We discuss a simple mathematical mechanism that produces a variety of striking images of great complexity and subtlety. We briefly explain this approach and present a selection of attractive images obtained using this technique

Burkhardt, Paula, \u2716

Currier, Gabriel, \u2716

de Langis, Mathieu, \u2715

Garcia, Stephan Ramon

Lutz, Bob, \u2713

Suh, Hong, \u2716

Keck Graduate Institute

Claremont Colleges
Scholarship @ Claremont
Pomona Faculty Publications and Research Pomona Faculty Scholarship
1-1-2015
An Exhibition of Exponential Sums: Visualizing
Supercharacters
Paula Burkhardt '16
Pomona College
Gabriel Currier '16
Pomona College
Stephan Ramon Garcia
Pomona College
Mathieu de Langis '15
Pomona College
Bob Lutz '13
Pomona College
See next page for additional authors
This Article is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted for
inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please
contact scholarship@cuc.claremont.edu.
Recommended Citation
Burkhardt, P., Currier, G., Garcia, S.R., de Langis, M., Lutz, B., Suh, H., An Exhibition of Exponential Sums: Visualizing
Supercharacters, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 475-478.
Authors
Paula Burkhardt '16, Gabriel Currier '16, Stephan Ramon Garcia, Mathieu de Langis '15, Bob Lutz '13, and
Hong Suh '16
This article is available at Scholarship @ Claremont: http://scholarship.claremont.edu/pomona_fac_pub/441
An Exhibition of Exponential Sums:
Visualizing Supercharacters
Paula Burkhardt, Gabriel Currier, Stephan Ramon Garcia, Mathieu de Langis, Bob Lutz, Hong Suh
Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA
stephan.garcia@pomona.edu
http://pages.pomona.edu/~sg064747
Abstract
We discuss a simple mathematical mechanism that produces a variety of striking images of great complexity and
subtlety. We briefly explain this approach and present a selection of attractive images obtained using this technique.
In this short note we discuss “supercharacters” on abelian groups, which provide a simple mathematical
mechanism that produces a variety of striking images of great complexity and subtlety. We briefly explain
this approach and present a selection of attractive images obtained using this technique.
Our original motivation stems from the recent (2008) work of P. Diaconis and I. M. Isaacs [3] in combi-
natorial representation theory. Here we only consider supercharacters on abelian groups. In this setting, the
details are simpler and we do not require any knowledge of representation theory. A more general approach
would require too much machinery and take us too far afield. A recent treatment of supercharacter theory on
abelian groups can be found in [2], while some useful remarks on the general case can be found in [5].
In what follows, we consider an abelian group G = (Z/nZ)d whose typical elements will be denoted
x = (x1, x2, . . . , xd), y = (y1, y2, . . . , yd), and so forth. A homomorphism ϕ : G → C× is called a
character ofG; that is, ϕ is a nonzero, complex-valued function onG that satisfies ϕ(x+y) = ϕ(x)ϕ(y) for
all x,y ∈ G. If ζn = e2pii/n, then the characters of G are precisely those functions of the form ϕ(x) = ζx·yn
for some y ∈ G. Here x · y = x1y1 + x2y2 + · · ·+ xdyd denotes the formal dot product of x,y ∈ G.
The values assumed by a character are nth roots of unity; plotting these values in the complex plane
yields little of interest. However, if one considers certain sums of characters, more interesting patterns often
emerge. We are concerned here with several families of character sums that display some truly remarkable
graphical behavior.
To construct our character sums, we need to select a group of automorphisms of G. Suppose that Γ is
a group of d × d invertible matrices with entries in Z/nZ. For certain technical reasons, we insist that Γ is
closed under the transpose operation. In the images that follow, Γ is constructed (in whole or in part) using
the cyclic subgroup 〈ω〉 of the unit group (Z/nZ)× generated by some unit ω ∈ Z/nZ.
The natural action (A,x) 7→ Ax of Γ on G yields a partition of G into Γ-orbits. For instance, the orbit
of x ∈ G is the set {Ax : A ∈ Γ} and the orbit of 0 = (0, 0, . . . , 0) is {0}. For each Γ-orbit X in G, the
associated supercharacter is the function σX : G→ C defined by
σX(y) =
∑
x∈X
ζx·yn . (1)
It can be shown that a supercharacter is constant on each Γ-orbit in G; that is, σX(y′) = σX(y) if y′ = Ay
for some A ∈ Γ. Thus, to plot the values {σX(y) : y ∈ G} in the complex plane, we need only select one
representative from each Γ-orbit.
We now present a gallery of supercharacter plots obtained by choosing suitable G, Γ, and X . The
resulting images are often visually striking and exhibit a variety of unexpected phenomena. Colors may be
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture
475
(A) n = 1767, ω = 88 (B) n = 1776, ω = 295 (C) n = 2442, ω = 73
Figure 1: The images above depict the values in C of the supercharacter σX : G→ C defined by (1). Here
G = (Z/nZ)2 is the direct sum of two copies of Z/nZ. We let Γ = S2 × 〈ω〉 be the direct product of the
symmetric group S2 and a cyclic subgroup 〈ω〉 ⊆ (Z/nZ)× generated by some unit ω. That is, the group Γ
that acts on G consists of all multiples of 2× 2 permutation matrices by a power of the generator ω. Finally,
X = {Ax : A ∈ Γ} is the Γ-orbit of x = (0, 1) in G. The color schemes are ad hoc; the color of the point
σX(y) depends upon some particular arithmetic properties of the input y.
added to these images in an ad-hoc manner by considering various arithmetic properties of the input y. For
instance, one might assign one of five different colors to σX(y) depending upon the value of y1+y2+· · ·+yd
modulo five. The choice of coloring scheme, while largely an aesthetic consideration, often clarifies the
details of our plots and hints at underlying arithmetic properties of the corresponding supercharacters.
The proofs that certain combinations of parameters result in particular types of images (e.g., the nested
five-cusped hypocycloids in Figure 2D) typically involve nontrivial tools from algebraic or analytic number
theory. In addition, the ad hoc analysis of some associated multivariate Laurent polynomials is often required.
We are therefore unable to give a satisfactory account of the mechanism behind any of these patterns in the
confines of this short note. However, the interested reader is invited to consult the papers [1, 4, 6] in which
the details are fully worked out.
References
[1] J. L. Brumbaugh, Madeleine Bulkow, Luis Alberto Garcia German, Stephan Ramon Garcia, Matt Michal, and Andrew P. Turner.
The graphic nature of the symmetric group. Experimental Mathematics, 22(4):421–442, 2013.
[2] J.L. Brumbaugh, Madeleine Bulkow, Patrick S. Fleming, Luis Alberto Garcia German, Stephan Ramon Garcia, Gizem Karaali,
Matt Michal, Andrew P. Turner, and Hong Suh. Supercharacters, exponential sums, and the uncertainty principle. Journal of
Number Theory, 144(0):151–175, 2014.
[3] Persi Diaconis and I. M. Isaacs. Supercharacters and superclasses for algebra groups. Transactions of the American Mathemat-
ical Society, 360(5):2359–2392, 2008.
[4] William Duke, Stephan Ramon Garcia, and Bob Lutz. The graphic nature of Gaussian periods. Proceedings of the American
Mathematical Society, 2015.
[5] Christopher F. Fowler, Stephan Ramon Garcia, and Gizem Karaali. Ramanujan sums as supercharacters. The Ramanujan
Journal, 35(2):205241, 2014.
[6] Stephan Ramon Garcia, Trevor Hyde, and Bob Lutz. Gauss’ hidden menagerie: from cyclotomy to supercharacters. Notices of
the American Mathematical Society, 2015. (accepted pending revision) http://arxiv.org/abs/1501.07507.
Burkhardt et al.
476
(A) n = 455175, ω = 13043 (B) n = 68913, ω = 88 (C) n = 91205, ω = 2337
(D) n = 352655, ω = 54184 (E) n = 273615, ω = 184747 (F) n = 328549, ω = 9247
(G) n = 255255, ω = 254 (H) n = 455175, ω = 3599
Figure 2: The images above depict the values in C of the supercharacter σX : G→ C defined by (1). Here
G = Z/nZ is a cyclic group (i.e., d = 1) and Γ = 〈ω〉 is the cyclic subgroup of (Z/nZ)× generated by an
invertible element ω in Z/nZ. The group Γ acts on G by multiplication. We let X = Γ denote the orbit of
1 under the action of Γ. The color schemes are ad hoc; the color of the point σX(y) depends upon some
particular arithmetic properties of the input y.
An Exhibition of Exponential Sums: Visualizing Supercharacters
477
(A) n = 9015, ω = 2284 (B) n = 890, ω = 479 (C) n = 9015, ω = 577
(D) n = 13309, ω = 7 (E) n = 2221, ω = 71 (F) n = 3571, ω = 47
(G) n = 4378, ω = 291 (H) n = 3020, ω = 1089
Figure 3: The images above depict the values in C of the supercharacter σX : G→ C defined by (1). Here
G = (Z/nZ)2 is the direct sum of two copies of Z/nZ. We let Γ = {diag(u, u−1) : u ∈ 〈ω〉} be a group
of diagonal matrices that acts on G by multiplication. Here ω is an element of the unit group (Z/nZ)× that
generates the cyclic subgroup 〈ω〉 of (Z/nZ)×. Finally, X = {Ax : A ∈ Γ} is the orbit in G of x = (1, 1)
under the action of Γ. The color schemes are ad hoc; the color of the point σX(y) depends upon some
particular arithmetic properties of the input y.
Burkhardt et al.
478


English

An Exhibition of Exponential Sums: Visualizing Supercharacters

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontPomona Faculty Publications and Research Pomona Faculty Scholarship1-1-2015An Exhibition of Exponential Sums: VisualizingSupercharactersPaula Burkhardt '16Pomona CollegeGabriel Currier '16Pomona CollegeStephan Ramon GarciaPomona CollegeMathieu de Langis '15Pomona CollegeBob Lutz '13Pomona CollegeSee next page for additional authorsThis Article is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted forinclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, pleasecontact scholarship@cuc.claremont.edu.Recommended CitationBurkhardt, P., Currier, G., Garcia, S.R., de Langis, M., Lutz, B., Suh, H., An Exhibition of Exponential Sums: VisualizingSupercharacters, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 475-478.AuthorsPaula Burkhardt '16, Gabriel Currier '16, Stephan Ramon Garcia, Mathieu de Langis '15, Bob Lutz '13, andHong Suh '16This article is available at Scholarship @ Claremont: http://scholarship.claremont.edu/pomona_fac_pub/441An Exhibition of Exponential Sums:Visualizing SupercharactersPaula Burkhardt, Gabriel Currier, Stephan Ramon Garcia, Mathieu de Langis, Bob Lutz, Hong SuhDepartment of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USAstephan.garcia@pomona.eduhttp://pages.pomona.edu/~sg064747AbstractWe discuss a simple mathematical mechanism that produces a variety of striking images of great complexity andsubtlety. We briefly explain this approach and present a selection of attractive images obtained using this technique.In this short note we discuss “supercharacters” on abelian groups, which provide a simple mathematicalmechanism that produces a variety of striking images of great complexity and subtlety. We briefly explainthis approach and present a selection of attractive images obtained using this technique.Our original motivation stems from the recent (2008) work of P. Diaconis and I. M. Isaacs [3] in combi-natorial representation theory. Here we only consider supercharacters on abelian groups. In this setting, thedetails are simpler and we do not require any knowledge of representation theory. A more general approachwould require too much machinery and take us too far afield. A recent treatment of supercharacter theory onabelian groups can be found in [2], while some useful remarks on the general case can be found in [5].In what follows, we consider an abelian group G = (Z/nZ)d whose typical elements will be denotedx = (x1, x2, . . . , xd), y = (y1, y2, . . . , yd), and so forth. A homomorphism ϕ : G → C× is called acharacter ofG; that is, ϕ is a nonzero, complex-valued function onG that satisfies ϕ(x+y) = ϕ(x)ϕ(y) forall x,y ∈ G. If ζn = e2pii/n, then the characters of G are precisely those functions of the form ϕ(x) = ζx·ynfor some y ∈ G. Here x · y = x1y1 + x2y2 + · · ·+ xdyd denotes the formal dot product of x,y ∈ G.The values assumed by a character are nth roots of unity; plotting these values in the complex planeyields little of interest. However, if one considers certain sums of characters, more interesting patterns oftenemerge. We are concerned here with several families of character sums that display some truly remarkablegraphical behavior.To construct our character sums, we need to select a group of automorphisms of G. Suppose that Γ isa group of d × d invertible matrices with entries in Z/nZ. For certain technical reasons, we insist that Γ isclosed under the transpose operation. In the images that follow, Γ is constructed (in whole or in part) usingthe cyclic subgroup 〈ω〉 of the unit group (Z/nZ)× generated by some unit ω ∈ Z/nZ.The natural action (A,x) 7→ Ax of Γ on G yields a partition of G into Γ-orbits. For instance, the orbitof x ∈ G is the set {Ax : A ∈ Γ} and the orbit of 0 = (0, 0, . . . , 0) is {0}. For each Γ-orbit X in G, theassociated supercharacter is the function σX : G→ C defined byσX(y) =∑x∈Xζx·yn . (1)It can be shown that a supercharacter is constant on each Γ-orbit in G; that is, σX(y′) = σX(y) if y′ = Ayfor some A ∈ Γ. Thus, to plot the values {σX(y) : y ∈ G} in the complex plane, we need only select onerepresentative from each Γ-orbit.We now present a gallery of supercharacter plots obtained by choosing suitable G, Γ, and X . Theresulting images are often visually striking and exhibit a variety of unexpected phenomena. Colors may beProceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture475(A) n = 1767, ω = 88 (B) n = 1776, ω = 295 (C) n = 2442, ω = 73Figure 1: The images above depict the values in C of the supercharacter σX : G→ C defined by (1). HereG = (Z/nZ)2 is the direct sum of two copies of Z/nZ. We let Γ = S2 × 〈ω〉 be the direct product of thesymmetric group S2 and a cyclic subgroup 〈ω〉 ⊆ (Z/nZ)× generated by some unit ω. That is, the group Γthat acts on G consists of all multiples of 2× 2 permutation matrices by a power of the generator ω. Finally,X = {Ax : A ∈ Γ} is the Γ-orbit of x = (0, 1) in G. The color schemes are ad hoc; the color of the pointσX(y) depends upon some particular arithmetic properties of the input y.added to these images in an ad-hoc manner by considering various arithmetic properties of the input y. Forinstance, one might assign one of five different colors to σX(y) depending upon the value of y1+y2+· · ·+ydmodulo five. The choice of coloring scheme, while largely an aesthetic consideration, often clarifies thedetails of our plots and hints at underlying arithmetic properties of the corresponding supercharacters.The proofs that certain combinations of parameters result in particular types of images (e.g., the nestedfive-cusped hypocycloids in Figure 2D) typically involve nontrivial tools from algebraic or analytic numbertheory. In addition, the ad hoc analysis of some associated multivariate Laurent polynomials is often required.We are therefore unable to give a satisfactory account of the mechanism behind any of these patterns in theconfines of this short note. However, the interested reader is invited to consult the papers [1, 4, 6] in whichthe details are fully worked out.References[1] J. L. Brumbaugh, Madeleine Bulkow, Luis Alberto Garcia German, Stephan Ramon Garcia, Matt Michal, and Andrew P. Turner.The graphic nature of the symmetric group. Experimental Mathematics, 22(4):421–442, 2013.[2] J.L. Brumbaugh, Madeleine Bulkow, Patrick S. Fleming, Luis Alberto Garcia German, Stephan Ramon Garcia, Gizem Karaali,Matt Michal, Andrew P. Turner, and Hong Suh. Supercharacters, exponential sums, and the uncertainty principle. Journal ofNumber Theory, 144(0):151–175, 2014.[3] Persi Diaconis and I. M. Isaacs. Supercharacters and superclasses for algebra groups. Transactions of the American Mathemat-ical Society, 360(5):2359–2392, 2008.[4] William Duke, Stephan Ramon Garcia, and Bob Lutz. The graphic nature of Gaussian periods. Proceedings of the AmericanMathematical Society, 2015.[5] Christopher F. Fowler, Stephan Ramon Garcia, and Gizem Karaali. Ramanujan sums as supercharacters. The RamanujanJournal, 35(2):205241, 2014.[6] Stephan Ramon Garcia, Trevor Hyde, and Bob Lutz. Gauss’ hidden menagerie: from cyclotomy to supercharacters. Notices ofthe American Mathematical Society, 2015. (accepted pending revision) http://arxiv.org/abs/1501.07507.Burkhardt et al.476(A) n = 455175, ω = 13043 (B) n = 68913, ω = 88 (C) n = 91205, ω = 2337(D) n = 352655, ω = 54184 (E) n = 273615, ω = 184747 (F) n = 328549, ω = 9247(G) n = 255255, ω = 254 (H) n = 455175, ω = 3599Figure 2: The images above depict the values in C of the supercharacter σX : G→ C defined by (1). HereG = Z/nZ is a cyclic group (i.e., d = 1) and Γ = 〈ω〉 is the cyclic subgroup of (Z/nZ)× generated by aninvertible element ω in Z/nZ. The group Γ acts on G by multiplication. We let X = Γ denote the orbit of1 under the action of Γ. The color schemes are ad hoc; the color of the point σX(y) depends upon someparticular arithmetic properties of the input y.An Exhibition of Exponential Sums: Visualizing Supercharacters477(A) n = 9015, ω = 2284 (B) n = 890, ω = 479 (C) n = 9015, ω = 577(D) n = 13309, ω = 7 (E) n = 2221, ω = 71 (F) n = 3571, ω = 47(G) n = 4378, ω = 291 (H) n = 3020, ω = 1089Figure 3: The images above depict the values in C of the supercharacter σX : G→ C defined by (1). HereG = (Z/nZ)2 is the direct sum of two copies of Z/nZ. We let Γ = {diag(u, u−1) : u ∈ 〈ω〉} be a groupof diagonal matrices that acts on G by multiplication. Here ω is an element of the unit group (Z/nZ)× thatgenerates the cyclic subgroup 〈ω〉 of (Z/nZ)×. Finally, X = {Ax : A ∈ Γ} is the orbit in G of x = (1, 1)under the action of Γ. The color schemes are ad hoc; the color of the point σX(y) depends upon someparticular arithmetic properties of the input y.Burkhardt et al.478

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An Exhibition of Exponential Sums: Visualizing Supercharacters

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