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 ‘finite fields’

## The representation theory of the additive group scheme

Posted in math.AG, math.RT, tagged finite fields, group actions on November 26, 2017| 5 Comments »

## 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.

## Hecke algebras and the Kazhdan-Lusztig polynomials

Posted in math.CO, math.RT, tagged Coxeter groups, finite fields, Hecke algebras, q-analogues on July 12, 2010| 2 Comments »

The **Hecke algebra** attached to a Coxeter system is a deformation of the group algebra of defined as follows. Take the free -module with basis , and impose the multiplicative relations

if , and

otherwise. (For now, ignore the square root of .) Humphreys proves that these relations describe a unique associative algebra structure on with as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)

The motivation behind this definition is a somewhat long story. When is the Weyl group of an algebraic group with Borel subgroup , the above relations describe the algebra of functions on which are bi-invariant with respect to the left and right actions of under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group , and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing and with and gives the Hecke operators in the theory of modular forms.

## The arithmetic plane

Posted in math.NT, tagged finite fields on January 4, 2010| 1 Comment »

If you haven’t seen them already, you might want to read John Baez’s week205 and Lieven le Bruyn’s series of posts on the subject of spectra. I especially recommend that you take a look at the picture of to which Lieven le Bruyn links before reading this post. John Baez’s introduction to week205 would probably also have served as a great introduction to this series before I started it:

There’s a widespread impression that number theory is about

numbers, but I’d like to correct this, or at least supplement it. A large part of number theory – and by the far the coolest part, in my opinion – is about a strange sort ofgeometry. I don’t understand it very well, but that won’t prevent me from taking a crack at trying to explain it….

Before we talk about localization again, we need some examples of rings to localize. Recall that our proof of the description of also gives us a description of :

**Theorem:** consists of the ideals where is irreducible, and the maximal ideals where is prime and is irreducible in .

The upshot is that we can think of the set of primes of a ring of integers , where is a monic irreducible polynomial with integer coefficients, as an “algebraic curve” living in the “plane” , which is exactly what we’ll be doing today. (When isn’t monic, unfortunate things happen which we’ll discuss later.) We’ll then cover the case of actual algebraic curves next.

## The cyclotomic identity and Lyndon words

Posted in math.CO, math.NT, tagged finite fields, MaBloWriMo, zeta functions on November 3, 2009| 5 Comments »

In number theory there is a certain philosophy that is a good toy model for the integers . The two rings share an important property: they are basically the canonical examples of Euclidean domains, hence PIDs, hence UFDs. However, many number-theoretic questions involving prime factorization over are much easier than their corresponding questions over . One way to see this is to look at their zeta functions.

The usual zeta function reflects the structure of prime factorization through its **Euler product**

where the product runs over all primes; this is essentially equivalent to unique factorization. Since we know that monic polynomials over also enjoy unique factorization, it’s natural to ask whether there’s a sum over monic polynomials that would give a similar Euler product.

In fact, there is such a product, and investigating it leads naturally to a seemingly unrelated subject: the combinatorics of words.

## Young diagrams, q-analogues, and one of my favorite proofs

Posted in math.CO, tagged finite fields, q-analogues, Young tableaux on June 11, 2009| 10 Comments »

I’ve decided to start blogging a little more about the algebraic combinatorics I’ve learned over the past year. In particular, I’d like to present one of my favorite proofs from Stanley’s Enumerative Combinatorics I.

The theory of Young tableaux is a great example of the richness of modern mathematics: although they can be defined in an elementary combinatorial fashion, interest in the theory is primarily driven (as I understand it) by their applications to representation theory and other fields of mathematics. The standard application is in describing the irreducible representations of the symmetric group in characteristic zero. I’ll be describing a more elementary aspect of the theory: the relationship between Young diagrams and -binomial coefficients.

Young diagrams can be defined as a visual representations of partitions. A partition of is a weakly decreasing sequence such that . Partitions uniquely describe cycle decompositions, hence conjugacy classes, of the symmetric group, which is a hint at the connection to representation theory. A Young diagram of the partition consists of rows of boxes where the row has boxes. Let denote the poset of Young diagrams that fit into an box, ordered by inclusion: equivalently, one can define the full Young lattice and define to be the elements less than or equal to the partition. (One can then define a standard Young tableau to be a chain in the Young lattice, but we will not need this notion.) One can easily check that the following is true.

**Proposition:** .

The rest of this post explains the -analogue of this result.

## The magic of the Frobenius map II

Posted in math.CO, math.NT, tagged finite fields, walks on graphs on June 9, 2009| 4 Comments »

Once upon a time I discussed some interesting uses of the Frobenius map to solve some Putnam-style problems. Unfortunately, I wrote that post before becoming really interested in combinatorics, so I neglected to develop that particular side of the story, which I’d like to do now.

The beginning of this story is the folklore combinatorial proof of Fermat’s little theorem: for any positive integer and prime , we have

.

The proof is simple but powerful. Consider the set of possible ways to color beads with colors. The cyclic group acts on this set in the obvious way. (One says that the beads are “on a necklace.”) The great thing about actions of a prime cyclic group is that elements arrange themselves into two kinds of orbits: fixed points and orbits of size . (This is just a special case of the orbit-stabilizer theorem.) The total number of colorings is , but if we only want to work it suffices to count the number of fixed points under the action of . These are precisely the colorings using only one color, of which there are , and the result follows.

The standard generalization of this result to finite fields is as follows: and is generated by the Frobenius map . *Does this result also have a combinatorial proof?*