The Conway-Norton monstrous moonshine conjecture set off a quest to discover the connection between the Monster and the J-function. The goal of this poster is to give an overview of the components of the conjecture, the conjecture itself, and some of the ideas that led to its solution. Special focus is given to Klein\u27s J-function

Riley, Catherine

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Cedarville University DigitalCommons@Cedarville The Research and Scholarship Symposium The 2021 Symposium An Overview of Monstrous Moonshine Catherine Riley Cedarville University, catherineriley@cedarville.edu Follow this and additional works at: https://digitalcommons.cedarville.edu/rs_symposium Riley, Catherine, "An Overview of Monstrous Moonshine" (2021). The Research and Scholarship Symposium. 18. https://digitalcommons.cedarville.edu/rs_symposium/2021/poster_presentations/18 This Poster is brought to you for free and open access by DigitalCommons@Cedarville, a service of the Centennial Library. It has been accepted for inclusion in The Research and Scholarship Symposium by an authorized administrator of DigitalCommons@Cedarville. For more information, please contact digitalcommons@cedarville.edu. An Overview of Monstrous MoonshineCatherine Riley, Department of Science and Mathematics, Cedarville University, Cedarville, OH 45314, catherineriley@cedarville.eduApril 14, 2021AbstractThe Conway-Norton monstrous moonshine conjecture set off a quest todiscover the connection between the Monster and the J-function. The goal ofthis poster is to give an overview of the components of the conjecture, theconjecture itself, and some of the ideas that led to its solution. Special focusis given to Klein’s J-function.The Monster GroupIn 1973, Fischer and Griess independently conjectured that the Monstergroup, denoted M, existed. In 1980, Griess constructed it by hand as theautomorphism group of a 196,883-dimensional commutative nonassociativealgebra [2]. The Monster is the largest of a collection of groups called thesporadic groups. The order of the Monster (the number of elements in theMonster considered as a set) is|M| = 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71≈ 8× 1053.We want to think about ways this group can be represented.Representation and Character TheoryWe can consider any finite group G as a finite subgroup of the groupGL(n,C) of n × n invertible matrices with entries that are complex numbersvia a special function f which takes each element g in the group G to an nby n matrix in GL(n,C). The function f is called a representation and is ahomomorphism. We can break a homomorphism down into multiplehomomorphisms. Concretely, since our homomorphism sends the elements ofthe group to matrices, this looks like breaking down a matrix into smallermatrices along the diagonal [3].As an example, let us think about A5, the group of even permutations of afive element set. Suppose each element of A5 maps to an 18 by 18 matrix.An example of what such a matrix would look like is                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                 ~~                                  ,where the black squares represent elements in C (the complex numbers).Given that A5 has irreducible representations of dimension 5, dimension 4,two different representations of dimension 3, and one of dimension 1, thisrepresentation can be broken down into irreducibles, perhaps with one blockof dimension 4, one block of dimension 3, two blocks of the otherrepresentation of dimension 3, and five blocks of dimension 1. This gives usRepresentation and Character Theory (cont.)    0 0 0 0 0 0 0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0    0 0 0 0 0 0 0 0 0 0 00 0 0 0    0 0 0 0 0 0 0 0 0 0 00 0 0 0    0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 00 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 00 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 00 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 00 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ,where the first representation of dimension 3 is shown by the black squaresand the other representation is shown by the asterisks.The Monster has 194 equivalence classes of irreducible representations. In1978, Fischer, Livingstone, and Thorne determined the dimensions of theserepresentations. Starting with the trivial representation, the dimensions of thefirst few irreducible representations are(rn)n=1,··· ,194 = (1, 196 883, 21 296 876, 842 609 326, 18 538 750 076, · · · )and the rest of the sequence can be found in the ATLAS of Finite Groups orthe Online Encyclopaedia of Integer Sequences [4].Preliminary Notation for the J-FunctionTo understand the J-function, we need to understand elliptic functions. For afunction to be elliptic it must be doubly periodic, which means it has twoperiods, ω1 and ω2, both of which are complex numbers, with ratio ω2/ω1,where this ratio is not real. A pair of periods (ω1, ω2) for a function f is afundamental pair if all the periods of f are in the setΩ = {mω1 + nω2|m, n ∈ Z}. We can create a parallelogram by considering0, ω1, ω2, and ω1 + ω2.Figure 1.2.a from [1] More of the LatticeThis parallelogram is important because it is part of a larger lattice. If weunderstand what is happening in this one parallelogram, for instance, if it hasa singularity at p, we will know what is happening in all the otherparallelograms in the lattice.The J-function is an elliptic function defined as quotient of two otherfunctions. We define invariants g2 = 60G4 and g3 = 140G6, whereGn =∑ω∈Ω, ω 6=01ωn. One more function, called the discriminant, is defined by∆ = g 32 − 27g 23 .Definition of the J-FunctionWe now define Klein’s J-function. This function is a combination of g2 and∆. For ω2/ω1 6∈ R (the real numbers), we define Klein’s function asJ(ω1, ω2) =g 32 (ω1, ω2)∆(ω1, ω2)= J(λω1 + λω2)for λ 6= 0. For τ = ω2/ω1 we have that J(1, τ ) = J(ω1, ω2) [1].The J-function can be written as a Fourier expansion, or a sum in terms ofexponentials. Formally, we have the Fourier expansion123J(τ ) = e−2πiτ + 744 +∞∑n=1c(n)e2πinτ ,where the c(n) are integers [1].The coefficients of this sum have been calculated for n ≤ 100. The first feware c(0) = 744, c(1) = 196 884, c(2) = 21 493 760, and c(3) = 864 299 970.In some versions of this formula the 744 is subtracted from both sides, andthe result is called J(τ ). For q = e2πiτ , that gives us123J(τ )− 744 = q−1 + 196 884q + 21 493 760q2 + 864 299 970q3 + · · · .From now on we will refer to this version of the formula as J(τ ).Monstrous MoonshineAt this point, we return to the Monster. Recall that the first few dimensionsof irreducible representations of the Monster are(rn)n=1,··· ,194 = (1, 196 883, 21 296 876, 842 609 326, 18 538 750 076, · · · ).We now notice that196 884 = 1 + 196 88321 493 760 = 1 + 196 883 + 21 296 876864 299 970 = 2 · 1 + 2 · 196 883 + 21 296 876 + 842 609 326,where the numbers on the left hand side of the equation are coefficients fromJ(τ ) and the numbers on the right are from the dimensions of the irreduciblerepresentations of the Monster. Although Conway and Norton’s conjecturethat there was a reason behind this connection was first seen as “moonshine”,or a crazy idea, in 1992 Borcherds proved that a connection existed using avertex operator algebra and a result from string theory. So the connectionbetween the J-function and the Monster group was formally shown. Hence,two different areas of mathematics were united in a beautiful way [2].References[1] Tom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics.Springer-Verlag New York, 1990. isbn: 978-0-387-97127-8.[2] T. Gannon. “Monstrous Moonshine: The first twenty-five years.”. In: Bulletin of the London MathematicalSociety 38.1 (2006), pp. 1–33. issn: 14692120. url: https://cedarville.ohionet.org/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=edselc&AN=edselc.2-52.0-31744448388&site=eds-live.[3] Victor E. Hill IV. Groups and Characters. Chapman Hall/CRC, 2000. isbn: 1-58488-038-4.[4] Valdo Tatitscheff. A short introduction to Monstrous Moonshine. 2019. arXiv: 1902.03118 [math.NT].Thank you to Dr. Darrin Frey for his advice and expertise on this project.Catherine Riley, Department of Science and Mathematics, Cedarville University, Cedarville, OH 45314, catherineriley@cedarville.eduAn Overview of Monstrous Moonshine

An Overview of Monstrous Moonshine

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An Overview of Monstrous Moonshine

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