Combinatorics of traces of Hecke operators. Academic Article

abstract

• We investigate the combinatorial properties of the traces of the nth Hecke operators on the spaces of weight 2k cusp forms of level N. We establish examples in which these traces are expressed in terms of classical objects in enumerative combinatorics (e.g., tilings and Motzkin paths). We establish in general that Hecke traces are explicit rational linear combinations of values of Gegenbauer (also known as ultraspherical) polynomials. These results arise from "packaging" the Hecke traces into power series in weight aspect. These generating functions are easily computed by using the Eichler-Selberg trace formula.

authors

• Papanikolas, Matthew

published proceedings

• Proc Natl Acad Sci U S A

author list (cited authors)

• Frechette, S., Ono, K., & Papanikolas, M.

citation count

• 8

complete list of authors

• Frechette, Sharon||Ono, Ken||Papanikolas, Matthew

publication date

• December 2004

publisher

• Proceedings of the National Academy of Sciences  Publisher

published in

• Proceedings of the National Academy of Sciences of the United States of America  Journal

keywords

• 0101 Pure Mathematics

PubMed Central ID

• 15563597

Digital Object Identifier (DOI)

• 10.1073/pnas.0407223101

URI

• https://hdl.handle.net/1969.1/178635

start page

• 17016

end page

• 17020

volume

• 101

issue

• 49

URL

• http://dx.doi.org/10.1073/pnas.0407223101