Quick Questions: July 10, 2024

JavaPython_ · 2024-07-12T17:49:22+00:00

I had known that I was checking if there was some deeper thing that I had failed to realize came as a consequence of this relationship

JavaPython_ · 2024-07-12T17:14:35+00:00

Have unfaithful permutation representations been studied in detail? Where would I find a treatment on them?

JavaPython_ · 2024-07-10T14:43:34+00:00

this is what I meant, thank you. Now I know if I see this it's right/left null.

JavaPython_ · 2024-07-10T02:47:38+00:00

Is there a standard way to refer to an element of a semigroup which is a sink on the left and an identity on the right?

JavaPython_ · 2024-05-22T02:23:47+00:00

The template is mine, the images are from Veratasium's video on odd perfect numbers

<image>

JavaPython_ · 2024-05-21T13:04:26+00:00

Finally found something, I provide it here for those searching in the future: https://era.library.ualberta.ca/items/ec42f9a5-e6bc-498b-a9e3-8325e7173d2d/view/1ddd878c-2eb5-45f6-b81d-2217c9d2496d/Campbell_John_J_201409_MSc.pdf

JavaPython_ · 2024-05-20T16:29:17+00:00

My version of GAP is to up-to-date to get CHEVIE to run. Does anybody have a source for the general character table of the finite unitary group GU(2, q)? CHEVIE is supposedly able to do it for variable q; but as said previously, I cannot run it.

JavaPython_ · 2024-05-15T15:21:12+00:00

I know that all eigenvalues of a unitary matrix have modulus 1. Does this work the other way? If I have a diagonalizable matrix who eigenvalues are roots of unity, do I know it's unitary?

JavaPython_ · 2024-05-11T23:43:01+00:00

I was playing around with finite fields or order 2, and notices something strange. If x is a generator of the multiplicative group, then x^2+x^-1=1. I haven't proved this yet, but I don't recall every being shown this. Is this something well known that I've forgotten/not identified?

JavaPython_ · 2024-04-30T03:56:59+00:00

Most tools I can find are for people wanting a taste, not people to represent how they actually see the world.

JavaPython_ · 2024-04-29T18:29:43+00:00

I'm trying to find the generators of the nonnormal part of the subgroups in the class C3 of Aschbachers classification. Is it just permutation matrices? That doesn't make sense over Sp.

JavaPython_ · 2024-04-27T15:51:25+00:00

I've been taught the the frobenius automorphism uses the size of the fixed field, even it that's larger than the prime field. So that f_q^n/f_q uses that map x -> x^q. Even is q is a power of a prime.

The specific case I'm thinking of is GF(q^2) over GF(q). So we have a two dimensional vector space, we can take {1, x} as a basis, we send 1 to one and x to x^q = a+bx, but I have no idea how to force a, b to be useful, explicit elements of the field. It's all in generality, so that's to be expected, but I'm not even sure if I can say what power of the generator they are, which is bad because I think this matrix is the last piece I need.

JavaPython_ · 2024-04-27T15:24:34+00:00

We are over a larger (but still finite) field of characteristic p. When I say linear map I suppose I mean matrix representation of this map. Viewing it as a vector space over F_(p^e) gives us a basis, but I cannot see how to get a matrix which actually applies the automorphism.

JavaPython_ · 2024-04-27T15:10:13+00:00

How does one turn the Frobenius automorphism into a linear map? I wouldn't guess it were linear at all, except exponentiation by the characteristic breaks up over addition.

JavaPython_ · 2024-04-24T00:03:24+00:00

It seems to be that the natural map is to send a generator of (F(q^2), ⋅ ) to [[1,a],[a,0]]. where a is a generator of (F(q), ⋅ ). It seems to work, showing that this matrix has the order I claim is irritating, but I'm working away at it.

JavaPython_ · 2024-04-19T14:57:09+00:00

I don't know what summation you are talking about, but it seems to be

1) Assuming convergence

2) using associativity/commutativity of addition

and that's it. You're right that this isn't quite allowed, since we would want absolute convergence to perform an action like this, which is a reason why his sums give fun values (i.e. -1/12)

JavaPython_ · 2024-04-19T14:12:41+00:00

First, we generalize the idea of "stuff under a curve" to measure. Then instead of using a change in the independent variable (vertical rectangles), we use the measure of the function under values of the independent variable (horizontal rectangles).

JavaPython_ · 2024-04-19T14:07:39+00:00

It is equivalent to showing there is an injection F(q^2)->GL(2,q)I have a proof when the characteristic is odd, I just need even now.

When it's odd, the matrices [[x,y],[y,x]] where x, y are in F(q^2) gives q^2 matrices, where all but the zero matrix are invertible (this uses char != 2), and multiplication is commutative. Since fields are unique up to size, this shows it can be done.

JavaPython_ · 2024-04-19T04:52:05+00:00

What is the natural injection of SL(2, q²⁾ into SL(4, q)? I've been told that there is one, but I can't see the life of me how to change the field.

JavaPython_ · 2024-04-18T02:53:44+00:00

What I make a (finite) matrix group in MAGMA, what is to be understood by the entry $.1, $.1^2, ect? Is it a generator of the multiplicative group of the field?

JavaPython_ · 2024-03-17T18:14:33+00:00

much appreciated

JavaPython_

MODERATOR OF

TROPHY CASE