The Character of the Infinite Wedge Representation

✍ Scribed by Spencer Bloch; Andrei Okounkov

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 440 KB
• Volume: 149
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1845

✦ Synopsis

A man's character is his fate.

Heraclitus, 540 480 B.C.

We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions. These correlation functions are sums of determinants built from genus 1 theta functions and their derivatives. 2000 Academic Press Contents 0. Introduction. 1. The infinite wedge representation. 2. Elliptic transformation on the character 0. 3. Quasimodular forms.

4. Quasimodularity for characters 0 and V. 5. Preliminaries on partitions. 6. The formula for correlation functions. 7. Beginning of the proof of Theorem 6.1. 8. Difference equations for the correlation functions. 9. Singularities of correlation functions. 10. Difference equations for the RHS in (6.5). 11. Singularities of the RHS in (6.5). 12. Conclusion of the proof of Theorem 6.1. 13. Another example.