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.

## Archive for the ‘math.RT’ Category

## 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 »

## 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.

## A transcript of my qualifying exam

Posted in math.AT, math.RT on December 13, 2013| 11 Comments »

I passed my qualifying exam last Friday. Here is a copy of the syllabus and a transcript.

Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of their quals, namely Christopher Wong and Eric Peterson. It would be nice if more people did this.

## Four flavors of Schur-Weyl duality

Posted in math.RT, tagged MaBloWriMo, symmetric functions on November 13, 2012| 21 Comments »

If is a finite-dimensional complex vector space, then the symmetric group naturally acts on the tensor power by permuting the factors. This action of commutes with the action of , so all permutations are morphisms of -representations. This defines a morphism , and a natural question to ask is whether this map is surjective.

Part of Schur-Weyl duality asserts that the answer is yes. The double commutant theorem plays an important role in the proof and also highlights an important corollary, namely that admits a canonical decomposition

where runs over partitions, are some irreducible representations of , and are the Specht modules, which describe all irreducible representations of . This gives a fundamental relationship between the representation theories of the general linear and symmetric groups; in particular, the assignment can be upgraded to a functor called a Schur functor, generalizing the construction of the exterior and symmetric products.

The proof below is more or less from Etingof’s notes on representation theory (Section 4.18). We will prove four versions of Schur-Weyl duality involving , and (in the special case that is a complex inner product space) .

## The double commutant theorem

Posted in math.RA, math.RT, tagged adjoint functors, Hecke algebras, MaBloWriMo on November 11, 2012| 4 Comments »

Let be an abelian group and be a collection of endomorphisms of . The **commutant** of is the set of all endomorphisms of commuting with every element of ; symbolically,

.

The commutant of is equal to the commutant of the subring of generated by the , so we may assume without loss of generality that is already such a subring. In that case, is just the ring of endomorphisms of as a left -module. The use of the term commutant instead can be thought of as emphasizing the role of and de-emphasizing the role of .

The assignment is a contravariant Galois connection on the lattice of subsets of , so the **double commutant** may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

## Noncommutative probability and group theory

Posted in math.GR, math.PR, math.RT, tagged Catalan numbers on September 24, 2012| Leave a Comment »

There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group . The kind which is covariantly functorial is some variation on the group algebra , which is the free -module on with multiplication inherited from the multiplication on . The kind which is contravariantly functorial is some variation on the algebra of functions with pointwise multiplication.

When and when is respectively either a discrete group or a compact (Hausdorff) group, both of these algebras can naturally be endowed with the structure of a random algebra. In the case of , the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of , while in the case of , the corresponding state is by definition integration with respect to normalized Haar measure on .

In general, some nontrivial analysis is necessary to show that the normalized Haar measure exists, but for compact groups equipped with a faithful finite-dimensional unitary representation it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on using representation theory. This construction will in some sense explain why the category of (finite-dimensional continuous unitary) representations of behaves like an inner product space (with being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on .

## The Jacobson radical

Posted in math.RA, math.RT, tagged adjoint functors on May 30, 2012| 4 Comments »

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a **semiprimitive** or Jacobson semisimple ring, one with the property that every element acts nontrivially in some simple (left) module .

Said another way, let the **Jacobson radical** of a ring consist of all elements of which act trivially on every simple module. By definition, this is an intersection of kernels of ring homomorphisms, hence a two-sided ideal. A ring is then semiprimitive if it has trivial Jacobson radical.

The goal of this post will be to discuss some basic properties of the Jacobson radical. I am again working mostly from Lam’s *A first course in noncommutative rings*.