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Supercharacters of Algebra Groups Benjamin Otto February 13, 2009

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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

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

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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.

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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.

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Algebra Groups

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There is no general description of the characters.

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Operations in an algebra group

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Actions left right conjugate

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Actions left right conjugate

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Kirillov Functions

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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

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Supercharacters

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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

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Elementary Properties

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Superdegrees and Superclass Sizes

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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.

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Ln

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An Argument Examine this

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The Argument Continued

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

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

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