Binary Determinantal Complexity

code, mathematics 2014-11-11
My good friend and colleague Christian Ikenmeyer and I wrote this cute preprint about polynomials and how they can be written as the determinant of a matrix with entries equal to zero, one and indeterminantes. Go ahead and read it if you know even just a little math, it's quite straightforward. The algorithm described in section 3 has been implemented and you can download the code from my website at the TU Berlin. Compilation instructions are in ptest.c, but you will need to get nauty to perform the entire computerized proof.

A determinant for reductive groups

mathematics 2014-05-17
For the group $G=\mathrm{GL}_n(\mathbb{K})$ of invertible matrices over any field, there is the well-known determinant character $\det:G\to\mathbb{K}^\times$ which, well, maps any matrix to its determinant. It has the property that \[ [G,G]=\mathrm{SL}_n(\mathbb{K})=\{ g\in G \mid \det(g)=1 \}, \] where $[G,G]$ denotes the derived subgroup of $G$. In more geometric terms, $[G,G]$ is the vanishing of the regular function $\det-1$ on $G$. Because I find it rather cute, I will show that you can find a similar function for all linear, reductive groups. (more…)
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