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I am currently working on something related to the character theory of the group of unipotent upper triangular matrices with elements in a finite field. I have seen in many papers on the topic the statement that determining the irreducible characters and conjugacy classes of these groups is a "wild" problem, but never with a proof or any additional information as to what this might mean. I have asked the professor that I am working with, and he didn't know the precise statement of this property either, only that it meant that it was impossible to solve (in some sense).

Seeing as how I have been unable to find any information about this on the internet, I am wondering if anyone could either explain it to me, or point me to a useful source. Thanks!

Here is an example paper that shows what I am talking about: http://www-stat.stanford.edu/~cgates/PERSI/papers/PatternGroups_1.pdf

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2 Answers 2

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The precise definition is discussed at this MO question. A classification problem is said to be wild if it contains the classification of representations of $F_2$ (equivalently, the classification of a pair of linear operators up to simultaneous conjugation) as a subproblem.

One reason for this to be a reasonable definition: there is a remarkable theorem of Drozd which asserts that the representation theory of a finite-dimensional algebra is either wild or, for fixed dimension $d$, the indecomposable $d$-dimensional representations decompose into a finite number of $1$-parameter families with finitely many exceptions. And as Mariano says in the linked question,

One of the reasons that make this theorem so amazing is that one can show that if $A$ is wild then $\text{mod}_A$ contains copies of the module categories of all finite dimensional algebras; in other words, wild algebras are really wild...

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This is exactly what I was looking for, thank you for your exposition as well. I find it amusing that I originally thought the same thing as the asker in the MO question, that the word wild was simply a colorful name for a problem the author found particularly exciting or difficult. Of course, when I saw it used by different authors in multiple papers, I began to suspect otherwise... –  Daniel Bragg Jun 5 '11 at 0:05

In linear algebra, a "wild" problem is one that contains the problem of classifying pairs of (noncommuting) matrices up to simultaneous similarity. Problems of this type are considered to be virtually hopeless. See this arXiv paper for more information.

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