Wikipedia:Reference desk/Archives/Mathematics/2013 July 3

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July 3

In simple terms, why are spin representations "projective" ?

(I originally posted this at Talk:Projective representation, but on second thoughts maybe it's more appropriate here)

I'm familiar that rotations can be computed using rotors/quaternions/spin-matrices, x → R x R^-1 instead of rotation matrices x → M x; and with Spin(n) as the group that describes the one-sided combination of those rotors/quaternions/spin-matrices, R_BA = R_B R_A.

I also have a very rudimentary idea of the construction of projective spaces for the lowest dimension Euclidean spaces.

Given that starting point, is it possible to understand in simple terms how the spin representations are "projective representations" of the orthogonal or special orthogonal groups? What is the meaning of "projective" here, and how does it come about?

I've read the lead and the introduction of the article, but I don't understand how it applies. (And without much of an algebra background, I don't really have much feel for most of the terminology, or the diagram).

So is it possible to explain in a more concrete way why, say, a representation of Spin(3) can be described as a "projective" representation of SO(3)? Is there some obvious thing that I'm missing? Jheald (talk) 10:06, 3 July 2013 (UTC)

Let me see if I can get a bit of the way. I know that

$SO(3)={\mathit {Spin}}(3)\,/\,Z_{2}$

as Z₂ is a quotient group of Spin(3) because there is a homomorphism h from Spin(3) to SO(3), i.e. a mapping that satisfies h(R_BA) = h(R_B) h(R_A), the kernel of h being {-1, +1}.

This is a surjection ("onto") because for every element in SO(3) there are corresponding elements in Spin(3).

So any representation of Spin(3) is also a representation of SO(3).

Now according to the article a representation is projective if it lives in PGL(V,F) = GL(V,F)/F*.

GL(V, F) means I need to think about a representation of Spin(3) in the group of matrices (or something equivalent). Then, for the representation to live in GL(V,F)/F*, I need to think about the significance of factoring out the group of scalar transformations F*. The word "projective" seems to be saying that a representation of Spin(3) can meet this requirement, but ordinary rotation matrices can't. Ordinary rotation matrices are constrained to have determinant 1, which one might think was factoring out their scale freedom. But they are not "projective" representations, so evidently that is not what's needed here. So what does it mean to ask for the representation to live in GL(V,F)/F*, and what consequences does it imply? Jheald (talk) 12:44, 3 July 2013 (UTC)

Projective representation means only the the group (in this case SO(3)) acts linearly on the projective space, but this representation is not associated to a linear representation. Sławomir Biały (talk) 18:42, 3 July 2013 (UTC)

Does that work? Surely the representation of Spin(3) *is* a linear representation, and *is* a representation of SO(3) ?

And how does what you're suggesting fit with the definition given in the article? Sorry if I'm being slow and not really underestanding, but this isn't home territory for me. Could you unpack it a little? Jheald (talk) 18:56, 3 July 2013 (UTC)

The reps of Spin(3) are all linear reps, but they are not linear reps of SO(3): instead they are projective reps of SO(3), meaning that SO(3) acts on the associated projective space (the Riemann sphere for SO(3)). Sławomir Biały (talk) 19:02, 3 July 2013 (UTC)

Okay. I'm still not seeing (1) how this fits with the definition given in the article; and (2) exactly what you're getting at. Is it that the spin matrices can be thought of as describing transformations of a 2d complex vector, but the rotation matrices act on a 3d real vector, and though the spin matrices can be mapped to the rotation matrices, there's no straightforward map from the 2d complex vector to a 3d real vector? Jheald (talk) 19:17, 3 July 2013 (UTC)

Rotations act on the sphere $\mathbb {C} \mathbb {P} ^{1}$, but not on $\mathbb {C} ^{2}$. As to the definition given in the article, an action of a group on projective space is a colineation if and only if the representation is induced by a group homeomorphism into the projective general linear group. Sławomir Biały (talk) 19:30, 3 July 2013 (UTC)

Okay, so to see if I can unpack the notation, $\mathbb {C} \mathbb {P} ^{1}$ is the Complex projective space with n=1, which is the Riemann sphere. That is equivalent to $\mathbb {C} ^{1}$, but with all the points at infinity in every direction identified as a single point. On the other hand the spin matrices of SU(2) = Spin(3) act on a vector of two complex numbers; so I'm not sure what that has to do with CP¹.

As to the next sentence, I recognise "representation induced by a group homomorphism into the projective general linear group" as the article's definition of a projective representation. I still don't understand what it is in that definition that makes Spin(3) fit the criterion but not SO(3). A collineation is a mapping that preserves collinearity. So you're saying that if a representation fits the definition in the article (like Spin(3) but not SO(3)), then its action on projective space will preserve collinearity. Okay, if you say so. But then what is the projective space here? CP¹? And why CP¹ if SU(2) is a set of 2x2 complex matrices? Jheald (talk) 23:16, 3 July 2013 (UTC)

CP1 is the space of one dimensional complex linear sub spaces in C2. SU(2) acts in the usual way one C2, and this induces an action on the set CP1. Sławomir Biały (talk) 01:21, 4 July 2013 (UTC)

Okay, I can sort of see how that might hold together -- though I'm not sure why the conjugation and the double cover would now come in, if you want to use Spin(3) to actually effect a rotation. But is this a relationship that only works for Spin(3), or is there something guaranteed similar for all Spin(n)?

I'm a bit surprised, because I had been starting to think that the "projective" quality in the name might be referring to a purely algebraic property, without such a direct relation to actual geometric projective spaces. So why does SU(2) satisfy the requirement of belonging to some GL(V,F)/F*, and SO(3) not satisfy it? Jheald (talk) 08:45, 4 July 2013 (UTC)

Well, SO(n) will always act on the complex projective space of the associated spin representation of Spin(n). So yes, this is true in every dimension. However, since the dimension of the spin representations grows exponentially, only in low dimensions will there be an easy way to understand the action of SO(n) on the projective space in terms of the fundamental representation of SO(n). For instance, with n=3, the spinors have two components, the projective space is CP1 and the action of SO(3) is just the ordinary rotations of the sphere. Sławomir Biały (talk) 17:30, 4 July 2013 (UTC)

After further thought, is the key thing here the "sandwich" conjugation product, x → R x R^-1 ?

If a transformation is effected in this way, we can scale the operator R by any scalar factor (including a complex phase), and the effect of that scaling will be neutralised by the corresponding effect on R^-1.

So we can apply any multiplying factor to the spin matrices and still represent the same transformation, whereas that is not true for the rotation matrices of SO(3).

Of course conventionally the spin matrices are taken to be the unitary matrices of SU(2), but unlike the rotation matrices we don't need to insist on that determinant 1.

This re-scaleability would be a necessary prerequisite for interpretations of the one-sided effect of an m by m complex matrix R in terms of its effect on a geometric complex projective space, eg the space of all the lines in C^m

It would be interesting to know what (or whether) interpretations/depictions of this kind for Spin(n) generally exist. I am a bit dubious as to whether the action of Spin(n) generally can be usefully visualised in this way. I'm not even sure I see SU(2) meaningfully working in this way for Spin(3) -- does anyone have animations? Jheald (talk) 12:48, 4 July 2013 (UTC)

No, this is not relevant here. While the action of the spin group on the fundamental representation of SO(n) can be realized as something like $RxR^{-1}$, this is a bona fide representation of SO(n) (and Spin(n)), not just a projective representation of SO(n). Sławomir Biały (talk) 17:30, 4 July 2013 (UTC)

Surely every spin representation of the rotation group is a "bona fide" representation, because one can define a group homomorphism to map each rotation to a corresponding spin matrix R -- which is the definition of a group representation.

I can't pretend to understand much on our page Fundamental representation, but unless I'm misunderstanding horribly, it says that for each orthogonal group, "[either] the spin representation... [or] the two half-spin representations... are fundamental representations". So the spin-matrices R are a fundamental representation of the rotation group.

And according to our article projective representation, whether a representation is projective or not purely depends on the kind of linear space its elements R live in (or perhaps the equivalence class of possible equivalent elements for R).

I think I asked at the top why a projective representation of a rotation was "projective".

It's entirely probable that there's more to understand about the representations of Spin(n) than just that they are projective representations of the rotation group, and I'm certainly interested to know more. But as you yourself wrote above, for n>3 I'm not sure that relating them to CP^m-1 for some increasingly growing value of m is so informative about them in the narrower context of them as representations of the rotation group. Jheald (talk) 00:52, 5 July 2013 (UTC)

No, spin representations of the rotation group are not bona fide representations of the rotation group. They are instead projective representations, which is a strictly weaker requirement. A group representation associates to each group element a linear operator on a vector space. That's not what we have here. Instead, the spin representation associates to every group element an automorphism of a projective space. But there is no linear action of the rotation group on spinors. There is a linear action of the Lie algebra associated to the rotation group on spinors (via the isomorphism of so(3) with su(3)), but this action does not exponentiate to a representation of the rotation group. (Although sometimes, especially in physics, the distinction between a group and its Lie algebra are sometimes ignored.) Sławomir Biały (talk) 12:34, 6 July 2013 (UTC)

I'm not trying to contradict you for the sake of contradiction, but it seems to me that (1) according to our projective representation page, the requirement for the is a stronger requirement, because it requires such a representation not only to be a representation, but also to fulfill the projectivity requirement. (2) We have a linear set of matrices SU(2), which act linearly on a 2d complex vector, and combine linearly in a way that exactly matches the way rotations combine. So that does associate to each group element a linear operator on a vector space. Even though it's a different vector space to the vectors of SO(3) act on. (3) Each of these SU(2) matrices can be written as a matrix exponential exp(B θ/2), where B is the matrix exponential of a bivector; so they do correspond to exponentiated generators. Jheald (talk) 14:05, 6 July 2013 (UTC)

Bur SU(2) and SO(3) are not isomorphic groups. Elements of SU(2) are not in 1-1 correspondence with rotations. The space C2 is a bona fide representation of SU(2), but only a projective representation of SO(3). Sławomir Biały (talk) 14:53, 6 July 2013 (UTC)

1-to-1 correspondence is not a necessary requirement for a representation. Per our article Group_representation#Definitions, a representation of a group G is a group homomorphism from G to GL(V), where V is a vector space called the representation space, and GL(V) is the general linear group over it.

Such a representation does not require the mapping to be either injective or surjective, so long as it is a homomorphism. Indeed, an important part of the definition is that it allows representations which are not faithful, i.e. many-to-one.

Though I'm interested by this repeated claim that one can define a subset of representations called "bona fide" representations. Does this subset have a more formal name, and is it more formally defined? Jheald (talk) 23:02, 6 July 2013 (UTC)

Yes, but you seem to be convinced that the representation of SU(2) on C2 is a representation of SO(3). This is not true. These are different groups, they cannot be identified woth each other as you seem to want to do, and they have different representations. A bona fide representation is known as a group representation. Every group representation is a projective representation, but not every projective representation is a group representation. Sławomir Biały (talk) 07:39, 7 July 2013 (UTC)

So what part of the definition of group representation is the representation of SU(2) on C2 not fulfilling if we would like to think of it as a representation of SO(3)?

As I understand it, we need a homomorphism, so h(R_BA) = h(R_B)h(R_A); and that homomorphism needs to be into a linear space.

Does a mapping from elements of SO(3) to appropriate elements of the representation of SU(2) on C2 not comply with both of those requirements? Jheald (talk) 08:30, 8 July 2013 (UTC)

That's my point, there is no such map from SO(3) into SU(2) (the map goes the other way). Sławomir Biały (talk) 10:25, 8 July 2013 (UTC)

Aaaaah! (facepalm).

I can arbitrarily choose half of SU(2) to map SO(3) into, but then the elements I've chosen aren't closed. For the algebra to be closed, I need the whole of SU(2). But then there's no longer a mapping from SO(3) to the all of the target algebra.

I can make a mapping if I declare elements R and -R of SU(2) to be equivalent. But then I'm not mapping into GL(V) any more, I'm mapping into PGL(V). And that means it's not a bona fide representation, only a projective one.

Thank you to everybody for being so persistent and patient with me. I hope I see it now. Jheald (talk) 13:34, 8 July 2013 (UTC)

Name of this relationship

What do you call the number that when summed with a given number equals one? For example for 0.2 it would be 0.8, or for 34% it would be 66% or for 3/12 it would be 9/12. Is there a word for that relationship? 182.11.170.204 (talk) 23:01, 3 July 2013 (UTC)

Fuzzy negation, maybe? Although there's little chance you'd be understood saying that without further clarification. « Aaron Rotenberg « Talk « 00:01, 4 July 2013 (UTC)

The J (programming language) has the notation 'minus dot' for this useful operation: "-.y is 1-y ; for a boolean argument it is the complement (not); for a probability, it is the complementary probability." Bo Jacoby (talk) 04:07, 4 July 2013 (UTC).

I'd call it normalization; however, that's usually the more general case where you have n unknowns and look for a solutions where the n values add up to 1. This is especially the case in problems where constant factors are irrelevanant, i.e. the class of problems where (1 , 2 , 3 , 4) is seen as "the same solution" as (.1 , .2 , .3 , .4). Mixing problems in chemistry come to mind. Is your problem one of those? - ¡Ouch! (^{hurt me} / _{more pain}) 07:52, 4 July 2013 (UTC)

No, I just saw someone refer to this relationship as "inverse", and I corrected them, but that got me thinking whether this relationship actually has a name. 202.155.85.18 (talk) 08:45, 4 July 2013 (UTC)

I'd call it the complement. You might like to read Method of complements and Ones' complement (if you haven't already done so), though the latter is usually applied to numbers written in base 2. Dbfirs 07:38, 7 July 2013 (UTC)

sum of two squares is a cube

Is there a way to find the solutions of x^2 + y+2 = z^3 x² + y² = z³, (x, y, and z are integers) that is significantly better than brute force? Bubba73 ^{You talkin' to me?} 23:59, 3 July 2013 (UTC)

x² + y + 2 = z³ ? If so, then all you have to do is take a random cube, and a random square smaller than the cube, then subtract 2 from their difference to get y. Unless, of course, you meant x² + y² = z³ — 79.113.211.75 (talk) 01:35, 4 July 2013 (UTC)

Whoops, typo, I did mean x² + y² = z³. Bubba73 ^{You talkin' to me?} 02:02, 4 July 2013 (UTC)

Let p be an integer, and $x=(1+p^{2})$, then it follows that $x^{2}+(xp)^{2}=x^{2}(1+p^{2})=(1+p^{2})^{3}$. That gives a set of solutions that can be generated from every integer, though I don't know if all solutions will necessarily have that form (probably not, I would guess). Dragons flight (talk) 02:14, 4 July 2013 (UTC)

Also, given any solution $(x,y,z)$, then it follows that $(p^{3}x,p^{3}y,p^{2}z)$ is also a solution for all integers p. Dragons flight (talk) 02:22, 4 July 2013 (UTC)

See Fermat's theorem on sums of two squares. Any z can be chosen where each prime factor of the form 4n+3 occur an even number of times as that is equivalent to the cube being of that form, though some of those will have x or y zero. Dmcq (talk) 11:57, 4 July 2013 (UTC)

There is a fairly fast way, based on the unique prime factorisation of Gaussian integers. x² + y² = z³ if and only if the square of the absolute value of x+iy is z³. This is soluble for a particular z if and only if there are integers a and b with a² + b² = z. If there are such ones, then just take the real and imaginary parts of (a+ib)³ to get your x and y. JoergenB (talk) 21:24, 5 July 2013 (UTC)

Yes, exactly. A number is the sum of two squares if and only if at least one of its divisors is also the sum of two squares. Euler said it more than three centuries ago, but I doubt that numbers have changed their properties since then. :-) Actually, it can be written as the sum of two squares in as many ways as there are numbers among its divisors which can be written in that way. The first such solution for strictly positive integers is obviously 2² + 2² = 2³. See Pythagorean triples for more information. — 79.113.230.30 (talk) 22:22, 5 July 2013 (UTC)