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Posts Tagged ‘group actions’

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let G be a finite p-group acting on a finite set X. Let X^G denote the subset of X consisting of those elements fixed by G. Then |X^G| \equiv |X| \bmod p; in particular, if p \nmid |X| then G has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

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A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

  • replace the study of a group G with the study of its category G\text{-Rep} of linear representations,
  • replace the study of a ring R with the study of its category R\text{-Mod} of R-modules,
  • replace the study of a topological space X with the study of its category \text{Sh}(X) of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if G is a finite group, then as a category, the only data that can be recovered from G\text{-Rep} is the number of conjugacy classes of G, which is not much information about G. We get considerably more data if we also have the monoidal structure on G\text{-Rep}, which gives us the character table of G (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of G. It turns out that to recover G we need the symmetric monoidal structure on G\text{-Rep}; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

Theorem: A group G can be recovered from its category G\text{-Set} of G-sets.

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Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.

There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.

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My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.

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One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation by hand, but it is conceptually cleaner to use the symmetries of the situation and representation theory. Deducing these results using representation theory has the added benefit that it identifies which parts of the situation depend only on symmetry and which parts depend on the particular form of the Hamiltonian. This is nicely explained in Singer’s Linearity, symmetry, and prediction in the hydrogen atom.

For awhile now I’ve been interested in finding a toy model to study the basic structure of the arguments involved, as well as more generally to get a hang for quantum mechanics, while avoiding some of the mathematical difficulties. Today I’d like to describe one such model involving finite graphs, which replaces the infinite-dimensional Hilbert spaces and Lie groups occurring in the analysis of the hydrogen atom with finite-dimensional Hilbert spaces and finite groups. This model will, among other things, allow us to think of representations of finite groups as particles moving around on graphs.

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Nationals 2010

MIT is hosting the United States Rubik’s Cube Championships this summer, August 6-8. All ages welcome! Normally I wouldn’t post about such things, but

  • I happen to be a member of the Rubik’s Cube club here, and
  • Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and there’s a lot to learn from the cube; for example, you quickly understand that groups are not in general commutative. The Rubik’s cube itself is also a good example of a torsor.

Actually, just so this post has some mathematical content, there’s something about the Rubik’s cube group that is probably very simple to explain, but which I don’t completely understand. It’s a common feature of Rubik’s cube algorithms that they need to switch around some parts of the cube without disturbing others; in other words, the corresponding permutation needs to have a lot of fixed points. This seems to be done by writing down a lot of commutators, but I’m not familiar with any statements in group theory of the form “commutators tend to have fixed points.” Can anyone explain this?

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The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even).

Lemma 1: Fix a field F. The group of fractional linear transformations PGL_2(F) acts triple transitively on \mathbb{P}^1(F) and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on \mathbb{P}^1(\mathbb{C}) preserving the upper half plane \mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \} is PSL_2(\mathbb{R}).

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).

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