## Nagoya Mathematical Journal

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

Satoshi Naito

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631954

Mathematical Reviews number (MathSciNet)
MR2019518

Zentralblatt MATH identifier
1074.11026

#### Citation

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. https://projecteuclid.org/euclid.nmj/1114631954

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