# Posts Tagged: permutation representation

• groups

## the monster graph and McKay’s observation

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While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s group-theoretic construction of the icosahedron from its rotation-symmetry group $A_5$. The alternating group $A_5$ has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say $C$ and… Read more »

• geometry, noncommutative

## noncommutative F_un geometry (2)

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We use Kontsevich’s idea of thin varieties to define complexified varieties over F\_un.

• featured

## what does the monster see?

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The Monster is the largest of the 26 sporadic simple groups and has order 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71. It is not so much the size… Read more »

• groups

## Galois’ last letter

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“Ne pleure pas, Alfred ! J’ai besoin de tout mon courage pour mourir à vingt ans!” We all remember the last words of Evariste Galois to his brother Alfred. Lesser known are the mathematical results contained in his last letter, written to his friend Auguste Chevalier, on the eve of his fatal duel. Here the… Read more »

• featured

## Farey symbols of sporadic groups

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John Conway once wrote : There are almost as many different constructions of $M_{24}$ as there have been mathematicians interested in that most remarkable of all finite groups. In the inguanodon post Ive added yet another construction of the Mathieu groups $M_{12}$ and $M_{24}$ starting from (half of) the Farey sequences and… Read more »

• featured

## Quiver-superpotentials

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It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group $\Gamma = PSL_2(\mathbb{Z})$ is determined by the conjugacy class of a cofinite subgroup $\Lambda \subset \Gamma$, or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a… Read more »

• featured

## quivers versus quilts

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We have associated to a subgroup of the modular group $PSL_2(\mathbb{Z})$ a quiver (that is, an oriented graph). For example, one verifies that the fundamental domain of the subgroup $\Gamma_0(2)$ (an index 3 subgroup) is depicted on the right by the region between the thick lines with the identification of edges as indicated…. Read more »

• featured

## recycled : dessins

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In a couple of days I’ll be blogging for 4 years… and I’m in the process of resurrecting about 300 posts from a database-dump made in june. For example here’s my first post ever which is rather naive. This conversion program may last for a couple of weeks and I apologize for all unwanted pingbacks… Read more »

• featured

## the modular group and superpotentials (1)

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Here I will go over the last post at a more leisurely pace, focussing on a couple of far more trivial examples. Here’s the goal : we want to assign a quiver-superpotential to any subgroup of finite index of the modular group. So fix such a subgroup $\Gamma’$ of the modular group $\Gamma=PSL_2(\mathbb{Z})$… Read more »

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## Superpotentials and Calabi-Yaus

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Yesterday, Jan Stienstra gave a talk at theARTS entitled “Quivers, superpotentials and Dimer Models”. He started off by telling that the talk was based on a paper he put on the arXiv Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d’Enfants but that he was not going to say a thing about dessins but… Read more »

• featured

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