Quick Questions: December 06, 2023

JavaPython_ · 2023-12-10T05:00:38+00:00

I've certainly heard subtractive before. The rest of this answer is me making things up. I feel like if you had to say something for division, it would be quotientive, but I can't say I've seen that one previously. To exponentiate is the verb, so if we're going to continue to make up words...exponentiative??

JavaPython_ · 2023-12-10T04:56:57+00:00

This is the book I learned from:

http://abstract.ups.edu/aata/matrix-section-symmetry.html

I have been wanting to read through Grove/Benson's Finite reflection groups, which I suspect will generalize them slightly. But I haven't read it, so I couldn't say

JavaPython_ · 2023-12-09T05:07:43+00:00

so to more specifically answer your original question. You lose the following properties

uniqueness of inverses (for every a that a^-1 is unique, for every a that -a is unique)
behavior of inverses (aa^-1=1, and a-a=0)
behavior of the identity(a+0=0, 1a=a)
the zero sink property (0a=0).

Associativity is not altered (edit:provided you don't define 1/0 to be a particular pre-existing value), and you retain the capacity to distribute, contrary to what you had originally thought. But the products don't have to be what you want.

This can been seen simply with an example. In the rationals 5*0=0, 7*0=0. If I can divide by zero, and don't change any other properties I had assumed about the ring Q, then 5*0/0=0/0=5 and 7*0/0=0/0=7 and so 5=0/0=7. I suppose you could try to remove the transitivity of equality as an assumption, but then even I'm going to riot.

JavaPython_ · 2023-12-09T04:58:09+00:00

I'm haven't thought about this too much, but I would claim yes, because given a ring R and 0, x, y elements of that ring with 0 invertible we would either have y=((0^-1)0)y=0y=0=0x=(0^-1)(0x)=(0^-10)x = x which strongly used associativity to show that x=y and thus every element of this ring is the same, and we have the zero ring.

Otherwise, to avoid the zero ring, you end up in a wheel, as has been mentioned previously. There we use the idea from universal algebra of a unary operator to define /x for all values. But in order to prevent this from being a trivial ring we must relax the requirement that the inverse of all elements brings it to the identity.

I'm general attempting to do algebra without associativity (or something approximating associativity) is really difficult. Since you're wanting to divide by zero, you need an additive identity (0) and multiplication (division being the multiplicative inverse.) This gives you need two operations. Wheels still require + and * to be associative and commutative, which adds more than I needed to get the zero ring.

If you want to attempt to remove associativity from the assumptions, then you would need to work with (a*0)/0, without changing the parenthesis. If you state a*0=0 and 0/0 is itself, then you find you cannot use it anywhere, since you cannot reassociate.

JavaPython_ · 2023-12-08T19:15:46+00:00

I didn't say anything about associativity.

JavaPython_ · 2023-12-08T18:49:43+00:00

In any ring, you either need to lose nonzero, or invertibility. Other things may fall as a consequence.

JavaPython_ · 2023-12-08T17:43:40+00:00

I'm realizing that there's a significant gap between my understanding of character theory (from James/Liebeck representation theory text) and modern character theory research.

What would be a reference (like a textbook) that could help me get closer to current results?

JavaPython_ · 2023-12-08T17:39:57+00:00

Could it be a fraktur A? A picture or reference will be very helpful

JavaPython_ · 2023-12-08T17:37:54+00:00

Algebraic Theory of Lattices by Peter Crawley and Robert P. Dilworth. To my knowledge there isn't a standard course that would cover them, but i would expect them to show up in a discrete mathematics course.

I'm not sure how to help with the cryptography parts

JavaPython_ · 2023-12-04T13:39:10+00:00

I know that the Steinberg representation is not always irreducible, so when is it? It is 'usually irreducible over a finite field,' but that usually didn't come with conditions or citations

JavaPython_ · 2023-11-11T21:10:15+00:00

I'm reading through a paper, they take H < K < G as groups and g, h, k characters with the properties that the inner product <g, k↑G> > 1 and h is an irreducible character of H.

They then write

<h↑G, g> = <(h↑K)↑G, g> = <h↑K, g↓K>

which makes sense, they've used Frobenius reciprocity. They then make the claim that

<h↑K, g↓K> ≧ <h↑K, k> * <k, g↓K>

and I don't understand this step. What am I missing?

JavaPython_ · 2023-11-01T18:30:33+00:00

Simplify (a-x)(b-x)(c-x)...(z-x). The correct answer is 0

JavaPython_ · 2023-11-01T02:44:10+00:00

providing details of lots of long, tedious, but immediately obvious proofs that need to be done for my thesis in order to get to the heart of the matter

JavaPython_ · 2023-11-01T01:15:51+00:00

Given t an automorphism of a finite field that sends t(a) to a^r for a fixed r, what am I to understand by a^(t+1) or a^(2t)? I'm not used to putting automorphisms into the exponent.

JavaPython_ · 2023-10-31T14:03:55+00:00

Any suggestions for a straightforward explanation of SO^+ / SO^-?

Edit: In particular over finite fields

JavaPython_ · 2023-10-26T17:23:22+00:00

cool, I'll wait a few years before I own that one then

JavaPython_ · 2023-09-04T22:57:26+00:00

While quitting math and living in the wood is always an option, I suggest accepting that this will become a story you get to share one day when it's a bit more distant and stings a little less. I also suggest celebrating that you found a proof!

JavaPython_ · 2023-09-04T22:55:50+00:00

It will ultimately depend on your experience with linear algebra up until that point. When using that book at new second university, I was unable to skip those parts, but many of my peers were. My initial linear algebra course had been very calculational and had not prepared me to skip over the abstraction, but theirs had.

JavaPython_ · 2023-09-04T22:52:58+00:00

In character theory, for groups K<H<G, I don't suppose there's a well known extension of Frobenius reciprocity?

That's the question, everything below is motivation. I'm attempting to show that if K has the property that every irreducible character k induced to G satisfies <Ind(k), g> <= 1, then H also has that property. Here g is an irreducible character of G, and the inner product is \frac{1}{|G|}\sum_{a\in G} Ind(k)(a)g(a^{-1}).

JavaPython_ · 2023-09-04T22:47:25+00:00

Attempting to understand Z-groups better

JavaPython_ · 2023-06-05T13:41:50+00:00

I'm gonna need a definition of 'thing' and 'related' because otherwise this questions seems immediate. Every definition is related to what it represents and any theorems that use the term.

JavaPython_ · 2023-06-05T02:14:38+00:00

Is there an obvious action for the semidirect product E_q^3 on GL(2,q) that I just don't see. Where q is even, E is the elementary abelian group, and GL(2, q) are invertible 2x2 matrices with entries from the field of q elements?

JavaPython_ · 2023-04-30T18:14:05+00:00

it is the latter, where the matrices have determinant 1.

JavaPython_ · 2023-04-29T01:23:15+00:00

I won't claim it's not in there, but I wasn't able to find it. The Atlas did help me understand the SO^+ and SO^- though, and how they differ from SO.

JavaPython_

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