6,314 reputation
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bio website boolesrings.org/nickgill
location San Jose, Costa Rica
age 38
visits member for 5 years, 5 months
seen 2 days ago

I'm a visiting professor at the Universidad de Costa Rica.


Mar
18
comment Maximal abelian subgroup of general linear groups
This might be helpful: mathnet.ru/php/…
Mar
11
comment Experimenting with the spider relator
This paper - ftp.mathe2.uni-bayreuth.de/axel/papers/ivanov:the_monster.ps.gz - seems to assert that ${Y222}\cong 3^5. O_5(3)$. By the way, your questions would be much easier to read if you used LaTeX!
Mar
6
comment Matrices congruent to each other via a permutation
... Actually, it's possible that you can then just consider super-diagonal entries and order these to be strictly increasing. You'd need to prove that this is (a) possible, and (b) unique (neither of which I'm sure about). Supposing this works, then iterating (considering super-super-diagonals etc) would yield a canonical form.
Mar
6
comment Matrices congruent to each other via a permutation
A first thought: Consider an integer matrix whose diagonal entries are all distinct. You can use permutation matrices to order these entries so that they are strictly increasing as you go down the diagonal. There will only be one such matrix in the equivalence class that you describe so you could consider this a canonical form for this type of matrix. (Note that this works fine for real matrices, not just integer matrices.) I'm not sure how to extend this to matrices with repeated diagonal entries...
Feb
26
comment How do small central extensions drop the dimension of a faithful representation?
Ben, you might be interested in this paper - arxiv.org/pdf/1408.1649.pdf - by Britnell, Saunders and Skyner. It looks at a similar phenomenon for $p$-groups - when the quotient of a $p$-group has minimal permutation representation larger than the original.
Feb
12
revised Can any finite distributive weighted lattice be realized by inclusion of groups?
typos fixed
Feb
11
answered Can any finite distributive weighted lattice be realized by inclusion of groups?
Feb
11
comment Can any finite distributive weighted lattice be realized by inclusion of groups?
Well that sounds promising. My thinking is that the three element lattice is equivalent to having an imprimitive permutation group on a set of size $mn$ with a unique system of imprimitivity. There are no doubt many of these, but the wreath product is certainly one such. And I can't see a reason why iterating wouldn't work, so that probably sorts chains out. I'll try and write a proper answer later today when I have time.
Feb
10
reviewed Approve Unable to find any information regarding this fact (Frey, elliptic curves)
Feb
10
comment Can any finite distributive weighted lattice be realized by inclusion of groups?
I could be wrong, but if you let $G=S_m \wr S_n$ and consider the natural action on $mn$ points, then the point-stabilizer has index $mn$ and there is a unique subgroup in between of index $n$. Sorry, no time to check this through but if it works it deals with $(mn,n,1)$.... And iterated wreath products will probably deal with chain lattices...
Feb
10
comment Can any finite distributive weighted lattice be realized by inclusion of groups?
Have you tried $(36,6,1)$? That seems like the first tricky $(n^2,n,1)$ case...
Feb
6
comment Simple Hurwitz Groups of order less than 10^7
Thomas, note that $B_2(q)$ and $C_2(q)$ are isomorphic.
Feb
6
comment Simple Hurwitz Groups of order less than 10^7
Thomas, I'm not sure why you are telling me about $M_{22}$ - did you mean to reply to Noam? Also, note that the results in Derek's question are for groups with representations of projective degree at most 6 (or, partially, 7) over any field whatsoever, so you can't use the ATLAS directly to check this - it only gives complex reps. Indeed, using the link given by Noam, one can see that $M_{22}$ has a projective rep in characteristic $2$ of dimension $6$.
Feb
4
comment Simple Hurwitz Groups of order less than 10^7
On another front: is it true that all groups of order at most $10^7$ (or $10^8$ or $10^9$) have projective representations of dimension at most $7$? If there are some that don't, these would not be covered by your answer, right?
Feb
4
comment Simple Hurwitz Groups of order less than 10^7
A query: according to the ATLAS, $J_1$ is a subgroup of $G_2(11)$ which in turn is a subgroup of $O_7(11)$ (I think? Not totally sure about that.) But Thomas says above that $J_1$ is a Hurwitz group. So shouldn't $J_1$ have cropped up in your list? Or is its omission covered by the caveat about dimension $7$?
Feb
4
comment Simple Hurwitz Groups of order less than 10^7
Oh, silly me! In which case you should check to see if this group is covered by results in the surveys.
Feb
4
comment Simple Hurwitz Groups of order less than 10^7
Thomas, you should check more carefully those Steinberg groups: When you write 2A2(3,9) I don't know if you mean ${^2A_2(9)}$ or ${^2A_3(9)}$ (I presume you mean one of these). The first is not Hurwitz by the surveys, the latter is not Hurwitz because its order is not divisible by $7$.
Feb
4
comment Simple Hurwitz Groups of order less than 10^7
Your notation for the $^2A_2$'s is a bit puzzling - I'm not sure what you mean here. However in general $^2A_2(q)$ are 3-dimensional unitaries and so are not Hurwitz. In any case, for a summary of the state of play (in 2000) you should look at the paper by Martino, Tamburini and Zalesski here: mathematik.uni-bielefeld.de/LAG/man/021.ps.gz
Dec
16
awarded  Nice Answer
Dec
15
reviewed Approve textbooks on modern algebraic geometry for 21st-century starters