"Hecke indefinite modular forms"의 두 판 사이의 차이

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1번째 줄: 1번째 줄:
 
==introduction==
 
==introduction==
−
Let $\mathfrak{g} = A_1^{(1)}$. Fix a dominant integral weight $\Lambda$ of $\mathfrak{g}$ of level $m \geq 1$,  
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Let <math>\mathfrak{g} = A_1^{(1)}</math>. Fix a dominant integral weight <math>\Lambda</math> of <math>\mathfrak{g}</math> of level <math>m \geq 1</math>,  
−
and let $\lambda$ be a maximal dominant weight of $L(\Lambda)$.
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and let <math>\lambda</math> be a maximal dominant weight of <math>L(\Lambda)</math>.
 
   
 
   
−
Let $N$ denote the quadratic form defined on $\mathbb{R}^2$ by:
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Let <math>N</math> denote the quadratic form defined on <math>\mathbb{R}^2</math> by:
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$$ N(x,y):= 2 (m+2)x^2  - 2m y^2 \;\;\;\; (x,y \in \mathbb{R})$$
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:<math> N(x,y):= 2 (m+2)x^2  - 2m y^2 \;\;\;\; (x,y \in \mathbb{R})</math>
−
and let $(\cdot|\cdot)$ denote the corresponding symmetric bilinear form.
+
and let <math>(\cdot|\cdot)</math> denote the corresponding symmetric bilinear form.
−
Let $M:=\mathbb{Z}^2$ and let $M^*$ denote the lattice dual to $M$ with respect to this form.
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Let <math>M:=\mathbb{Z}^2</math> and let <math>M^*</math> denote the lattice dual to <math>M</math> with respect to this form.
  
−
Let $O(N)$ denote the group of invertible linear operators on $\mathbb{R}^2$
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Let <math>O(N)</math> denote the group of invertible linear operators on <math>\mathbb{R}^2</math>
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preserving $N$, and $SO_0(N)$ be the connected component of  
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preserving <math>N</math>, and <math>SO_0(N)</math> be the connected component of  
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$O(N)$ containing the identity. We then have the groups  
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<math>O(N)</math> containing the identity. We then have the groups  
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$G := \{g \in SO_0(N): g M =M \}$ and $G_0 := \{g \in G: g \text{ fixes } M^*/M \text{ pointwise}\}$.
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<math>G := \{g \in SO_0(N): g M =M \}</math> and <math>G_0 := \{g \in G: g \text{ fixes } M^*/M \text{ pointwise}\}</math>.
−
The set $U^+:=\{(x,y) \in \mathbb{R}^2: N(x,y) >0\}$ is preserved under the action of $O(N)$ on $\mathbb{R}^2$.
+
The set <math>U^+:=\{(x,y) \in \mathbb{R}^2: N(x,y) >0\}</math> is preserved under the action of <math>O(N)</math> on <math>\mathbb{R}^2</math>.
−
We let $A:=\frac{\langle{\Lambda + \rho,\check{\alpha}_1}\rangle}{2(m+2)}$ and   
+
We let <math>A:=\frac{\langle{\Lambda + \rho,\check{\alpha}_1}\rangle}{2(m+2)}</math> and   
−
$B:= \frac{\langle{\lambda, \check{\alpha}_1}\rangle}{2m}$ where $\check{\alpha}_1$ is the coroot corresponding to the underlying finite type diagram ($\mathfrak{sl}_2$ in this case), and $\rho$ is the Weyl vector.  Then, $(A,B) \in M^*$, and we set $L:= (A,B) + M$.
+
<math>B:= \frac{\langle{\lambda, \check{\alpha}_1}\rangle}{2m}</math> where <math>\check{\alpha}_1</math> is the coroot corresponding to the underlying finite type diagram (<math>\mathfrak{sl}_2</math> in this case), and <math>\rho</math> is the Weyl vector.  Then, <math>(A,B) \in M^*</math>, and we set <math>L:= (A,B) + M</math>.
  
 
The Hecke indefinite modular form is the following sum:
 
The Hecke indefinite modular form is the following sum:
−
$$\theta_L(\tau) := \sum_{\substack{(x,y) \in L \cap U^+ \\ (x,y) \text{ mod } G_0}} \mathrm{sign}(x,y) \,  e^{\pi i \tau N(x,y)},$$
+
:<math>\theta_L(\tau) := \sum_{\substack{(x,y) \in L \cap U^+ \\ (x,y) \text{ mod } G_0}} \mathrm{sign}(x,y) \,  e^{\pi i \tau N(x,y)},</math>
−
where $\mathrm{sign}(x,y) = 1$ for $x \geq 0$ and $-1$ for $x<0$.  
+
where <math>\mathrm{sign}(x,y) = 1</math> for <math>x \geq 0</math> and <math>-1</math> for <math>x<0</math>.  
−
This is an absolutely convergent sum for $\tau$ in the upper half plane $\mathbb{H}$, and defines a cusp form of weight 1.
+
This is an absolutely convergent sum for <math>\tau</math> in the upper half plane <math>\mathbb{H}</math>, and defines a cusp form of weight 1.
  
 
===examples===
 
===examples===
 
* Hecke
 
* Hecke
−
$$
+
:<math>
 
\prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2}
 
\prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2}
−
$$
+
</math>
 
* Kac-Peterson
 
* Kac-Peterson
−
$$
+
:<math>
 
\prod_{n=1}^{\infty}(1-q^n)(1-q^{2n})=\sum_{\substack{m,n=-\infty \\ n\geq 3|m|}}^{\infty}(-1)^{n+m}q^{(n^2-8m^2)/2+n/2}
 
\prod_{n=1}^{\infty}(1-q^n)(1-q^{2n})=\sum_{\substack{m,n=-\infty \\ n\geq 3|m|}}^{\infty}(-1)^{n+m}q^{(n^2-8m^2)/2+n/2}
−
$$
+
</math>
  
 
==string functions==
 
==string functions==
 
;thm (Kac-Peterson)
 
;thm (Kac-Peterson)
−
Let $\mathfrak{g} = A_1^{(1)}$. Let $\Lambda$ be a dominant integral weight of $\mathfrak{g}$, and $\lambda$ be a maximal dominant weight of $L(\Lambda)$. Then  
+
Let <math>\mathfrak{g} = A_1^{(1)}</math>. Let <math>\Lambda</math> be a dominant integral weight of <math>\mathfrak{g}</math>, and <math>\lambda</math> be a maximal dominant weight of <math>L(\Lambda)</math>. Then  
−
$$c^{\Lambda}_{\lambda}(\tau) = \theta_L(\tau) \, \eta(\tau)^{\scriptstyle{s} -3}.$$
+
:<math>c^{\Lambda}_{\lambda}(\tau) = \theta_L(\tau) \, \eta(\tau)^{\scriptstyle{s} -3}.</math>
−
Here $\theta_L(\tau)$ is a Hecke indefinite modular form and $\eta(\tau)$ is the Dedekind eta function.
+
Here <math>\theta_L(\tau)</math> is a Hecke indefinite modular form and <math>\eta(\tau)</math> is the Dedekind eta function.
  
 
==memo==
 
==memo==
50번째 줄: 50번째 줄:
 
==articles==
 
==articles==
 
* Westerholt-Raum, Martin. “H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Indefinite Theta Series.” arXiv:1207.5603 [math], July 24, 2012. http://arxiv.org/abs/1207.5603.
 
* Westerholt-Raum, Martin. “H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Indefinite Theta Series.” arXiv:1207.5603 [math], July 24, 2012. http://arxiv.org/abs/1207.5603.
−
* Sharma, Sachin S., and Sankaran Viswanath. ‘The $t$-Analogs of String Functions for $A_1^{(1)}$ and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
+
* Sharma, Sachin S., and Sankaran Viswanath. ‘The <math>t</math>-Analogs of String Functions for <math>A_1^{(1)}</math> and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
 
* Polishchuk, Alexander. ‘A New Look at Hecke’s Indefinite Theta Series’. arXiv:math/0012005, 1 December 2000. http://arxiv.org/abs/math/0012005.
 
* Polishchuk, Alexander. ‘A New Look at Hecke’s Indefinite Theta Series’. arXiv:math/0012005, 1 December 2000. http://arxiv.org/abs/math/0012005.
 
* Hiramatsu, Toyokazu, Noburo Ishii, and Yoshio Mimura. ‘On Indefinite Modular Forms of Weight One’. Journal of the Mathematical Society of Japan 38, no. 1 (January 1986): 67–83. doi:10.2969/jmsj/03810067.
 
* Hiramatsu, Toyokazu, Noburo Ishii, and Yoshio Mimura. ‘On Indefinite Modular Forms of Weight One’. Journal of the Mathematical Society of Japan 38, no. 1 (January 1986): 67–83. doi:10.2969/jmsj/03810067.

2020년 11월 16일 (월) 05:33 기준 최신판

introduction

Let \(\mathfrak{g} = A_1^{(1)}\). Fix a dominant integral weight \(\Lambda\) of \(\mathfrak{g}\) of level \(m \geq 1\), and let \(\lambda\) be a maximal dominant weight of \(L(\Lambda)\).

Let \(N\) denote the quadratic form defined on \(\mathbb{R}^2\) by: \[ N(x,y):= 2 (m+2)x^2 - 2m y^2 \;\;\;\; (x,y \in \mathbb{R})\] and let \((\cdot|\cdot)\) denote the corresponding symmetric bilinear form. Let \(M:=\mathbb{Z}^2\) and let \(M^*\) denote the lattice dual to \(M\) with respect to this form.

Let \(O(N)\) denote the group of invertible linear operators on \(\mathbb{R}^2\) preserving \(N\), and \(SO_0(N)\) be the connected component of \(O(N)\) containing the identity. We then have the groups \(G := \{g \in SO_0(N): g M =M \}\) and \(G_0 := \{g \in G: g \text{ fixes } M^*/M \text{ pointwise}\}\). The set \(U^+:=\{(x,y) \in \mathbb{R}^2: N(x,y) >0\}\) is preserved under the action of \(O(N)\) on \(\mathbb{R}^2\). We let \(A:=\frac{\langle{\Lambda + \rho,\check{\alpha}_1}\rangle}{2(m+2)}\) and \(B:= \frac{\langle{\lambda, \check{\alpha}_1}\rangle}{2m}\) where \(\check{\alpha}_1\) is the coroot corresponding to the underlying finite type diagram (\(\mathfrak{sl}_2\) in this case), and \(\rho\) is the Weyl vector. Then, \((A,B) \in M^*\), and we set \(L:= (A,B) + M\).

The Hecke indefinite modular form is the following sum: \[\theta_L(\tau) := \sum_{\substack{(x,y) \in L \cap U^+ \\ (x,y) \text{ mod } G_0}} \mathrm{sign}(x,y) \, e^{\pi i \tau N(x,y)},\] where \(\mathrm{sign}(x,y) = 1\) for \(x \geq 0\) and \(-1\) for \(x<0\). This is an absolutely convergent sum for \(\tau\) in the upper half plane \(\mathbb{H}\), and defines a cusp form of weight 1.

examples

  • Hecke

\[ \prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2} \]

  • Kac-Peterson

\[ \prod_{n=1}^{\infty}(1-q^n)(1-q^{2n})=\sum_{\substack{m,n=-\infty \\ n\geq 3|m|}}^{\infty}(-1)^{n+m}q^{(n^2-8m^2)/2+n/2} \]

string functions

thm (Kac-Peterson)

Let \(\mathfrak{g} = A_1^{(1)}\). Let \(\Lambda\) be a dominant integral weight of \(\mathfrak{g}\), and \(\lambda\) be a maximal dominant weight of \(L(\Lambda)\). Then \[c^{\Lambda}_{\lambda}(\tau) = \theta_L(\tau) \, \eta(\tau)^{\scriptstyle{s} -3}.\] Here \(\theta_L(\tau)\) is a Hecke indefinite modular form and \(\eta(\tau)\) is the Dedekind eta function.

memo

  • see Appendix of Jimbo, Miwa and Okado, 1986

related items

articles

  • Westerholt-Raum, Martin. “H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Indefinite Theta Series.” arXiv:1207.5603 [math], July 24, 2012. http://arxiv.org/abs/1207.5603.
  • Sharma, Sachin S., and Sankaran Viswanath. ‘The \(t\)-Analogs of String Functions for \(A_1^{(1)}\) and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
  • Polishchuk, Alexander. ‘A New Look at Hecke’s Indefinite Theta Series’. arXiv:math/0012005, 1 December 2000. http://arxiv.org/abs/math/0012005.
  • Hiramatsu, Toyokazu, Noburo Ishii, and Yoshio Mimura. ‘On Indefinite Modular Forms of Weight One’. Journal of the Mathematical Society of Japan 38, no. 1 (January 1986): 67–83. doi:10.2969/jmsj/03810067.
  • Jimbo, Michio, Tetsuji Miwa, and Masato Okado. ‘Solvable Lattice Models with Broken ZN Symmetry and Hecke’s Indefinite Modular Forms’. Nuclear Physics B 275, no. 3 (24 November 1986): 517–45. doi:10.1016/0550-3213(86)90611-5.
  • Jimbo, Michio, and Tetsuji Miwa. ‘A Solvable Lattice Model and Related Rogers-Ramanujan Type Identities’. Physica D: Nonlinear Phenomena 15, no. 3 (April 1985): 335–53. doi:10.1016/S0167-2789(85)80003-8.
  • Andrews, George E. "Hecke modular forms and the Kac-Peterson identities." Transactions of the American Mathematical Society (1984): 451-458.
  • Kac, V. G., and D. H. Peterson. ‘Affine Lie Algebras and Hecke Modular Forms’. Bulletin (New Series) of the American Mathematical Society 3, no. 3 (November 1980): 1057–61. http://projecteuclid.org/euclid.bams/1183547694
  • Über einen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen
    • E. Hecke, Mathematische Werke, Vandenhoeck and Ruprecht, Góttingen, 1959, pp. 418-427
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