neverendingbooks

Tag: monster

  • Monstrous dessins 1

    Dedekind’s Psi-function $\Psi(n)= n \prod_{p |n}(1 + \frac{1}{p})$ pops up in a number of topics: $\Psi(n)$ is the index of the congruence subgroup $\Gamma_0(n)$ in the modular group $\Gamma=PSL_2(\mathbb{Z})$, $\Psi(n)$ is the number of points in the projective line $\mathbb{P}^1(\mathbb{Z}/n\mathbb{Z})$, $\Psi(n)$ is the number of classes of $2$-dimensional lattices $L_{M \frac{g}{h}}$ at hyperdistance $n$ in […]

  • the Riemann hypothesis and 5040

    Yesterday, there was an interesting post by John Baez at the n-category cafe: The Riemann Hypothesis Says 5040 is the Last. The 5040 in the title refers to the largest known counterexample to a bound for the sum-of-divisors function \[ \sigma(n) = \sum_{d | n} d = n \sum_{d | n} \frac{1}{d} \] In 1983,…

  • the monster dictates her picture

    The monstrous moonshine picture is a sub-graph of Conway’s Big Picture on 218 vertices. These vertices are the classes of lattices needed in the construction of the 171 moonshine groups. That is, moonshine gives us the shape of the picture. (image credit Friendly Monsters) But we can ask to reverse this process. Is the shape…

  • A forgotten type and roots of unity (again)

    The monstrous moonshine picture is the finite piece of Conway’s Big Picture needed to understand the 171 moonshine groups associated to conjugacy classes of the monster. Last time I claimed that there were exactly 7 types of local behaviour, but I missed one. The forgotten type is centered at the number lattice $84$. Locally around…

  • What we (don’t) know

    Do we know why the monster exists and why there’s moonshine around it? The answer depends on whether or not you believe that vertex operator algebras are natural, elegant and inescapable objects. the monster Simple groups often arise from symmetries of exceptionally nice mathematical objects. The smallest of them all, $A_5$, gives us the rotation…

  • the moonshine picture – at last

    The monstrous moonshine picture is the subgraph of Conway’s big picture consisting of all lattices needed to describe the 171 moonshine groups. It consists of: – exactly 218 vertices (that is, lattices), out of which – 97 are number-lattices (that is of the form $M$ with $M$ a positive integer), and – 121 are proper…

  • Moonshine’s green anaconda

    The largest snake in the moonshine picture determines the moonshine group $(24|12)$ and is associated to conjugacy class $24J$ of the monster. It contains $70$ lattices, about one third of the total number of lattices in the moonshine picture. The anaconda’s backbone is the $(288|1)$ thread below (edges in the $2$-tree are black, those in…

  • the 171 moonshine groups

    Monstrous moonshine associates to every element of order $n$ of the monster group $\mathbb{M}$ an arithmetic group of the form \[ (n|h)+e,f,\dots \] where $h$ is a divisor of $24$ and of $n$ and where $e,f,\dots$ are divisors of $\frac{n}{h}$ coprime with its quotient. In snakes, spines, and all that we’ve constructed the arithmetic group…

  • Snakes, spines, threads and all that

    Conway introduced his Big Picture to make it easier to understand and name the groups appearing in Monstrous Moonshine. For $M \in \mathbb{Q}_+$ and $0 \leq \frac{g}{h} < 1$, $M,\frac{g}{h}$ denotes (the projective equivalence class of) the lattice \[ \mathbb{Z} (M \vec{e}_1 + \frac{g}{h} \vec{e}_2) \oplus \mathbb{Z} \vec{e}_2 \] which we also like to represent…

  • The defining property of 24

    From Wikipedia on 24: “$24$ is the only number whose divisors, namely $1, 2, 3, 4, 6, 8, 12, 24$, are exactly those numbers $n$ for which every invertible element of the commutative ring $\mathbb{Z}/n\mathbb{Z}$ is a square root of $1$. It follows that the multiplicative group $(\mathbb{Z}/24\mathbb{Z})^* = \{ \pm 1, \pm 5, \pm…