Post on 26-Mar-2015
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