Posts Tagged: Brauer

  • 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 »

  • web

    the future of this blog (2)

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    is decided : I’ll keep maintaining this URL until new-year’s eve. At that time I’ll be blogging here for 5 years… The few encounters I’ve had with architects, taught me this basic lesson of life : the main function of several rooms in a house changes every 5 years (due to children and yourself getting… Read more »

  • 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 »

  • absolute

    Looking for F_un

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    There are only a handful of human activities where one goes to extraordinary lengths to keep a dream alive, in spite of overwhelming evidence : religion, theoretical physics, supporting the Belgian football team and … mathematics. In recent years several people spend a lot of energy looking for properties of an elusive object : 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

    Iguanodon series of simple groups

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    Bruce Westbury has a page on recent work on series of Lie groups including exceptional groups. Moreover, he did put his slides of a recent talk (probably at MPI) online. Probably, someone considered a similar problem for simple groups. Are there natural constructions leading to a series of finite simple groups including some sporadic groups… Read more »

  • featured

    The Mathieu groupoid (1)

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    Conway’s puzzle M(13) is a variation on the 15-puzzle played with the 13 points in the projective plane $\mathbb{P}^2(\mathbb{F}_3) $. The desired position is given on the left where all the counters are placed at at the points having that label (the point corresponding to the hole in the drawing has label 0). A typical… Read more »

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    neverendingbooks-geometry (2)

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    Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.

  • featured

    NeverEndingBooks-groups

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    Here a collection of pdf-files of NeverEndingBooks-posts on groups, in reverse chronological order.

  • stories

    down with determinants

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    The categorical cafe has a guest post by Tom Leinster Linear Algebra Done Right on the book with the same title by Sheldon Axler. I haven’t read the book but glanced through his online paper Down with determinants!. Here is ‘his’ proof of the fact that any n by n matrix A has at least… Read more »

  • stories

    THE rationality problem

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    This morning, Esther Beneish arxived the paper The center of the generic algebra of degree p that may contain the most significant advance in my favourite problem for over 15 years! In it she claims to prove that the center of the generic division algebra of degree p is stably rational for all prime values… Read more »

  • featured

    anabelian geometry

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    Last time we saw that a curve defined over $\overline{\mathbb{Q}} $ gives rise to a permutation representation of $PSL_2(\mathbb{Z}) $ or one of its subgroups $\Gamma_0(2) $ (of index 2) or $\Gamma(2) $ (of index 6). As the corresponding monodromy group is finite, this representation factors through a normal subgroup of finite index, so it… Read more »

  • featured

    permutation representations of monodromy groups

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    Today we will explain how curves defined over $\overline{\mathbb{Q}} $ determine permutation representations of the carthographic groups. We have seen that any smooth projective curve $C $ (a Riemann surface) defined over the algebraic closure $\overline{\mathbb{Q}} $ of the rationals, defines a _Belyi map_ $\xymatrix{C \ar[rr]^{\pi} & & \mathbb{P}^1} $ which is only ramified over… Read more »

  • web

    lulu neverendingbooks

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    Half a year ago, it all started with NeverEndingBooks in which I set out a rather modest goal : Why NeverEndingBooks ? We all complain about exaggerated prices of mathematical books from certain publishers, poor quality of editing and refereeing offered, as well as far too stringent book-contracts. Rather than lamenting about this, NeverEndingBooks gives… Read more »

  • stories

    Jacobian update

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    One way to increase the blogshare-value of this site might be to give readers more of what they want. In fact, there is an excellent guide for those who really want to increase traffic on their site called 26 Steps to 15k a Day. A somewhat sobering suggestion is rule S : “Think about what… Read more »

  • stories

    reading backlog

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    One of the things I like most about returning from a vacation is to have an enormous pile of fresh reading : a week's worth of newspapers, some regular mail and much more email (three quarters junk). Also before getting into bed after the ride I like to browse through the arXiv in search for… Read more »

  • featured

    hyper-resolutions

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    [Last time][1] we saw that for $A$ a smooth order with center $R$ the Brauer-Severi variety $X_A$ is a smooth variety and we have a projective morphism $X_A \rightarrow \mathbf{max}~R$ This situation is very similar to that of a desingularization $~X \rightarrow \mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$. The top variety $~X$ is a… Read more »

  • featured

    smooth Brauer-Severis

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    Around the same time Michel Van den Bergh introduced his Brauer-Severi schemes, [Claudio Procesi][1] (extending earlier work of [Bill Schelter][2]) introduced smooth orders as those orders $A$ in a central simple algebra $\Sigma$ (of dimension $n^2$) such that their representation variety $\mathbf{trep}_n~A$ is a smooth variety. Many interesting orders are smooth : hereditary orders, trace… Read more »

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    Brauer-Severi varieties

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    ![][1] Classical Brauer-Severi varieties can be described either as twisted forms of projective space (Severi\’s way) or as varieties containing splitting information about central simple algebras (Brauer\’s way). If $K$ is a field with separable closure $\overline{K}$, the first approach asks for projective varieties $X$ defined over $K$ such that over the separable closure $X(\overline{K})… Read more »

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    Galois and the Brauer group

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    Last time we have seen that in order to classify all non-commutative $l$-points one needs to control the finite dimensional simple algebras having as their center a finite dimensional field-extension of $l$. We have seen that the equivalence classes of simple algebras with the same center $L$ form an Abelian group, the Brauer group. The… Read more »

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    Brauer’s forgotten group

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    Non-commutative geometry seems pretty trivial compared to commutative geometry : there are just two types of manifolds, points and curves. However, nobody knows how to start classifying these non-commutative curves. I do have a conjecture that any non-commutative curve can (up to non-commutative birationality) be constructed from hereditary orders over commutative curves by universal methods… Read more »