What kind of symmetry does the Galois group of an equation show between the root sets?

I know that the Galois group of an equation on $mathbb{Q}$ can be determined according to the rational relation determined by Vieta’s theorem.

Page 393 of this book also points out that these relations are to keep all polynomial relations between root sets (the result are all rational numbers) unchanged:

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Taking the equation $f(x)=x^{4}-4x^{2}-5$ as an example, its four roots are $sqrt{5}$, $-sqrt{5}$, $i$ and $-i$ in turn.

f1(x1_, x2_, x3_, x4_) := x1 + x2
f2(x1_, x2_, x3_, x4_) := x3 + x4
f3(x1_, x2_, x3_, x4_) := x1*x2(*Rational relations between root sets determined by Vieta theorem*)
G = SymmetricGroup(4);
sol = Select({Permute(Inactive(f1)(x1, x2, x3, x4), G),
    Permute(Inactive(f2)(x1, x2, x3, x4), G), 
    Permute(Inactive(f3)(x1, x2, x3, x4), 
     G)}(Transpose), ((((Activate(#) /. {x1 -> 
            Root(#1^4 - 4 #1^2 - 5 &, 1), 
           x2 -> Root(#1^4 - 4 #1^2 - 5 &, 2), 
           x3 -> Root(#1^4 - 4 #1^2 - 5 &, 3), 
           x4 -> Root(#1^4 - 4 #1^2 - 5 &, 4)})) // 
       FullSimplify) == {0, 0, -5}) &)
FindPermutation(#, Inactive(f1)(x1, x2, x3, x4)) & /@ sol((All, 1))
PermutationList(#, 4) & /@ %(*The group is isomorphic to the Klein's four-group and is a subgroup of group S4*)

From the above code, we can see that by keeping the polynomial relationship between the root sets such as $x1 + x2 = 0$, $x3 +x4 = 0$, $x1*x2=-5$, which results with rational numbers unchanged, we find the Galois group of the equation $f(x)=x^{4}-4x^{2}-5$ on $mathbb{Q}$.

If the Galois group G is solvable, then the equation $f(x)$ on $mathbb{Q} $ is radical solvable and the group has a normal subgroup sequence $mathrm{G}=mathrm{G}_{0} triangleright mathrm{G}_{1} triangleright ldots triangleright mathrm{G}_{mathrm{r}}={mathrm{e}}$. Let every quotient group $G_{i} / G_{i+1}, i=0,1,2, ldots, r-1$ be commutative.

I want to know what symmetric property of the root set of the representation of this Galois group G determines that it can have radical solutions. And what property of the Galois group described by the symmetry of the root set determines that it can be solved by radical. In other words, which of the more in-depth properties between the root sets represented by the Galois group determine that the equation can be solved by radical. It’s better to show this relationship clearly with MMA.

probability – Generate Symmetry consistent combinations from equivalent positions

I want to generate all possible substitutions from a simple molecule, for example Naphthalene

Naphthelene

Now i have three groups, a,b,c; how can I generate all possible unique molecules from a,b,c?

example: if taken in following notation (group:position, group:position …)

Then,
(a:8, a:1, b:2) = (a:8,a:1,b:7)

whereas
(a:8,c:1,b:2), (a:8,c:1,b:7) are different unique molecules.

How to systematically generate all possible unique combinations here, or in general?

Symmetry of fractional laplacian

Let $Omegasubsetmathbb{R}^n$, let $sin (1/2,1)$, let $uin C^{1,2s-1+epsilon}(Omega)$ such that: $u=0$ on $mathbb{R}^nsetminusOmega$, and: $uin C^{0,s}(mathbb{R}^n)$, is true that:
$$int_{mathbb{R}^n}phi(-Delta)^su,dx=int_{mathbb{R}^n}u(-Delta)^sphi,quadforallphiin C^infty_c(mathbb{R}^n)?$$
I kwon only that:
$$ int_{mathbb{R}^n}phi(-Delta)^sf,dx=int_{mathbb{R}^n}f(-Delta)^sphi,dx,quadforall f,phiin mathcal{S}(mathbb{R}^n).$$
I have no idea on how to go on, any help is appreciated.

Symmetry of Command Dot

I would like to work with the command Dot assuming symmetry. Specifically with the order

D[v

v

But I would like to get this one

2 v

Thanks in advice.

chemistry – Symmetry unique atom coordinates

One thing I love about Mathematica is how easily I can go from the name of a molecule to estimated coordinates of its atoms, with a command like

AtomList(Molecule(Entity("Chemical", "Toluene")), All, {"AtomicNumber", "AtomCoordinates"})

(although, oddly enough, “AtomCoordinates” does not appear in the “AtomList” documentation)

I can also easily get the point group:

Molecule(Entity("Chemical", "Toluene"))("PointGroup")

This is exciting because this is exactly the input I need to run GAMESS and do quantum chemistry calculations (starting with a geometry optimization, of course, since JM has informed me that these coordinates are heuristic guesses).

But, really, this is not exactly the input that I need: what I really need are coordinates of only the symmetry-unique atoms.

I don’t suppose there’s a way to get coordinates of symmetry-unique atoms, which I can use for GAMESS input? I know there’s some functions related to point group symmetry, but I haven’t thought of how to do it.

nt.number theory – Symmetry in Hardy-Littlewood k-tuple conjecture

Assuming Hardy-Littlewood $k$-tuple conjecture, do the “dual” prime constellations $(0,m_1, m_2,cdots, m_i,cdots, m_{k-1}=d)$ and $(d, m’_1, m’_2,cdots,m’_i=d-m_i,cdots,0)$ have the same distribution?

If yes, does it imply that the function $f(n):=dfrac{log g_n}{loglog p_n}$ and the function $f'(n)$ obtained through the substitution $g_nmapsto g’_n:=dfrac{log^{2} p_n}{g_n}$ reach the same values an asymptotically equal number of times? Is it related to the functional equation of zeta with which it would then share the same type of symmetry?

mirror symmetry – Embedding Calabi-Yau manifolds in projective space

When studying homological mirror symmetry, a lot of work is done not in the setting of complex manifolds, but of smooth (quasi-)projective varieties, see e.g. a paper from Orlov. However, the actual case one wants to consider is usually that of Calabi-Yau-manifolds.

It is usually not explained why the former case includes the latter, although it seems to be used very often that it does. My question is therefore: Can every (possibly non-compact) Calabi-Yau be embedded into projective space, giving a smooth quasiprojective variety (I read that it sometimes can’t be included as a projective one)? I assume that to show this one would have to (like for Fano manifolds) construct an ample line bundle over it, as that would already imply the result. If it is possible, how can it be shown, and more importantly if not, why is Orlov’s approach even justified?

Maybe a few further remarks: It seems like to embed a complex manifold $X$ into projective space, it must neccessarily be Kähler, as the Kähler structure of $mathbb{C} P^n$ restricts to one on $X$, but this is obviously fulfilled here. Also, I read the article on ncatlab about Calabi-Yau-varieties, and to me it seems like it also supposes that these are equivalent to Calabi-Yau manifolds, although it only describes the analytification direction.

symmetry: decomposition into irreducible modules $ S_n $, also known as Specht modules.

Leave $ S _ pi} $ where $ pi $ is an entire partition of $ n $, denote the Specht module corresponding to $ pi $.

I am trying to decompose the set of all homogeneous polynomials into $ x_1, x_2, …, x_n $ linearly generated (over any zero characteristic field) by the monomials of the form $ x_i ^ 2x_jx_k $ ($ i, j, k $ are different), in Specht modules. I managed to do it for the polynomials generated by each of the following monomial classes with $ i, j, k, l $ different: $ x_i ^ 3x_j, x_i ^ 2x_j ^ 2, x_i ^ 4, x_ix_jx_kx_l $.

First $ x_i ^ 3x_j $ $ (i ne j) $: We know $ x_i ^ 3x_j = displaystyle frac {x_i ^ 3x_j + x_ix_j ^ 3} 2+ frac {x_i ^ 3x_j-x_ix_j ^ 3} 2 $. The terms of the form $ displaystyle frac {x_i ^ 3x_j + x_ix_j ^ 3} 2 $ linearly generate an isomorphic space like $ S_n $-modules (modulus action is by index permutation) to square-free homogeneous degree 2 polynomials. This is isomorphic to $ S (n-2,2)} oplus S (n-1,1)} oplus S (n)} $. The terms of the form $ displaystyle frac {x_i ^ 3x_j-x_ix_j ^ 3} 2 $ linearly generate an isomorphic space like $ S_n $-modules (the module's action is through the permutation of indices) to the second external power of a vector space generated by $ {x_1, x_2, …, x_n } $ where $ x_i wedge x_j $ is in one-to-one correspondence with $ displaystyle frac {x_i ^ 3x_j-x_ix_j ^ 3} 2 $. So, this is isomorphic to $ S (n-2,1,1) oplus S (n-1,1)} $.
So the decomposition is

$ displaystyle S _ {(n-2,2)} oplus S_ {(n-2,1,1)} oplus 2S _ {(n-1,1)} oplus S _ {(n) } $

where "$ 2 $"indicates that we have two copies of S (n-1,1)}.

Second $ x_i ^ 4 $: This is simply a vector space generated by $ x_i ^ 4 $, and is a direct sum of the standard and trivial representations of $ S_n $ that is to say S (n-1) and S (n). Therefore, decomposition is $ S (n-1,1)} oplus S (n) $.

Third $ x_ix_jx_kx_l $: Generate the isomorphic module to the module $ M_ lambda $ like in Bruce Sagan's book "The Symmetric Group" where $ lambda = (n-4,4) $ which figure is just S (n-4,4) oplus S (n-3,3)} oplus S (n-2,2)} oplus S (n-1,1) } oplus S_ {(n)} $.

Room $ x_i ^ 2x_j ^ 2 $: Generate the isomorphic module to the module $ M_ lambda $ where $ lambda = (n-2,2) $ which figure is just $ S (n-2,2)} oplus S (n-1,1)} oplus S (n)} $.

The reason behind showing above is that these are alarmingly simple deductions, although I can't seem to find such a skillful one for the class. $ x_i ^ 2x_jx_k $. I also tried dimension count, because grade 4 homogeneous polynomials are dimension $ {n + 3 choose 4} $. But this counting technique leads to many possible decompositions. I'm a bit out of ideas about this. Could anyone help?

How to establish assumptions of non-trivial tensor symmetry?

For example dddg, the third derivative of the metric tensor, in dimension $ 4 $.

I need to say that it is symmetric in the first two indices and symmetrical in the last three indices.

sg.symplectic geometry – Mirror symmetry for $ C ^ * $

The Liouville variety $ T ^ * S ^ 1 $ it is said to be "mirror" of the complex variety $ C ^ * $. (see for example Lesson 7 here: http://math.columbia.edu/~topology/Eilenberg_lectures_fall_2016)

This is manifested in the fact that the category Fukaya wrapped in $ T ^ * S ^ 1 $ it is quasi equivalent to (a dg upgrade) from the coherent pulley category at $ C ^ *.

Given a fiber $ F_x subset T ^ * S ^ 1 $, one can calculate that $ operatorname {Hom} _ { operatorname {Fuk}} (F_x, F_x) = mathbb {C} (x, x ^ {- 1}) = operatorname {Ext} ( mathcal {O} _ { mathbb {C} ^ *}, mathcal {O} _ { mathbb {C} ^ *}) $.

So we say that the fibers are "mirror" to the sheaf structure.

Two (related) questions that have been bothering me:

(i) is there a systematic / functorial way to build a functor from $ operatorname {Fuk} (T ^ * S ^ 1) to operatorname {Coh} (C ^ *) $? As I understand it, the "Family Floer Theory" program builds this in the context of compact toric varieties, but this story doesn't seem to apply here on the nose (for example, it involves a rigid analytical geometry and not they imply wrapped Fukaya categories, at least not in the documents I found.)

(ii) The above story seems to depend a lot on the fact that the Fukaya category is defined on $ mathbb {C} ^ * $. In general, the Fukaya category is defined on a Novikov ring; as $ T ^ * S ^ 1 $ it is exact, one can also use $ mathbb {C} $-coefficients that use a rescaling "trick" described at the top of p7 of these notes: https://arxiv.org/pdf/1301.7056.pdf

Does the story still work with Novikov coefficients? What changes?