The Graphic Nature of the Symmetric Group

Abstract

We investigate a remarkable class of exponential sums that are derived from the symmetric groups and that display a diverse array of visually appealing features. Our interest in these expressions stems not only from their astounding visual properties, but also from the fact that they represent a novel and intriguing class of supercharacters.

2000 AMS Subject Classification:

• 11L99
• 11L03
• 20B30
• 20C30
• 20K01

Keywords:

• supercharacter
• symmetric group
• exponential sum
• abelian group
• dihedral group
• orbits
• group action

ACKNOWLEDGMENTS

This research was was supported in part by NSF Grants DMS-1001614 and DMS-1265973. We also gratefully acknowledge the support of the Fletcher Jones Foundation and Pomona College's SURP Program.

Notes

¹If , then , where denotes the permanent of a matrix.

²One can also view this endeavor in terms of the classical character theory of the semidirect product (sometimes referred to as a generalized symmetric group). However, the supercharacter approach is cleaner and more natural, since is highly nonabelian and possesses a large number of conjugacy classes, whereas is abelian and, by comparison, has relatively few superclasses. Moreover, many of the irreducible characters of are uninteresting for our purposes (e.g., assuming only 0 or nth roots of unity as values).

³In order to facilitate the work of other researchers, we have included in an appendix the Mathematica code for generating supercharacter plots.