Supercharacters of Algebra Groups Benjamin Otto February 13, 2009.
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Transcript of Supercharacters of Algebra Groups Benjamin Otto February 13, 2009.
![Page 1: Supercharacters of Algebra Groups Benjamin Otto February 13, 2009.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5514144d550346d8488b5220/html5/thumbnails/1.jpg)
Supercharacters of Algebra Groups
Benjamin Otto
February 13, 2009
![Page 2: Supercharacters of Algebra Groups Benjamin Otto February 13, 2009.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5514144d550346d8488b5220/html5/thumbnails/2.jpg)
![Page 3: Supercharacters of Algebra Groups Benjamin Otto February 13, 2009.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5514144d550346d8488b5220/html5/thumbnails/3.jpg)
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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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
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Elementary Properties
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Superdegrees and Superclass Sizes
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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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Ln
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An Argument
Examine this
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The Argument Continued
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The Argument’s 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