## Tuesday, September 9, 2008

### Matrix Automorphism Groups

Suppose you are given an arbitrary matrix, and you wish to know the set of permutations of the columns, such that the (unordered) set of rows remains the same. This is often called the automorphism group of the matrix, and can be computed as follows, using the ability to compute the automorphism groups of hypergraphs.

First, suppose there are N symbols in your matrix. Construct N matrices of the same size as your original matrix, one for each symbol in the original matrix. Put a 1 in any position which contains the corresponding symbol in the original, and you get a collection of (0,1)-matrices. Think of the rows of these N matrices as the blocks of N hypergraphs. Each permutation in the matrix automorphism group must also be an automorphism of each of these hypergraphs. Note that you can freely delete any zero rows in these matrices without harm.

Finally, if you have refinement methods for computing automorphism groups of hypergraphs, you can easily construct a refinement method for the matrix automorphism group, by repeatedly refining against these hypergraphs until the partition stabilizes. The other two functions needed for a specific implementation are very easy to do for matrix automorphism, and there you have it.

Hypergraphs are due to appear sometime tomorrow (this is the plan anyway) in Sage 3.1.2, and matrix automorphism groups are currently coded, based on this. They should appear in Sage 3.1.3. Thanks once again to Google for funding this work.