Supercharacters of Algebra Groups

Benjamin Otto

February 13, 2009

Overview

• Characters are important tools for studying groups. There is no general description for the characters of algebra groups

• Supercharacters and Kirillov functions are two suggested stand-ins

• Some results

• A quick proof

Group Theory

• A group is a number system that encodes symmetry.

• It is a set with multiplication and inverses.

• The dihedral group of order 8 is the collection of actions that leave a square fixed.

• There are 4 rotations and 4 flips. Any can be undone, and combining any two results in one of the original actions.

Character Theory

• Character theory is a powerful tool for studying groups.

• A character is a certain kind of map from a group to the complex numbers

• Knowing certain important characters allows one to recover the size of the group, the normal subgroups, the number of conjugacy classes, and more.

Algebra Groups

• There is no general description of the characters.

Operations in an algebra group

Actions

left

right

conjugate

Actions

left

right

conjugate

Kirillov Functions

The Intuition Behind Kirillov Functions

functions from a group to a field

functions from a group to the complex numbers

functions from the group to the complex numbers

orthonormal basis for space of class functions

orthogonal basis for space of class functions

Supercharacters

Supercharactersvs

Kirillov FunctionsSupercharacters

+ Mutually orthogonal

- May not span class functions

+ Partition irreducible characters

+ Are characters

Kirillov Functions

+ Orthonormal basis for class functions

- May not be class functions

Elementary Properties

Superdegrees and Superclass Sizes

Superdegrees and Superclass Sizes

Interplay

• Every irreducible constituent of a Kirillov function is also a constituent of the supercharacter arising from the same functional.

• Two Kirillov functions that share a linear constituent must arise from functionals in the same two-sided orbit.

Ln

Ln

An Argument

Examine this

The Argument Continued

The Argument’s Conclusion

In other words, no polynomial (including Ln) can improve the supercharacters.

Hence,

Thank You

Slides available at www.math.wisc.edu/~otto