Mercio_and_Rudi - 2018-09-09

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8:38 AM

@mercio In case you intend to make any changes to the MS it would be good if you give me some time horizon. Otherwise I'll rework it a bit. But as mentioned it would be quite good to start some discussion s.t. we as soon as possible get to some consensual understanding.

2 hours later…

mercio

10:46 AM

@Rudi_Birnbaum so I will pretty much have to write in the supporting information section ?

are you sure about the "exactly 3" in your first sentence in 0.1 ?

(also I am still irky about your use of "contains", and you are butchering that subset symbol later)

for example in C4, E² = A+A+B+B has 4 pieces

and 4 > 3

it's still really really confusing to read

your first sentence in 0.2 doesn't read right at all

you might have many "germanicisms" in your writing ?

it actually doesn't have a verb that is not in a subordinate proposition

the word "any" should be banned because noone ever knows if it means "forall" or "there exists"

I don't think I am able to differentiate while reading between what is it that you are observing and that is supposedly new, and what is it that is common knowledge in chemistry

Rudi_Birnbaum

11:11 AM

@mercio No I was not sure. In case you detect any errors just go on correct it, its our paper.

@mercio also formulations feel free to do whatever you consider has to be done.

@mercio I think boundaries are floating here. In that resp. we just have to rely on my feeling.

@mercio yes that complete one in a way that it interfaces with the main text. but moreover you have to control the rest of the paper as well ... :-)

mercio

yes

1 hour later…

mercio

12:46 PM

@Rudi_Birnbaum i sent you a mail

Rudi_Birnbaum

@mercio I just read version 2. 1) The more we can put into the paper and the less in to the supp part the better. The level of abstraction and formality seems to be still ok to keep it in this form in the main part

2) on the "angular momentum" definition. This is something that is so clear to the physical chemist as anything. I think for many readers it would be the most "concrete" term apperaing in that section at all.

So in my feeling we should not treat it with "spitze Zeigerfinger" but rather as something other can firmly hold on ;-)

mercio

that is basically the complete opposite view of a mathematician reading this

Rudi_Birnbaum

1:01 PM

@mercio yes I suspected that. The question is what can we offer to the readers to show why \Gamma_l_z transforms like Alt^2(X).

it will be in the cross product

and that stuff we have at the moment in the supp part

l=r \times p is something everyone knows

mercio

it is something I would just add next

Rudi_Birnbaum

Yes, good idea. OK.

mercio

also do the "this is new" things correspond to the observations you are making ?

Rudi_Birnbaum

I need some nifty short "statement" at the end of that part of the technical section, what would be the interface to the Discussion part and that is: "In G' \Gamma_l_z is contained in E x E" and after restriction will stay 'there' (nice wording for 'there' still required). And in all other cases its not there."

@mercio Yes, that looks all nice and sound to me.

"But it could be in T x T"

...

Thats important since one important JT class is E in Oh, for example. And in there you do not observe paramagnetic response (l_z being in the product after distortion),.

4) (related) "The last family of point groups are the ones whose representation decomposes into A1 ⊕ A2 ⊕ A3 and we don’t talk about them." we have to talk abou them. Since I intend to give calculations on examples molecules for all principal cases. This adds lot of relevance for the general reader.

So for all cases there is always the question is l_z in any square of a degenarate irrep and will it end after distortion in some product of two irreps (which ultimatly means the particular magnetic repsonde).

But the main point is the in G its exactly in E^2. While in the rest it depends on the specific case.

That gives a certain aha-effect ...

In essence it shows why in certain rings shaped molecules paramagneticity is inherently connected to distortion. While in octaheron-shaped molecules this is much rarer.

And that in turn shows some previously hidden inner logics of the term "anti-aromaticity".

mercio

1:17 PM

I was about to merge them into the first case

by removing the assumption that E is irreducible

lemme just continue what i was doing then

is it actually important that the symmetric part has occurences of the trivial iirep ?

Rudi_Birnbaum

Only when my "primoid" JT effect comes into play. So yes in the next section, about which we yet didn't talk too much

I noticed its very badly written, yet. But I hoped that it would be a better start for discussion.

than what we had on MSE

mercio

I haven't read those parts yet

Rudi_Birnbaum

I think it would be good if you understand that as well, since only then you can see we the journey goes to. Actually only the last (primoid) part is at the moment not that relevant.

That one can be sorted out separately.

But the JT you should understand as good as possible.

already now

mercio

I am bothered that it is super easy to show that the trivial representation occurs in a tensor square, but that is occurs in the symmetric part sounds harder

Rudi_Birnbaum

1:34 PM

@mercio I am not sure that we will need that, that its specifically in the symm. part. At the moment definitely not, but when we can show that it would be a matter of style to also "take it".

Its btw the very same with l_z, we actually don't need that its in the anti ... but then for you its even the definition ...

mercio

1:49 PM

how do i skip a line

it is a wall of text now

Rudi_Birnbaum

I just know \\

mercio

sent a new one, I hope it still reads well for you

Rudi_Birnbaum

2:06 PM

reads very good! (up to some little typos)

9 hours later…

mercio

10:37 PM

suppose $T \in GL_3(k)$ such that the $(x\wedge y)$-line and the $(y\wedge z,x\wedge z)$-plane are stable by $Alt^2(T)$.
write its matrix $\begin{pmatrix}a &b &c\\d &e &f\\g &h &i\end{pmatrix}$.
Then $ah=bg, dh=eg, af=cd, bf=ce$
$\det(T) = aei+bfg+cdh-afh-bdi-ceg = aei+afh+ceg-afh-bdi-ceg = (ae-bd)i$
$c\det(T) = c(ae-bd)i = (ace - bcd)i = (abf - abf)i = 0$
so $c=0$, and a similar computation shows $b=d=g=0$, and so
the $z$-line and the $(x,y)$-plane are stable by $T$

this really looks like $Alt^2 \circ Alt^2 = Alt^3 \otimes Id \oplus Alt^4$

or something

but searching for the value of this plethysm is hard, especially because plethysm also means the decomposition of tensor products

or maybe something else instead of the first Alt²

I meant $c=f=g=h=0$ how did I mess this up lol

mercio

11:02 PM

correction, $Alt^4 \oplus Alt^2 \circ Alt^2 = Alt^3 \otimes Id$

(but in dimension 3, $Alt^4$ vanishes so we don't really care)

also I checked that in formal power series with schur functors coefficients, $(\sum_{k \ge 0} Sym^k t^k) \otimes (\sum_{k \ge 0} Alt^k (-t)^k) = 1$, I wonder if this is known

the Molien formula falls out of this equality nicely

and in all of the stuff I have read those last few days I have never seen a discussion of the decompositions of $F(A \oplus B)$ for a Schur functor $F$

mercio

11:43 PM

though that might be the branching rules now that I think about it

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