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Posts Tagged ‘group actions’

Today we’ll resolve half the puzzle of why the cohomology group H^2(BG, k^{\times}) appears both when classifying projective representations of a group G over a field k and when classifying k-linear actions of G on the category \text{Mod}(k) of k-vector spaces by describing a functor from the former to the latter.

(There is a second half that goes in the other direction.)

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Continuing yesterday’s story about relative positions, let G be a finite group and let X and Y be finite G-sets. Yesterday we showed that G-orbits on X \times Y can be thought of as “atomic relative positions” of “X-figures” and “Y-figures” in some geometry with symmetry group G, and further that if X \cong G/H and Y \cong G/K are transitive G-sets then these can be identified with double cosets H \backslash G / K.

Representation theory provides another interpretation of G-orbits on X \times Y as follows. First, if \mathbb{C}[X] is any permutation representation, then the G-fixed points \mathbb{C}[X]^G have a natural basis given by summing over G-orbits. (This is a mild categorification of Burnside’s lemma.) Next, consider the representations \mathbb{C}[X], \mathbb{C}[Y]. Because \mathbb{C}[X] is self-dual, we have

\displaystyle \text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y]) \cong (\mathbb{C}[X] \otimes \mathbb{C}[Y])^G \cong \mathbb{C}[X \times Y]^G

and hence \text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y]) has a natural basis given by summing over G-orbits of the action on X \times Y.

Definition: The G-morphism \mathbb{C}[X] \to \mathbb{C}[Y] associated to a G-orbit of X \times Y via the above isomorphisms is the Hecke operator associated to the G-orbit (relative position, double coset).

Below the fold we’ll write down some details about how this works and see how we can use the idea that G-morphisms between permutations have a basis given by Hecke operators to work out, quickly and cleanly, how some permutation representations decompose into irreducibles. At the end we’ll state another puzzle.

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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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