I’m actually getting a lot of research done these days to post much, but in case you guys are still dropping by, happy holidays. =)
Some personal good news: we finally got that epic (in the “epically long” sense and not the “epically awesome” sense, but I’m still happy it is done) paper about counting invertible matrices over finite fields up on the ArXiv: http://arxiv.org/abs/1011.4539 . However, despite the season, I’m not in the most cheery mood about the paper, because something I wanted to qualitatively understand remains at large:
Take some finite field, and some even . Consider the number of invertible skew-symmetric matrices and the number of invertible symmetric matrices. They equal. Furthermore, they both equal the number of invertible symmetric matrices with zeroes on the diagonal.
We proved this (it is Propostion 3.6 as of the latest version), but I don’t think we quite understand why (for either of the equivalences). I didn’t feel like the proof we used, a “twisty” bijection, showed us anything (we mention a “high-brow” way, but I don’t know if it is too helpful in its current form). I guess during the editing process I just wanted to get the damn thing finished, but now it is bugging me like all sorts of other things tend to do near holiday midnights. If anyone has ideas, please let me know. Extra points for using the words “Lie Superalgebras” because then I probably owe Steven Sam a beer.