1. Supercharacters of Algebra Groups Benjamin Otto February 13, 2009

3. 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

4. Group Theory A group is a number system that encodes symmetry. It is a set with multiplication and inverses.

5. 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.

6. 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.

7. Algebra Groups

8. There is no general description of the characters.

9. Operations in an algebra group

10. Actions left right conjugate

11. Actions left right conjugate

12. Kirillov Functions

13. 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

14. Supercharacters

15. Supercharacters vs Kirillov Functions Supercharacters + Mutually orthogonal - May not span class functions + Partition irreducible characters + Are characters Kirillov Functions + Orthonormal basis for class functions - May not be class functions

16. Elementary Properties

17. Superdegrees and Superclass Sizes

19. 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.

20. Ln

22. An Argument Examine this

23. The Argument Continued

24. The Arguments Conclusion In other words, no polynomial (including Ln) can improve the supercharacters. Hence,

25. Thank You Slides available at www.math.wisc.edu/~otto