In this post we’ll describe the representation theory of the additive group scheme over a field . The answer turns out to depend dramatically on whether or not has characteristic zero.

## Posts Tagged ‘group actions’

## The representation theory of the additive group scheme

Posted in math.AG, math.RT, tagged finite fields, group actions on November 26, 2017| 2 Comments »

## Projective representations give categorical representations

Posted in math.CT, math.GR, tagged cohomology, group actions, MaBloWriMo on November 13, 2015| 1 Comment »

Today we’ll resolve half the puzzle of why the cohomology group appears both when classifying projective representations of a group over a field and when classifying -linear actions of on the category of -vector spaces by describing a functor from the former to the latter.

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

## Hecke operators are also relative positions

Posted in math.GR, math.RT, tagged group actions, Hecke algebras, MaBloWriMo on November 7, 2015| Leave a Comment »

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

Representation theory provides another interpretation of -orbits on as follows. First, if is any permutation representation, then the -fixed points have a natural basis given by summing over -orbits. (This is a mild categorification of Burnside’s lemma.) Next, consider the representations . Because is self-dual, we have

and hence has a natural basis given by summing over -orbits of the action on .

**Definition:** The -morphism associated to a -orbit of via the above isomorphisms is the **Hecke operator** associated to the -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 -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.

## The p-group fixed point theorem

Posted in math.CO, math.GR, math.NT, tagged finite fields, fixed point theorems, group actions, walks on graphs on July 9, 2013| 12 Comments »

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 be a finite -group acting on a finite set . Let denote the subset of consisting of those elements fixed by . Then ; in particular, if then has a fixed point.

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

## Connected objects and a reconstruction theorem

Posted in math.CT, math.GR, tagged group actions on April 1, 2013| 8 Comments »

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 with the study of its category of linear representations,
- replace the study of a ring with the study of its category of -modules,
- replace the study of a topological space with the study of its category 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 is a finite group, then as a category, the only data that can be recovered from is the number of conjugacy classes of , which is not much information about . We get considerably more data if we also have the monoidal structure on , which gives us the character table of (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of . It turns out that to recover we need the *symmetric* monoidal structure on ; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

**Theorem:** A group can be recovered from its category of -sets.

## Groupoid cardinality

Posted in math.AT, math.CT, tagged group actions, groupoids, MaBloWriMo on November 8, 2012| 9 Comments »

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.

## Groupoids

Posted in math.AT, math.CT, math.GR, tagged 2-categories, group actions, groupoids, MaBloWriMo on November 1, 2012| 9 Comments »

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.