"Hecke indefinite modular forms"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| − | + | Let $\kma = A_1^{(1)}$. Fix a dominant integral weight $\Lambda$ of $\kma$ of level $m \geq 1$, | |
| − | + | and let $\lambda$ be a maximal dominant weight of $L(\Lambda)$. | |
| + | Let $N$ denote the quadratic form defined on $\reals^2$ by: | ||
| + | $$ N(x,y):= 2 (m+2)x^2 - 2m y^2 \;\;\;\; (x,y \in \reals)$$ | ||
| + | and let $\cform{\cdot}{\cdot}$ denote the corresponding symmetric bilinear form. | ||
| + | Let $M:=\integers^2$ 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 $\reals^2$ | |
| + | 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 $\Uplus:=\{(x,y) \in \reals^2: N(x,y) >0\}$ is preserved under the action of $O(N)$ on $\reals^2$. | ||
| + | We let $A:=\frac{\form{\Lambda + \rho,\check{\alpha}_1}}{2(m+2)}$ and | ||
| + | $B:= \frac{\form{\lambda, \check{\alpha}_1}}{2m}$ 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 that occurs in theorem \ref{kpthm} is the following sum: | ||
| + | $$\Hmf_L(\tau) := \sum_{\substack{(x,y) \in L \cap \Uplus \\ (x,y) \text{ mod } G_0}} \sign(x,y) \, e^{\pi i \tau N(x,y)},$$ | ||
| + | where $\sign(x,y) = 1$ for $x \geq 0$ and $-1$ for $x<0$. | ||
| + | This is an absolutely convergent sum for $\tau$ in the upper half plane $\uhp$, and defines a cusp form of weight 1. | ||
| − | |||
| − | + | ==string functions== | |
| + | ;thm (Kac-Peterson) | ||
| + | Let $\kma = A_1^{(1)}$. Let $\Lambda$ be a dominant integral weight of $\kma$, and $\lambda$ be a maximal dominant weight of $L(\Lambda)$. Then | ||
| + | $$\csf[\tau] = \Hmf_L(\tau) \, \eta(\tau)^{\s -3}.$$ | ||
| + | Here $\Hmf_L(\tau)$ is a {\em Hecke indefinite modular form} and $\eta(\tau)$ is the Dedekind eta function. | ||
| + | |||
| − | + | ==memo== | |
| + | * see the appendix of | ||
| + | ** Michio Jimbo, Tetsuji Miwa and Masato Okado [http://dx.doi.org/10.1016/0550-3213(86)90611-5 Solvable lattice models with broken symmetry and Hecke's indefinite modular forms], 1986 | ||
2014년 12월 15일 (월) 21:15 판
introduction
Let $\kma = A_1^{(1)}$. Fix a dominant integral weight $\Lambda$ of $\kma$ of level $m \geq 1$, and let $\lambda$ be a maximal dominant weight of $L(\Lambda)$.
Let $N$ denote the quadratic form defined on $\reals^2$ by:
$$ N(x,y):= 2 (m+2)x^2 - 2m y^2 \;\;\;\; (x,y \in \reals)$$ and let $\cform{\cdot}{\cdot}$ denote the corresponding symmetric bilinear form. Let $M:=\integers^2$ 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 $\reals^2$ 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 $\Uplus:=\{(x,y) \in \reals^2: N(x,y) >0\}$ is preserved under the action of $O(N)$ on $\reals^2$. We let $A:=\frac{\form{\Lambda + \rho,\check{\alpha}_1}}{2(m+2)}$ and $B:= \frac{\form{\lambda, \check{\alpha}_1}}{2m}$ 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 that occurs in theorem \ref{kpthm} is the following sum: $$\Hmf_L(\tau) := \sum_{\substack{(x,y) \in L \cap \Uplus \\ (x,y) \text{ mod } G_0}} \sign(x,y) \, e^{\pi i \tau N(x,y)},$$ where $\sign(x,y) = 1$ for $x \geq 0$ and $-1$ for $x<0$. This is an absolutely convergent sum for $\tau$ in the upper half plane $\uhp$, and defines a cusp form of weight 1.
string functions
- thm (Kac-Peterson)
Let $\kma = A_1^{(1)}$. Let $\Lambda$ be a dominant integral weight of $\kma$, and $\lambda$ be a maximal dominant weight of $L(\Lambda)$. Then $$\csf[\tau] = \Hmf_L(\tau) \, \eta(\tau)^{\s -3}.$$
Here $\Hmf_L(\tau)$ is a {\em Hecke indefinite modular form} and $\eta(\tau)$ is the Dedekind eta function.
memo
- see the appendix of
- Michio Jimbo, Tetsuji Miwa and Masato Okado Solvable lattice models with broken symmetry and Hecke's indefinite modular forms, 1986
articles
- 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.
- 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