
===asymptotic growth of coefficients=== | ===asymptotic growth of coefficients=== | ||
− | * modularity of | + | * modularity of <math>c_{\lambda }^{\Lambda}</math> implies |
;thm (Kac-Peterson) | ;thm (Kac-Peterson) | ||
− | Let | + | Let <math>\Lambda\in P_{k}^{+},\, \lambda\in \max(\Lambda)</math>. As <math>n\to \infty</math>, |
− | + | :<math> | |
− | \log (\operatorname{mult}_{\Lambda}(\lambda-n\delta))\sim (\frac{ | + | \log (\operatorname{mult}_{\Lambda}(\lambda-n\delta))\sim (\frac{2c(k)\pi^2n}{3})^{1/2} |
− | + | </math> | |
==conjectural formula for string functions== | ==conjectural formula for string functions== | ||
* [[Fermionic formula for string functions and parafermion characters]] | * [[Fermionic formula for string functions and parafermion characters]] | ||
− | * denote the level by | + | * denote the level by <math>\ell\in \mathbb{Z}</math> and assume <math>\ell\geq 2</math> |
− | * | + | * <math>H_\ell=\{(a,m)|a=1,\cdots, r, 1\leq m \leq t_a \ell-1\}</math> |
* let | * let | ||
− | + | :<math> | |
K^{m n}_{a b} = \Bigl(\hbox{min}(t_bm, t_an) - {m n\over \ell}\Bigr) | K^{m n}_{a b} = \Bigl(\hbox{min}(t_bm, t_an) - {m n\over \ell}\Bigr) | ||
(\alpha_a \vert \alpha_b) | (\alpha_a \vert \alpha_b) | ||
− | + | </math> | |
;conjecture '''[Kuniba-Nakanishi-Suzuki 93]''' | ;conjecture '''[Kuniba-Nakanishi-Suzuki 93]''' | ||
143번째 줄: | 152번째 줄: | ||
\begin{equation}\label{qkns} | \begin{equation}\label{qkns} | ||
c^{\ell\Lambda_0}_\lambda(q)\cdot \eta(\tau)^r= | c^{\ell\Lambda_0}_\lambda(q)\cdot \eta(\tau)^r= | ||
− | \sum_{\{N^{(a)}_m\}}\frac{q^{\frac{1}{2}\sum_{(a,m), (b,n) \in H_\ell} | + | \sum_{\{(N^{(a)}_m)\}}\frac{q^{\frac{1}{2}\sum_{(a,m), (b,n) \in H_\ell} |
K^{mn}_{ab}N^{(a)}_mN^{(b)}_n}} | K^{mn}_{ab}N^{(a)}_mN^{(b)}_n}} | ||
{\prod_{(a,m) \in H_\ell}(1-q)(1-q^2)\cdots (1-q^{N^{(a)}_m})} | {\prod_{(a,m) \in H_\ell}(1-q)(1-q^2)\cdots (1-q^{N^{(a)}_m})} | ||
\end{equation} | \end{equation} | ||
− | up to a rational power of | + | up to a rational power of <math>q</math>, where <math>\eta</math> is the Dedekind eta function . |
The outer sum is over | The outer sum is over | ||
− | + | <math>N^{(a)}_m \in \Z_{\ge 0}</math> | |
such that | such that | ||
− | + | :<math>\sum_{(a,m) \in H_\ell}mN^{(a)}_m\overline{\alpha_a} \equiv \overline{\lambda} | |
− | \mod \ell M. | + | \mod \ell M.</math> |
===example=== | ===example=== | ||
− | * let | + | * let <math>\mathfrak{g}=A_1</math> |
;thm '''[Lepowski-Primc 1985]''' | ;thm '''[Lepowski-Primc 1985]''' | ||
− | + | :<math> | |
c^{\ell\Lambda_0}_{\ell\Lambda_0}(\tau)\cdot \eta(\tau)=\sum_{(N_1,\dots,N_{\ell-1})\in \mathbb{Z}_{\geq 0}^{\ell-1}}\frac{q^{\sum_{n,m=1}^{\ell-1} N_n N_m (\min (n,m) -\frac{nm}{\ell})}} | c^{\ell\Lambda_0}_{\ell\Lambda_0}(\tau)\cdot \eta(\tau)=\sum_{(N_1,\dots,N_{\ell-1})\in \mathbb{Z}_{\geq 0}^{\ell-1}}\frac{q^{\sum_{n,m=1}^{\ell-1} N_n N_m (\min (n,m) -\frac{nm}{\ell})}} | ||
{\prod_{m=1}^{\ell-1}(1-q)\cdots(1-q^{N_m})} | {\prod_{m=1}^{\ell-1}(1-q)\cdots(1-q^{N_m})} | ||
− | + | </math> | |
where the sum is under the constraint | where the sum is under the constraint | ||
− | + | <math> \sum_{m=1}^{\ell-1} m N_m | |
\equiv 0 | \equiv 0 | ||
− | \ \mathrm{mod}\ \ell | + | \ \mathrm{mod}\ \ell</math>. |
+ | |||
+ | |||
+ | ===evidence=== | ||
+ | * compare the asymptotic behavior of \ref{qkns} as <math>t\to 0</math> with <math>q=e^{-t}</math> | ||
+ | * LHS of \ref{qkns} <math>\exp(\frac{\pi^2(c(\ell)-r)}{6t})</math> | ||
+ | * RHS of \ref{qkns} <math>\exp(\frac{\sum_{(a,m)\in H_\ell} L(x_{m}^{(a)})}{t})</math> | ||
+ | where <math>0<x_{m}^{(a)}<1</math> is the solution of the system of equations | ||
+ | :<math> | ||
+ | x_{m}^{(a)} = \prod_{(b,n)\in H_{\ell}}(1-x_{n}^{(b)})^{K_{ab}^{mn}},\, (a,m)\in H_{\ell} | ||
+ | </math> | ||
+ | and <math>L</math> is the Rogers dilogarithm function | ||
+ | :<math> | ||
+ | L(x) = \operatorname{Li}_ 2(x)+\frac{1}{2}\log x\log (1-x),\, 0<x<1 | ||
+ | </math> | ||
+ | :<math> | ||
+ | \operatorname{Li}_ 2(x)= \sum_{n=1}^\infty {x^n \over n^2},\, 0<x<1 | ||
+ | </math> | ||
+ | ;thm (Chapoton, Nakanishi) | ||
+ | :<math> | ||
+ | \sum_{(a,m)\in H_\ell} L(x_{m}^{(a)}) = \frac{\pi^2}{6}(c(\ell)-r) | ||
+ | </math> | ||
+ | * proof uses Y-systems and cluster algebras | ||
+ | |||
+ | ===example=== | ||
+ | * <math>\overline{\mathfrak{g}} = B_2</math>, level <math>\ell = 2</math>, rank <math>r=2</math> | ||
+ | * <math>t_1=1,t_2=2</math> | ||
+ | * <math>H_{\ell} = \{(1,1),(2,1),(2,2),(2,3)\}</math> | ||
+ | * dual Coxeter number : <math>h^{\vee}=3</math> | ||
+ | * <math>\dim \overline{\mathfrak{g}}=10</math> | ||
+ | * <math>c(\ell)-r = 4-2=2</math> | ||
+ | * <math>K = \left( | ||
+ | \begin{array}{cccc} | ||
+ | 2 & -1 & -2 & -1 \\ | ||
+ | -1 & 3 & 2 & 1 \\ | ||
+ | -2 & 2 & 4 & 2 \\ | ||
+ | -1 & 1 & 2 & 3 \\ | ||
+ | \end{array} | ||
+ | \right)/2</math> | ||
+ | * equation for <math>x^{(a)}_m</math> | ||
+ | :<math> | ||
+ | \begin{aligned} | ||
+ | x^{(1)}_1 & = (1-x^{(1)}_1)(1-x^{(2)}_1)^{-1/2}(1-x^{(2)}_2)^{-1}(1-x^{(2)}_3)^{-1/2} \\ | ||
+ | x^{(2)}_1 & = (1-x^{(1)}_1)^{-1/2}(1-x^{(2)}_1)^{3/2}(1-x^{(2)}_2)^{1}(1-x^{(2)}_3)^{1/2}\\ | ||
+ | x^{(2)}_2 & = \dots \\ | ||
+ | x^{(2)}_3 & = \dots \\ | ||
+ | \end{aligned} | ||
+ | </math> | ||
+ | * <math>x^{(1)}_1= 3/4,x^{(2)}_1= 2/5,x^{(2)}_2= 4/9,x^{(2)}_3= 2/5</math> | ||
+ | :<math> | ||
+ | L\left(\frac{3}{4}\right)+2 L\left(\frac{2}{5}\right)+L\left(\frac{4}{9}\right) = \frac{2\pi^2}{6} | ||
+ | </math> | ||
+ | |||
+ | ==quantum affine algebras and KR modules== | ||
+ | * Q. is there more representation theoretic way to describe <math>x^{(a)}_m</math>? | ||
+ | * A. these numbers can be obtained from the quantum dimensions of Kirillov-Reshetikhin modules | ||
+ | * <math>\exists</math> bij. between iso. classes of fin.-dim'l irr. reps of <math>\uqg</math> and the set of <math>I</math>-tuples <math>\mathbf{P}=(P_i)_{i\in I}</math> of polys <math>P_i\in \mathbb{C}[z]</math> with <math>P_i(0)=1</math>, called Drinfeld poly. | ||
+ | * KR module <math>W^{(a)}_{m}(u)</math> with <math>a\in I</math>, <math>m\in \mathbb{Z}_{\geq 0}</math> and <math>u\in \mathbb{C}^{\times}</math> is associated with Drinfeld polynomials <math>\mathbf{P}=(P_i)_{i\in I}</math> of the form | ||
+ | :<math> | ||
+ | P_i(z) = | ||
+ | \begin{cases} | ||
+ | \prod _{s=1}^m \left(1- z u q_{a}^{2(s-1)}\right), & \text{if <math>i=a</math>}\\ | ||
+ | 1, & \text{otherwise} \\ | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | where <math>q_{a} = q^{t/t_a}</math> and <math>t=\max_{a\in I}t_a</math>. | ||
+ | |||
+ | * The quantum dimension of irr. h.w. <math>U_q(\overline{\mathfrak{g}})</math>-modules <math>L(\lambda)</math> at level <math>k</math> is | ||
+ | :<math> | ||
+ | \frac{\prod_{\alpha\in \Delta_{+}}\sin \frac{\pi(\lambda+\rho|\alpha)}{h^{\vee}+k}}{ \prod_{\in \Delta_{+}}\sin \frac{\pi (\rho|\alpha)}{h^{\vee}+k}}. | ||
+ | </math> | ||
+ | * recovers dimension as <math>k\to \infty</math> (qdim is an alg. int. not necessarily positive) | ||
+ | * regarding <math>W^{(a)}_{m}(u)</math> as <math>U_q(\overline{\mathfrak{g}})</math>, obtain quantum dimension of a KR module | ||
+ | ;thm (L.) | ||
+ | Fix level <math>\ell\geq 2</math>. Let <math>Q_{m}^{(a)}</math> be the qdim of <math>W^{(a)}_{m}(u)</math> at level <math>\ell</math>. Then <math>Q_{m}^{(a)}</math> with <math>(a,m)\in H_{\ell}</math> is positive, <math>Q_{t_a\ell}^{(a)}=1</math>, and <math>x^{(a)}_m= 1-\frac{Q_{m-1}^{(a)}Q_{m+1}^{(a)}}{(Q_{m}^{(a)})^2}</math>. | ||
+ | * need fusion ring | ||
+ | ===example=== | ||
+ | * <math>\overline{\mathfrak{g}} = B_2</math>, level <math>\ell = 2</math>, rank <math>r=2</math> | ||
+ | * <math>Q_{m}^{(1)} = 1,2,1</math> for <math>m=0,1,2</math> | ||
+ | * <math>Q_{m}^{(2)} = 1,\sqrt{5},3,\sqrt{5},1</math> for <math>m=0,1,2,3,4</math> | ||
==memo== | ==memo== | ||
− | + | :<math> | |
\operatorname{mult}_{\Lambda}(\lambda-n\delta)\sim (\text{const})\times n^{-(1/4)(r+3)}e^{4\pi (a n)^{1/2}} | \operatorname{mult}_{\Lambda}(\lambda-n\delta)\sim (\text{const})\times n^{-(1/4)(r+3)}e^{4\pi (a n)^{1/2}} | ||
− | + | </math> | |
181번째 줄: | 269번째 줄: | ||
[[분류:Talks and lecture notes]] | [[분류:Talks and lecture notes]] | ||
[[분류:theta]] | [[분류:theta]] | ||
+ | [[분류:migrate]] |
2020년 11월 16일 (월) 05:31 기준 최신판
abstract
The character of an irreducible representation with dominant integral highest weight of an affine Lie algebra can be written as a linear combination of theta functions, with coefficients given by string functions which are modular forms. There are still many aspects of string functions that are not well-understood. In this talk I will review the basic properties of them, and explain certain connections with finite-dimensional representations of quantum affine algebras.
key message
- string functions know about Kirillov-Reshetikhin modules
- infinite vs. finite
\( \newcommand{\g}{\mathfrak{g}} \newcommand{\h}{\mathfrak{h}} \newcommand{\res}{\operatorname{res}} \newcommand{\uqg}{U_{q}(\g)} \newcommand{\ghat}{\widehat{\g}} \newcommand{\uqghat}{U_{q}(\ghat)} \)
review of affine Lie algebras and their integrable representations
affine Lie algebras
- Affine Kac-Moody algebra
- \(\overline{\mathfrak{g}}\) : complex simple Lie algebra of rank \(r\) assoc. to Cartan matrix \((a_{ij})_{i,j\in \overline{I}}\), \(\overline{I}=\{1,\cdots, r\}\)
- untwisted affine Kac-Moody algebra \(\mathfrak{g}\)
\[\mathfrak{g}=\overline{\mathfrak{g}}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d\]
- \((a_{ij})_{i,j\in I}\) : extended Cartan matrix \(I=\{0\}\cup \overline{I}\)
- can be also defined as a Lie algebra with generators \(e_i,h_i,f_i , (i=0,1,2,\cdots, r)\) and relations, for example,
- \(\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0\) (\(i\neq j\))
- basis of the Cartan subalgebra \(\mathfrak{h}\); \(h_0,h_ 1,\cdots,h_r,d\)
- dual basis for \(\mathfrak{h}^{*}\); \(\Lambda_0,\Lambda_1,\cdots,\Lambda_r,\delta\)
- we call \(\Lambda_0,\Lambda_1,\cdots,\Lambda_r\) the fundamental weights and \(\delta\) the imaginary root
- simple roots \(\alpha_0,\alpha_1,\cdots,\alpha_r\)
- \(a_i,\, i=0,1,\dots, r\) : marks
- \(a_i^{\vee},\, i=0,1,\dots, r\) : comarks
- distinguished elements
- longest root of \(\overline{\mathfrak{g}}\) \[\theta = \sum_{i=1}^{r}a_i\alpha_i\]
- central element \(c=\sum_{i=0}^{r}a_i^{\vee}h _i\)
- imaginary root \(\delta=\sum_{i=0}^{r}a_i\alpha_i\)
- Weyl vector \(\rho=\sum_{i=0}^{r}\Lambda_i\)
remarks on affine weights
- call \(k=\lambda(c)\) the level of \(\lambda\in \mathfrak{h}^{*}\)
- sometimes convenient to write \(\lambda\in \mathfrak{h}^{*}\) as \(\lambda=(k;\overline{\lambda};\xi)\in \mathbb{C}\times \overline{\mathfrak{h}}^{*}\times \mathbb{C}\) where \(k=\lambda(c)\), \(\overline{\lambda}\) is the restriction of \(\lambda\) on \(\overline{\mathfrak{h}}\), \(\xi=\lambda(\delta)\)
- \(\Lambda_0 = (a_0^{\vee};0;0)\)
- \(\Lambda_i = (a_i^{\vee};\omega_i;0)\), for \(i=1,\dots, r\) (\(\omega_i\) is fundamental weight for \(\overline{\mathfrak{g}}\))
- \(\delta = (0;0;0)\), for \(i=1,\dots, r\)
- \(\alpha_0 = (0;-\theta;1)\)
- \(\alpha_i = (0;\alpha_i;0)\), for \(i=1,\dots, r\) (\(\alpha_i\) simple root for \(\overline{\mathfrak{g}}\))
- bilinear form \((\cdot|\cdot)\) on \(\mathfrak{h}^{*}\)
- \(\left((k_1;\overline{\lambda}_1;\xi_1)|(k_2;\overline{\lambda}_2;\xi_2)\right) = k_1\xi_2+k_2\xi_1+(\overline{\lambda}_1|\overline{\lambda}_2)_{\overline{\mathfrak{h}}^{*}}\)
- normalize \((\cdot|\cdot)\) so that \((\theta|\theta)_{\overline{\mathfrak{h}}^{*}}=2\)
- sometimes write \(\overline{\lambda} = (0;\overline{\lambda};0)\) by abusing notation
- let \(Q=\sum_{i=1}^{r}\Z\, \alpha_i\subseteq \mathfrak{h}^{*}\) (root lattice of \(\overline{\mathfrak{g}}\))
- define \(M\subseteq Q\) by \(M=\{\sum_{i=1}^{r}\Z\, \alpha_i^{\vee}\}\) where \(\alpha_i^{\vee}=t_i\alpha_i\) where \(t_i=\frac{2}{(\alpha_i|\alpha_i)}\)
affine Weyl group
- Affine Weyl group
- The affine Weyl group \(W\) is generated by \(s_0,s_1,\cdots, s_r\in \operatorname{Aut}\,\mathfrak{h}^{*}\) defined by
\[s_{i}\lambda = \lambda -\lambda(h_i)\alpha_i\] for \(i=0,1, \cdots, r\).
- for \(\gamma\in \mathfrak{h}^{*}\), define \(t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}\) by
\[t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma|\lambda)\right)\delta \]
- thm
Let \(T=\{t_{\gamma}|\gamma\in M\}\). Then \(W=\overline{W} \ltimes T\)
integrable representations and characters
- Unitary representations of affine Kac-Moody algebras
- for each \(\lambda\in \mathfrak{h}^{*}\), \(\exists\) irreducible \(\mathfrak{g}\)-module \(L(\lambda)\) (quotient of Verma module)
- A \(\mathfrak{g}\)-module \(V\) is integrable if \(V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}\) and if \(e_i : V\to V\) and \(f_i : V\to V\) are locally nilpotent for all \(i=0,1,\cdots, r\)
- \(\Lambda\in \mathfrak{h}^{*}\) is dominant integral if \(\Lambda(\mathfrak{h}_i)\in \mathbb{Z}_{\geq 0},\, i=0,1,\cdots,r\)
- let \(P_{+}\) be the set of dominant integral weights, i.e. \(\{\Lambda\in \mathfrak{h}^{*}|\Lambda=\sum_{i=0}^{l}\lambda_{i}\Lambda_i+\xi \delta, \lambda_i \in\mathbb{Z}_{\geq 0},\xi \in \mathbb{C}\} \)
- thm
Let \(V\) be an irreducible \(\mathfrak{g}\)-module in a certain category \(\mathcal{O}\). Then \(V=L(\Lambda)\) for some \(\Lambda\in \mathfrak{h}^{*}\) and \(L(\Lambda)\) is integrable if and only if \(\Lambda\in P_{+}\)
- why care irreducible and integrable representation? Weyl's character formula holds
- character of \(L(\Lambda)\)
\[\operatorname{ch} L(\Lambda):=\sum_{\lambda\in \mathfrak{h}^{*}}\operatorname{mult}_{\Lambda}(\lambda) e^{\lambda}\]
- thm (Weyl-Kac formula)
Let \(\Lambda\in P_{+}\). Then \[ \operatorname{ch} L(\Lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\Lambda+\rho})}{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})} \]
- remark
For actual computation of \(m_{\lambda} = \operatorname{mult}_{\lambda}(\lambda)\), more practical to use Freudenthal multiplicity formula \[ (|\Lambda+\rho|^2-|\lambda+\rho|^2)m_{\lambda}=2\sum_{\alpha\in \Delta_{+}}\sum_{j\geq 1}(\operatorname{mult} \alpha)(\lambda+j\alpha|\alpha)m_{\lambda+j\alpha} \]
string functions
- String functions and branching functions
- Fix \(\Lambda\in P_{+}^{k}\)
- def
For each \(\lambda\in \mathfrak{h}^{*}\), the string function \(c_{\lambda }^{\Lambda}\) is \[ c_{\lambda }^{\Lambda}=e^{-m_{\Lambda,\lambda}\delta}\sum_{n=-\infty}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n \delta} \] where \(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}\) and \(m_{\Lambda,\lambda}=m_{\Lambda}-\frac{\lambda^2}{2k}\)
- note that \(m_{\Lambda}=h_{\Lambda}-\frac{c(k)}{24}+\xi\) where \(h_{\Lambda}=\frac{(\bar{\Lambda}+2\bar{\rho}|\bar{\Lambda})}{2(k+h^{\vee})}\) and \(c(k)=\frac{k}{k+h^{\vee}}\dim \mathfrak{\overline{g}}\) (these number frequently appear in rep. theory of Virasoro algebra)
- remarks
- modular form of weight \(-r/2\) after setting \(q:=e^{-\delta}\)
- an explicit expression for the string functions is not known in general
- the few that are known were guessed using the modular transformations
- \(c_{\lambda }^{\Lambda}=c_{w\lambda }^{\Lambda}\) for \(w\in W\)
- Theta functions in Kac-Moody algebras
- for each \(\lambda\in P^k\), define the theta function as
\[ \Theta_{\lambda}= e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)}=e^{k\Lambda_0}\sum_{\gamma\in M+\overline{\lambda}/k}e^{-\frac{1}{2}k|\gamma|^2 \delta + k \gamma} \]
- A weight \(\lambda\) of \(L(\Lambda)\) is maximal if \(\lambda+\delta\) is not a weight
- the set \(\max(\Lambda)\) of maximal weights is stable under \(W\)
- thm
\[ e^{-m_{\Lambda}\delta}\operatorname{ch} L(\Lambda)=\sum_{\lambda}c^{\Lambda}_{\lambda }\Theta_{\lambda} \]
- proof
\[ \begin{aligned} \operatorname{ch} L(\Lambda)&=\sum_{\lambda\in \max{L(\Lambda)}}\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{\lambda-n \delta} && \text{(any weight \(\mu\] is of the form <math>\lambda-n \delta\) for some unique \(\lambda, n\))} \\ &=\sum_{\lambda\in \max{L(\Lambda)}/T} \left(\sum_{\gamma\in M}e^{t_{\gamma}(\lambda)}\right)\left(\sum_{n=0}^{\infty}\operatorname{mult}_{\Lambda}(\lambda-n\delta)e^{-n\delta}\right)\\ &=\sum_{\lambda\in \max{L(\Lambda)}/T} e^{m_{\Lambda}\delta}c^{\Lambda}_{\lambda}\Theta_{\lambda} \end{aligned} </math> ■
asymptotic growth of coefficients
- modularity of \(c_{\lambda }^{\Lambda}\) implies
- thm (Kac-Peterson)
Let \(\Lambda\in P_{k}^{+},\, \lambda\in \max(\Lambda)\). As \(n\to \infty\), \[ \log (\operatorname{mult}_{\Lambda}(\lambda-n\delta))\sim (\frac{2c(k)\pi^2n}{3})^{1/2} \]
conjectural formula for string functions
- Fermionic formula for string functions and parafermion characters
- denote the level by \(\ell\in \mathbb{Z}\) and assume \(\ell\geq 2\)
- \(H_\ell=\{(a,m)|a=1,\cdots, r, 1\leq m \leq t_a \ell-1\}\)
- let
\[ K^{m n}_{a b} = \Bigl(\hbox{min}(t_bm, t_an) - {m n\over \ell}\Bigr) (\alpha_a \vert \alpha_b) \]
- conjecture [Kuniba-Nakanishi-Suzuki 93]
We have \begin{equation}\label{qkns} c^{\ell\Lambda_0}_\lambda(q)\cdot \eta(\tau)^r= \sum_{\{(N^{(a)}_m)\}}\frac{q^{\frac{1}{2}\sum_{(a,m), (b,n) \in H_\ell} K^{mn}_{ab}N^{(a)}_mN^{(b)}_n}} {\prod_{(a,m) \in H_\ell}(1-q)(1-q^2)\cdots (1-q^{N^{(a)}_m})} \end{equation} up to a rational power of \(q\), where \(\eta\) is the Dedekind eta function .
The outer sum is over \(N^{(a)}_m \in \Z_{\ge 0}\) such that \[\sum_{(a,m) \in H_\ell}mN^{(a)}_m\overline{\alpha_a} \equiv \overline{\lambda} \mod \ell M.\]
example
- let \(\mathfrak{g}=A_1\)
- thm [Lepowski-Primc 1985]
\[ c^{\ell\Lambda_0}_{\ell\Lambda_0}(\tau)\cdot \eta(\tau)=\sum_{(N_1,\dots,N_{\ell-1})\in \mathbb{Z}_{\geq 0}^{\ell-1}}\frac{q^{\sum_{n,m=1}^{\ell-1} N_n N_m (\min (n,m) -\frac{nm}{\ell})}} {\prod_{m=1}^{\ell-1}(1-q)\cdots(1-q^{N_m})} \] where the sum is under the constraint \( \sum_{m=1}^{\ell-1} m N_m \equiv 0 \ \mathrm{mod}\ \ell\).
evidence
- compare the asymptotic behavior of \ref{qkns} as \(t\to 0\) with \(q=e^{-t}\)
- LHS of \ref{qkns} \(\exp(\frac{\pi^2(c(\ell)-r)}{6t})\)
- RHS of \ref{qkns} \(\exp(\frac{\sum_{(a,m)\in H_\ell} L(x_{m}^{(a)})}{t})\)
where \(0<x_{m}^{(a)}<1\) is the solution of the system of equations \[ x_{m}^{(a)} = \prod_{(b,n)\in H_{\ell}}(1-x_{n}^{(b)})^{K_{ab}^{mn}},\, (a,m)\in H_{\ell} \] and \(L\) is the Rogers dilogarithm function \[ L(x) = \operatorname{Li}_ 2(x)+\frac{1}{2}\log x\log (1-x),\, 0<x<1 \] \[ \operatorname{Li}_ 2(x)= \sum_{n=1}^\infty {x^n \over n^2},\, 0<x<1 \]
- thm (Chapoton, Nakanishi)
\[ \sum_{(a,m)\in H_\ell} L(x_{m}^{(a)}) = \frac{\pi^2}{6}(c(\ell)-r) \]
- proof uses Y-systems and cluster algebras
example
- \(\overline{\mathfrak{g}} = B_2\), level \(\ell = 2\), rank \(r=2\)
- \(t_1=1,t_2=2\)
- \(H_{\ell} = \{(1,1),(2,1),(2,2),(2,3)\}\)
- dual Coxeter number \[h^{\vee}=3\]
- \(\dim \overline{\mathfrak{g}}=10\)
- \(c(\ell)-r = 4-2=2\)
- \(K = \left( \begin{array}{cccc} 2 & -1 & -2 & -1 \\ -1 & 3 & 2 & 1 \\ -2 & 2 & 4 & 2 \\ -1 & 1 & 2 & 3 \\ \end{array} \right)/2\)
- equation for \(x^{(a)}_m\)
\[ \begin{aligned} x^{(1)}_1 & = (1-x^{(1)}_1)(1-x^{(2)}_1)^{-1/2}(1-x^{(2)}_2)^{-1}(1-x^{(2)}_3)^{-1/2} \\ x^{(2)}_1 & = (1-x^{(1)}_1)^{-1/2}(1-x^{(2)}_1)^{3/2}(1-x^{(2)}_2)^{1}(1-x^{(2)}_3)^{1/2}\\ x^{(2)}_2 & = \dots \\ x^{(2)}_3 & = \dots \\ \end{aligned} \]
- \(x^{(1)}_1= 3/4,x^{(2)}_1= 2/5,x^{(2)}_2= 4/9,x^{(2)}_3= 2/5\)
\[ L\left(\frac{3}{4}\right)+2 L\left(\frac{2}{5}\right)+L\left(\frac{4}{9}\right) = \frac{2\pi^2}{6} \]
quantum affine algebras and KR modules
- Q. is there more representation theoretic way to describe \(x^{(a)}_m\)?
- A. these numbers can be obtained from the quantum dimensions of Kirillov-Reshetikhin modules
- \(\exists\) bij. between iso. classes of fin.-dim'l irr. reps of \(\uqg\) and the set of \(I\)-tuples \(\mathbf{P}=(P_i)_{i\in I}\) of polys \(P_i\in \mathbb{C}[z]\) with \(P_i(0)=1\), called Drinfeld poly.
- KR module \(W^{(a)}_{m}(u)\) with \(a\in I\), \(m\in \mathbb{Z}_{\geq 0}\) and \(u\in \mathbb{C}^{\times}\) is associated with Drinfeld polynomials \(\mathbf{P}=(P_i)_{i\in I}\) of the form
\[ P_i(z) = \begin{cases} \prod _{s=1}^m \left(1- z u q_{a}^{2(s-1)}\right), & \text{if \(i=a\]}\\
1, & \text{otherwise} \\
\end{cases} \) where \(q_{a} = q^{t/t_a}\) and \(t=\max_{a\in I}t_a\).
- The quantum dimension of irr. h.w. \(U_q(\overline{\mathfrak{g}})\)-modules \(L(\lambda)\) at level \(k\) is
\[ \frac{\prod_{\alpha\in \Delta_{+}}\sin \frac{\pi(\lambda+\rho|\alpha)}{h^{\vee}+k}}{ \prod_{\in \Delta_{+}}\sin \frac{\pi (\rho|\alpha)}{h^{\vee}+k}}. \]
- recovers dimension as \(k\to \infty\) (qdim is an alg. int. not necessarily positive)
- regarding \(W^{(a)}_{m}(u)\) as \(U_q(\overline{\mathfrak{g}})\), obtain quantum dimension of a KR module
- thm (L.)
Fix level \(\ell\geq 2\). Let \(Q_{m}^{(a)}\) be the qdim of \(W^{(a)}_{m}(u)\) at level \(\ell\). Then \(Q_{m}^{(a)}\) with \((a,m)\in H_{\ell}\) is positive, \(Q_{t_a\ell}^{(a)}=1\), and \(x^{(a)}_m= 1-\frac{Q_{m-1}^{(a)}Q_{m+1}^{(a)}}{(Q_{m}^{(a)})^2}\).
- need fusion ring
example
- \(\overline{\mathfrak{g}} = B_2\), level \(\ell = 2\), rank \(r=2\)
- \(Q_{m}^{(1)} = 1,2,1\) for \(m=0,1,2\)
- \(Q_{m}^{(2)} = 1,\sqrt{5},3,\sqrt{5},1\) for \(m=0,1,2,3,4\)
memo
\[ \operatorname{mult}_{\Lambda}(\lambda-n\delta)\sim (\text{const})\times n^{-(1/4)(r+3)}e^{4\pi (a n)^{1/2}} \]