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Supercharacters of Algebra Groups

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

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Supercharacters of Algebra Groups

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

  2. 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

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

  4. 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.

  5. 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.

  6. Algebra Groups

  7. There is no general description of the characters.

  8. Operations in an algebra group

  9. Actions left right conjugate

  10. Actions left right conjugate

  11. Kirillov Functions

  12. functions from a group to a field functions from a group to the complex numbers functions from the group to the complex numbers orthogonal basis for space of class functions orthonormal basis for space of class functions The Intuition Behind Kirillov Functions

  13. Supercharacters

  14. 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 SupercharactersvsKirillov Functions

  15. Elementary Properties

  16. Superdegrees and Superclass Sizes

  17. Superdegrees and Superclass Sizes

  18. 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.

  19. Ln

  20. Ln

  21. Examine this An Argument

  22. The Argument Continued

  23. The Argument’s Conclusion Hence, In other words, no polynomial (including Ln) can improve the supercharacters.

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

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