Nagoya Mathematical Journal

A combinatorial identity for the derivative of a theta series of a finite type root lattice

Satoshi Naito

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Let ${\mathfrak g}$ be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with ${\mathfrak h}$ the Cartan subalgebra and $Q \subset {\mathfrak h}^{*}$ the root lattice. Denote by $\Theta_{Q}(q)$ the theta series of the root lattice $Q$ of ${\mathfrak g}$. We prove a curious "combinatorial" identity for the derivative of $\Theta_{Q}(q)$, i.e.\ for $q \frac{d}{dq} \Theta_{Q}(q)$, by using the representation theory of an affine Lie algebra.

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Nagoya Math. J., Volume 172 (2003), 1-30.

First available in Project Euclid: 27 April 2005

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Primary: 11F22: Relationship to Lie algebras and finite simple groups
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A30: $q$-calculus and related topics [See also 33Dxx] 11E45: Analytic theory (Epstein zeta functions; relations with automorphic 11F27: Theta series; Weil representation; theta correspondences 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B69: Vertex operators; vertex operator algebras and related structures


Naito, Satoshi. A combinatorial identity for the derivative of a theta series of a finite type root lattice. Nagoya Math. J. 172 (2003), 1--30.

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  • I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models , Invent. Math., 62 (1980), 23–66.
  • V. G. Kac, Infinite-dimensional algebras, Dedekind's $\eta$-function, classical Möbius function and the very strange formula , Adv. Math., 30 (1978), 85–136.
  • V. G. Kac, An elucidation of “Infinite-dimensional algebras $\dots$ and the very strange formula.” $E_8^(1)$ and the cube root of the modular invariant $j$ , Adv. Math., 35 (1980), 264–273.
  • V. G. Kac, A remark on the Conway-Norton conjecture about the “Monster” simple group , Proc. Natl. Acad. Sci. U.S.A., 77 (1980), 5048–5049.
  • V. G. Kac, Infinite Dimensional Lie Algebras (3rd ed.), Cambridge Univ. Press, Cambridge (1990).
  • V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms , Adv. Math., 53 (1984), 125–264.
  • V. G. Kac and I. T. Todorov, Affine orbifolds and rational conformal field theory extensions of $W_1+\infty$ , Comm. Math. Phys., 190 (1997), 57–111.
  • V. G. Kac and M. Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras , Adv. Math., 70 (1988), 156–236.
  • Z.-X. Wan, Introduction to Kac-Moody Algebra, World Scientific, Singapore (1991).