The Galois Correspondence for Branched Covering Spaces and its Relationship to Hecke Algebras

Authors

Matthew Ong

Document Type

Article

Publication Date

11-27-2002

First Advisor

S. Allen Broughton

Abstract

There is a very beautiful correspondence between branched covers of the Riemann sphere P¹ and subgroups of the fundamental group π₁(P¹ − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of non- Galois covers S/P¹. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of the induced character 1_H^G , and that the character of the homology representation of a Galois group generalizes to one for Hecke algebras, the decomposition of which depends on certain double cosets in the group corresponding to the Galois closure of the cover S/P¹.

Comments

MSTR 02-08

Recommended Citation

Ong, Matthew, "The Galois Correspondence for Branched Covering Spaces and its Relationship to Hecke Algebras" (2002). Mathematical Sciences Technical Reports (MSTR). 89.
https://scholar.rose-hulman.edu/math_mstr/89