The Math Thread

[lpetrich]

As to why I do what I'm doing here, it's because of all the nice patterns that I find -- and underappreciated patterns, since one has to know lots of arcane math to appreciate them, it seems.

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What's a "quotient group"?

Let us consider a subgroup H of a group G. Lagrange's theorem states that (order of G)/(order of H) = (index of H in G) -- an integer. "Order" is the number of elements.

To prove it, construct the left cosets of the subgroup: a.(elements of H) for every a in G. It also works for right cosets: (elements of H).a . Each side of coset is a set of disjoint sets that span G, and the number of them, including H, is the index of H in G.

Now consider the conjugate of an element: conjugate of a by b = b.a.b^-1.

The conjugate of H by b is the conjugate of every element of it by b. It is another subgroup, and we get a set of subgroups related by conjugacy.

If there is only one such subgroup, then that subgroup is a "normal subgroup". It also has the interesting property that (left cosets) = (right cosets).

Construct a multiplication law for cosets of a normal subgroup: (element of coset C1) . (element of coset C2) = (element of coset C3). This is clearly a group, a "quotient group" or a "factor group". If a group has a quotient group, then every element of that group can be depicted as tagged with an element of that quotient group.

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Consider a group that has a subgroup with half its elements. Is that subgroup a normal one? What is its quotient group?

 

[beero1000]

As to why I do what I'm doing here, it's because of all the nice patterns that I find -- and underappreciated patterns, since one has to know lots of arcane math to appreciate them, it seems.

 

[lpetrich]

Here's an interesting curiosity about the reality of spinor representations of SO: the group Spin.

It has a periodicity of 8 in n, a "Bott periodicity".

SO(8k+0): RR
SO(8k+1): R
SO(8k+2): CC
SO(8k+3): H
SO(8k+4): HH
SO(8k+5): H
SO(8k+6): CC
SO(8k+7): R

R = real, H = pseudoreal, C = complex. For SO(2n+1), the spinor rep is irreducible, with dimension 2^n, while for SO(2n), the spinor rep splits into two equal-sized irreps, with dimension 2^(n-1). In the CC case, they are complex conjugates of each other, while in the RR and HH cases, they are separate.

For SO(n1,n2), its spinor-rep reality is like that for SO(|n1-n2|). Thus, the Lorentz group's two spinor reps are complex -- and complex conjugates of each other. This manifests itself in the chirality or handedness of some elementary particles and their interactions.

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There is a nice and interesting way of constructing spinors that uses a "Clifford algebra", a set of operators related by anticommutation instgead of commutation:
Ordinary commutator: [A,B] = A.B - B.A -- departure from being commutative
Anticommutator: {A,B} = A.B + B.A

Operators G(i) are related by {G(i),G(j)} = 2*δ(i,j)

For both Spin(2n) and Spin(2n+1), one constructs (2n+1) of the G's.

Their analogues of SO(2n) and SO(2n+1) operators one finds with L(i,j) = 1/(4*i)*[G(i),G(j)], with i and j ranging over 1 to 2n and 1to 2n+1 respectively.

So for Spin(2n), operator G(2n+1) splits the spinor rep into two irreps, but for Spin(2n+1), the spinor rep is an irrep.

There is a nice way of constructing the G's with Pauli matrices, as an outer product of n of them.

G(1) = {σ1,I,I,...,I}
G(2) = {σ2,I,I,...,I}
G(3) = {σ3,σ1,I,...,I}
G(4) = {σ3,σ2,I,...,I}
G(2n-1) = {σ3,σ3,σ3,...,σ1}
G(2n) = {σ3,σ3,σ3,...,σ2}
G(2n+1) = {σ3,σ3,σ3,...,σ3}

Thus, Spin(2n) has a spinor rep with dimension 2^n that splits into two irreps with dimension 2^(n-1) each, and Spin(2n) has a spinor rep that is irreducible.

 

[lpetrich]

My Science and Math Stuff -- it has math stuff in it as Mathematica notebooks

I especially like SemisimpleLieAlgebras.zip -- that archive contains code for doing semisimple Lie algebras and their representations. It constructs the algebras themselves and irreps of them as sets of root vectors. It finds product reps, symmetrized powers of reps ("plethysms" -- symmetric, antisymmetric, and mixed symmetry), and what a rep looks like in a subalgebra ("branching rules"). It's done in brute-force fashion, though it does use a speed-up that I found in the LIEART package: constructing Weyl orbits.

I wrote that code in Mathematica, then ported it to Python to make it more portable, and then to C++ to speed it up. For some reason, Python's built-in fraction code is very slow, so I had to integerize the root-vector calculations. I then carried over that integerized code to C++. I used a lot of Standard Template Library stuff for containers and the like, something that saved me a *lot* of work relative to plain C.

I could have done it in Fortran, the first-ever high-level language, but that's not much different from C.

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I also have some computational-geometry algorithms implemented in Mma: implementations of convex hulls and Delaunay triangulation with various algorithms, including multi-dimensional generalizations: Delaunay simplexes. I've neglected Voronoi diagrams, however

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I also have a Mma notebook on the Cayley-Dickson construction of division algebras, inspired by The Octonions.

 

[beero1000]

My Science and Math Stuff -- it has math stuff in it as Mathematica notebooks

I especially like SemisimpleLieAlgebras.zip -- that archive contains code for doing semisimple Lie algebras and their representations. It constructs the algebras themselves and irreps of them as sets of root vectors. It finds product reps, symmetrized powers of reps ("plethysms" -- symmetric, antisymmetric, and mixed symmetry), and what a rep looks like in a subalgebra ("branching rules"). It's done in brute-force fashion, though it does use a speed-up that I found in the LIEART package: constructing Weyl orbits.

I wrote that code in Mathematica, then ported it to Python to make it more portable, and then to C++ to speed it up. For some reason, Python's built-in fraction code is very slow, so I had to integerize the root-vector calculations. I then carried over that integerized code to C++. I used a lot of Standard Template Library stuff for containers and the like, something that saved me a *lot* of work relative to plain C.

I could have done it in Fortran, the first-ever high-level language, but that's not much different from C.

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I also have some computational-geometry algorithms implemented in Mma: implementations of convex hulls and Delaunay triangulation with various algorithms, including multi-dimensional generalizations: Delaunay simplexes. I've neglected Voronoi diagrams, however

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I also have a Mma notebook on the Cayley-Dickson construction of division algebras, inspired by The Octonions.

If you've done Delaunay then you've done Voronoi. Yay, duality!

 

[lpetrich]

Yes, but it's mainly a bookkeeping issue, especially in higher dimensions. I know how to calculate a Voronoi diagram from a Delaunay triangulation, but I got stuck on the issue of the bookkeeping involved in Voronoi diagrams, especially in higher dimensions. I might settle for 2D ones or maybe also 3D ones.

I've also been concerned about the issue of doing spherical Delaunay triangulations and Voronoi diagrams. Packages like qhull don't support such diagrams. Would one do a triangulation in full 3D and then find out which parts of it are on the surface of the sphere?

ETA: I also have Geometry Demo with Google Maps -- it constructs a convex hull, a Delaunay triangulation, and a Voronoi diagram from points that you click on the map. I'm sure about the convex hull and the Delaunay triangulation, but I think that the Voronoi diagram has some bugs in the part of the sphere away from the points when the points are close together.

 

[lpetrich]

I've also created a page on Symmetries:

Frieze Groups -- using ASCII art

Wallpaper Groups -- using ASCII art wherever possible

2D Point-Group Demo -- it uses SVG

Organism-Symmetry Demo -- it also uses SVG

Organism-Symmetry Gallery -- various point-group and space-group symmetries, and also a fractal.

Symmetry Explorer -- load pictures into it. It might only work on pictures on my site because of security restrictions.

 

[beero1000]

Yes, but it's mainly a bookkeeping issue, especially in higher dimensions. I know how to calculate a Voronoi diagram from a Delaunay triangulation, but I got stuck on the issue of the bookkeeping involved in Voronoi diagrams, especially in higher dimensions. I might settle for 2D ones or maybe also 3D ones.

I've also been concerned about the issue of doing spherical Delaunay triangulations and Voronoi diagrams. Packages like qhull don't support such diagrams. Would one do a triangulation in full 3D and then find out which parts of it are on the surface of the sphere?

ETA: I also have Geometry Demo with Google Maps -- it constructs a convex hull, a Delaunay triangulation, and a Voronoi diagram from points that you click on the map. I'm sure about the convex hull and the Delaunay triangulation, but I think that the Voronoi diagram has some bugs in the part of the sphere away from the points when the points are close together.

Ok, if the duality transform is a problem, you can always use the lifting map, which is how you probably computed the Delaunay anyway (at least implicitly). The Delaunay is the orthogonal projection of the lower hull, while the Voronoi is the orthogonal projection of the halfspace intersections.

You can get the spherical Voronoi by taking the spherical convex hull of the points on the sphere. Spherical delaunay is still dual to that.

 

[Kharakov]

I do stuff like:

 

[lpetrich]

Here's another mathematical curiosity: the  Cayley–Dickson construction. From a length-1 vector whose contents are some algebraic field, it constructs a series of algebras, each over a vector twice the length of the previous one.

Each step has addition, multiplication, and conjugation defined as possible on pairs of elements from the previous step:

Addition:
(a1,b1) + (a2,b2) = (a1+a2, b1+b2)

Multiplication:
c*(a,b) = (c*a, c*b) and (a,b)*c = (a*c, b*c) with scalar c
(a1,b1).(a2,b2) = (a1.a2 - q*cjg(b2).b1, b2.a1 + b1.cjg(a2))
where q is a constant, equal to 1 in the standard version.

Conjugation:
cjg((a,b)) = (cjg(a), -b)

On the initial member of the algebra sequence, cjg(a) = a

Properties:
Addition forms an abelian group
Multiplication is distributive over addition
Conjugation has properties
cjg(cjg(a)) = a
cjg(a+b) = cjg(a) + cjg(b)
cjg(a.b) = cjg(b).cjg(a)

a + cjg(a) and a.cjg(a) are both commutative and associative with every element of the algebra

Flattening out this nested construction, we get a vector with length 2^n, where the construction was applied n times. The algebra's multiplicative identity has a simple form in it: (1) = {1,0,0,...,0}.

Both a + cjg(a) and a.cjg(a) have the form (scalar) * (1). So we can make the analogy
First component ~ real part of a complex number ~ cjg(a) = a = (scalar)*(1)
Rest of the components ~ imaginary part of a complex number ~ cjg(a) = -a
a + cjg(a) = 2*(generalized real part)
a - cjg(a) = 2*(generalized imaginary part)
Scalar value for a.cjg(a): the norm of a

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This sequence of algebras has the interesting property that it loses properties as one goes.

• Real numbers: initial algebra -- can use other algebraic fields
• Complex numbers: loses self-conjugacy
• Quaternions: multiplication loses commutativity
• Octonions:
  • multiplication loses associativity
  • multiplication is still alternative (associative with two inputs equal)
• Sedenions:
  • multiplication loses alternativity
  • multiplication loses well-normed property (norm of product = product of norms)
  • multiplication is still power-associative (when taking a power, order of multiplication does not matter)
  • zero gets nonzero divisors (one can find pairs of nonzero values that multiply to give zero)
• All the rest: no changes of properties

This means that complex numbers can be represented by 2*2 real matrices, and quaternions by 2*2 complex matrices or 4*4 real matrices, but that octonions and higher 2^n-ions cannot be represented with matrices.

Alternativity: (a.a).b = a.(a.b), (a.b).a = a.(b.a), (b.a).a = b.(a.a)

Power-associativity: if a has generalized real part ar and generalized imaginary part ai, then a^n has generalized real and imaginary parts
P(n,ar,ai.ai)
ai*Q(n,ar,ai,ai)
where for n a nonnegative integer, P and Q are polynomials in ar and ai.ai that are related to the Chebyshev polynomials.

The reciprocal of a also has this form. It is cjg(a)/(norm of a)

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The automorphisms of these algebras are rather interesting.

• Real numbers: identity group
• Complex numbers: Z2 (complex conjugation)
• Quaternions: SO(3) (rotation of imaginary part)
• Octonions and the rest: G2 (rotation of the imaginary part)

The G2 that appears is the matrix group from the fundamental representation of the G2 algebra. It has dimension 7, and it's a subgroup of the group SO(7).

These groups have subgroups that operate on the units of these algebras: elements {0,0,...,+-1,...,0}. Those groups have a sign-flipping subgroup of all matrices diag({1,+-1,+-1,...,+-1}). This is Z2 for the complex numbers, Z2^(2^n-2) for the quaternions and higher. That group's quotient group is GL(n,2).
GL(1,2) = identity group
GL(2,2) = S3 = D3
GL(3,2) = PSL(2,7), a 168-element group that is the second smallest nonabelian simple group. The smallest one is A5 = PSL(2,4) = PSL(2,5).

 

[lpetrich]

Back to group representations.

A common sort of rep in physical applications is the product rep. For reps D1 and D2, the product rep D12 is their outer product:

D12(a)_{(i1,i2),
(j1,j2)} = D1(a)_i1,j1 * D2(a)_i2,j2

In general, product reps are reducible.

If D1 and D2 are the same rep, then the product rep can have various symmetries. For a 2-rep product,
D(a)_{(i1,i2),
(j1,j2),
(s,a)} = (1/2) * (D(a)_{(i1,i2),
(j1,j2)} +- D(a)_{(i1,i2),
(j2,j1)})
s = symmetric, a = antisymmetric

Notice that the second pair of indices got permuted. The permutations and their coefficients:
(1,2) : 1/2
(2,1): +1/2 or -1/2

In general, one can compose an operator for the combined effects of the permutations:
px = sum over p of w(p) * p

From what this operator does, px.px = px -- px must be "idempotent". This gives us a constraint on the w(p)'s:

w(p) = sum over p1,p2 such that p1.p2 = p of w(p1)*w(p2)

In this example, w12 = w12² + w21² and w21 = 2*w12*w21
The solutions: (w1, w2, w3) =
0, 0
1, 0
1/2, 1/2
1/2, -1/2
These solutions are sums of (0 or 1) times each of these solutions: (1/2,1/2) and (1/2,-1/2)

For a 3-rep product, the solutions get much more numerous and complicated, so I'll only do a few of them. I'll have w1 for (1,2,3), the identity permutation, w2 for (2,3,1) and (3,1,2), the two rotations of all three symbols, and w3 for (1,3,2), (3,2,1), and (2,1,3), the three interchanges of two symbols that leave the remaining one fixed. I get these equations:
w1 = w1² + 2*w2² + 3*w3², w2 = 2*w1*w2 + w2³ + 3*w2*w3, w3 = 2*w1*w3 + 4*w2*w3
The solutions: (w1, w2, w3) =
0,0,0
1,0,0
2/3, -1/3, 0
1/3, 1/3, 0
5/6, -1/6, 1/6
5/6, -1/6, -1/6
1/6, 1/6, 1/6
1/6, 1/6, -1/6
These solutions are sums of (0 or 1) times each of these solutions: (1/6,1/6,1/6), (1/6,1/6,-1/6), and (2/3,-1/3,0)

 

[lpetrich]

Finding the group-rep matrices can be a very difficult task, but one can still learn a lot about a group without doing so. A convenient tool for doing so is the traces of the rep matrices, or the "character" of the rep:
χ(a) = Tr(D(a))

Characters have the nice property of being independent of transforms on the rep matrices, and they have additional nice properties. The character of a rep that got decomposed into reps 1, 2, 3, ...
χ(a) = χ1(a) + χ2(a) + χ3(a) + ...

The character of a product rep:
χ12(a) = χ1(a) * χ2(a)

So one can see what irreps are in a product rep.

Looking at rep-power case, for the two-rep case,
χ_w(a) = w12*χ(a)² + w21*χ(a²)

For the three-rep case:
χ_w(a) = w1*χ(a)³ + w2*χ(a³) + w3*χ(a)*χ(a²)

Likewise for higher powers.

A theorem about rep matrices is
sum over a of D1(a)_ij*D2(a^-1)_kl = (N/n)*δ_il*δ_jk for D1 = D2, 0 otherwise

N = order of the group and n = matrix dimension, χ(identity)

For continuous groups, one must replace summing over elements by integrating over parameters. Since we can always change variables, we must also multiply be a quantity called the "Haar measure".

This theorem has an interesting result for groups with unitary representations, like finite groups. For irreps χ1 and χ2:

<χ1(a)*χ2^*(a)>_a = 1 if χ1 = χ2, 0 otherwise

There is a very interesting result about characters. They are vectors over a group's elements, but they are always identical for certain subsets of those elements.

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An interesting result about an irrep's reality: <χ(a²)>_a = 1 (real), -1 (pseudoreal), 0 (complex)

Also, for a character vector χ, its complex conjugate will also be a character vector.

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Let us return to group-element conjugates. Now take the the conjugate of a group element over every element. The result is that element's "conjugacy class". For an abelian group, it contains only that element, while for a nonabelian one, it may contain more. Its order always evenly divides the group's order, with the ratio being the class's index.

If a and b are in the same conjugacy class A, then χ(a) = χ(b) and we can call it χ(A) -- so a character can be interpreted as a vector over conjugacy classes.

There is a remarkable theorem for finite groups: the number of conjugacy classes = the number of irreps. That makes the character vectors form a square matrix. The characters have an additional bit of orthogonality:

sum over all the irreps of χ(A)*χ^*(B) = (index of A = order of group / order of A) if B = A, 0 otherwise.

 

[lpetrich]

For finite groups and some infinite ones, a character satisfies additional properties.

χ(a) = sum of eigenvalues of D(a). Those eigenvalues are roots of unity, and χ(a) is thus an "algebraic integer". An algebraic integer is the root of some polynomial with integer coefficients and with the highest coefficient equal to 1 -- an integer-coefficient "monic polynomial".

(order of A) * χ(A) / χ(identity) for conjugacy class A is also an algebraic integer.

χ(identity) evenly divides the order of the group.

χ(a) = χ(its quotient-group element for some normal subgroup) -- so one can carry over its character values.

Character values also work in subgroups, though an irrep may become reducible in a subgroup.

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Let's now see some character values.

Cyclic group Z:
χ(p,k) = w^p*k
for irrep p from 0 to n - 1
for element k from 0 to n - 1
w is a primitive nth root of unity -- no lower nonnegative power equals 1.

Thus, the identity group has character table {{1}}, Z2 has character table
e, a
1, 1
1, -1
Z3 has character table
e, a, a²
1, 1, 1
1, w, w²
1, w², w
where w³ = 1 and w != 1.

Every finite abelian group is the product of cyclic groups, Thus, Z2*Z2 has character table
e, a, b, a*b
1, 1, 1, 1
1, -1, 1, -1
1, 1, -1, -1
1, -1, -1, 1
This is sometimes called the Viergruppe or four-group.

Here are some characters calculated for some rotation and rotoreflection (rotation-reflection) matrices.

For angle a:
2D rotation: χ(a) = 2*cos(a)
2D rotoreflection: χ(a) = 0
3D rotation: χ(a) = 1 + 2*cos(a)
3D rotoreflection: χ(a) = - (1 + 2*cos(a))

Here are the complete characters for the dihedral group. They are somewhat different for Dih(2n) and Dih(2n+1), so I'll treat them separately. For Dih, the rotation angles a = 2*pi*k/n, and I'll denote them by their k values.

Even case: Dih(2n): conjugacy classes:
Identity
Rotations: k, -k for 0 < k < n
Rotations: n
Rotoreflections: k even
Rotoreflections: k odd
Thus, (n+3) classes and irreps

The table (irreps downward, classes rightward):
1, 1, 1, 1, 1
1, 1, 1, -1, -1
1, (-1)^k, (-1)ⁿ, 1, -1
1, (-1)^k, (-1)ⁿ, -1, 1
2, 2*cos(p*a), (-1)^p, 0, 0
for 0 < p < n

Odd case: Dih(2n+1): conjugacy classes:
Identity
Rotations: k, -k for 0 < k <= n
Rotoreflections
Thus, (n+2) classes and irreps

The table:
1, 1, 1
1, 1, -1
2, 2*cos(p*a), 0
for 0 < p <= n

Dih(1) ~ Z2
Dih(2) ~ Z2*Z2

Thus, Dih(3) has table
Identity, 120d rotations, rotoreflections
1, 1, 1
1, 1, -1
2, -1, 0

Dih(4) has table
Identity, 90d, 180d, even, odd
1, 1, 1, 1, 1
1, 1, 1, -1, -1
1, -1, 1, 1, -1
1, -1, 1, -1, 1
2, 0, 2, 0, 0

Dih(6) has table
Identity, 60d, 120d, 180d, even, odd
1, 1, 1, 1, 1, 1
1, 1, 1, 1, -1, -1
1, -1, 1, -1, 1, -1
1, -1, 1, -1, -1, 1
2, 1, -1, -2, 0, 0
2, -1, -1, 2, 0, 0

 

[lpetrich]

Quaternionic dihedral QDih: character table = Dih(2n) character table for n > 1, even though the groups are not isomorphic.
n = 1 is a special case:
QDih(1) ~ Z4, but Dih(2) ~ Z2*Z2
Character table for Z4:
e, a, a², a³
1, 1, 1, 1
1, i, -1, -i
1, -1, 1, -1
1, -i, -1, i

I'll now take on the tetrahedral, octahedral, and icosahedral groups and their quaternionic extensions.

Tetrahedral:
Identity, 180d (3), 120d (4), 120d (4) = 12
1. 1. 1. 1
1, 1, w, w²
1, 1, w², w
3, -1, 0, 0
w = cube root of unity, the 120d rotations are in two separate classes, depending on whether they are clockwise or counterclockwise with respect to the vertices, or vice versa for faces. The number in parentheses is the size of the conjugacy class.

Quaternionic tetrahedral
Identity, 180d (1), 90d (6), 60d (4), 60d (4), 120d (4), 120d (4) = 24
1, 1, 1, 1, 1, 1, 1
1, 1, 1, w, w², w, w²
1, 1, 1, w², w, w², w
3, 3, -1, 0, 0, 0, 0
2, -2, 0, 1, 1, -1, -1
2, -2, 0, w, w², -w, -w²
2, -2, 0, w², w, -w², -w

Octahedral:
Identity, 180d (3), 120d (8), 90d (6), 180d (6) = 24
1, 1, 1, 1, 1
1, 1, 1, -1, -1
2, 2, -1, 0, 0
3, -1, 0, 1, -1
3, -1, 0, -1, 1
The 180d rotations are divided into two classes: (first) rotations around the edges and (second) rotations around octahedron vertices or cube faces

Quaternionic octahedral:
Identity, 180d (1), 90d (6), 60d (8), 120d (8), 45d (6), 135d (6), 90d (12) = 48
1, 1, 1, 1, 1, 1, 1, 1
1, 1, 1, 1, 1, -1, -1, -1
2, 2, 2, -1, -1, 0, 0, 0
3, 3, -1, 0, 0, 1, 1, -1
3, 3, -1, 0, 0, -1, -1, 1
2, -2, 0, 1, -1, sq2, -sq2, 0
2, -2, 0, 1, -1, -sq2, sq2, 0
4, -4, 0, -1, 1, 0, 0, 0
sq2 = sqrt(2)

Icosahedral:
Identity, 180d (15), 60d (20), 72d (12), 144d (12) = 60
1, 1, 1, 1, 1
3, -1, 0, (1+sq5)/2, (1-sq5)/2
3, -1, 0, (1-sq5)/2, (1+sq5)/2
4, 0, 1, -1, -1
5, 1, -1, 0, 0
sq5 = sqrt(5)

Quaternionic icosahedral:
Identity, 180d (1), 90d (30), 60d (20), 120d(20), 36d (12), 144d (12), 72d (12), 108d (12) = 120
1, 1, 1, 1, 1, 1, 1, 1, 1
3, 3, -1, 0, 0, (1+sq5)/2, (1+sq5)/2, (1-sq5/2), (1-sq5)/2
3, 3, -1, 0, 0, (1-sq5)/2, (1-sq5)/2, (1+sq5/2), (1+sq5)/2
4, 4, 0, 1, 1, -1, -1, -1, -1
5, 5, 1, -1, -1, 0, 0, 0, 0
2, -2, 0, 1, -1, (1+sq5)/2, (-1-sq5)/2, (-1+sq5)/2, (1-sq5)/2
2, -2, 0, 1, -1, (1-sq5)/2, (-1+sq5)/2, (-1-sq5)/2, (1+sq5)/2
4, -4, 0, -1, 1, 1, -1, -1, 1
6, -6, 0, 0, 0, -1, 1, 1, -1

 

[lpetrich]

I'll now consider a complication that arises for some infinite groups. I will consider O(2) and O(1,1) for simplicity, two groups related by analytic continuation.

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Consider the O(2) elements:
D0(a) = {{cos(a), -sin(a)}, {sin(a), cos(a)}} -- rotation: SO(2)
D1(a) = {{cos(a), sin(a)}, {sin(a), -cos(a)}} -- rotoreflection

D0(a) and D1(a) are both unitary, since they are orthogonal, and their eigenvalues are, respectively, e^i*a,e^-i*a, and 1,-1.

The integration measure for integrating over parameters is simply da, where a runs from 0 to 2*pi. Rotations and rotoreflections are weighted the same.

Here are the characters of the irreps:
SO(2): e^i*n*a for all integers n
O(2): {D0, D1}
{1, 1}
{1, -1}
{2*cos(n*a), 0} for positive integers n

So far, so good. SO(2) and O(2) behave much like finite cyclic and dihedral groups. Thus, those groups are compact.

-

Now consider O(1,1).
D00(a) = {{cosh(a), sinh(a)}, {sinh(a), cosh(a)}} -- rotation: SO(1,1)
D01(a) = {{cosh(a), -sinh(a)}, {sinh(a), -cosh(a)}} -- 2nd-dimension reflection
D10(a) = {{-cosh(a), -sinh(a)}, {sinh(a), cosh(a)}} -- 1st-dimension reflection
D11(a) = {{-cosh(a), -sinh(a)}, {-sinh(a), -cosh(a)}} -- both-dimensions reflection: multiplication of rotations by {{-1,0},{0,-1}}
These matrices are only unitary for a = 0.

Their traces are 2*cosh(a), 0, 0, -2*cosh(a), and their eigenvalues are, respectively e^u,e^-u; 1,-1; 1,-1; -e^u,-e^-u

This is unlike finite groups, and this group is thus noncompact.

I'll now guess at the characters of possible irreps of these groups.
SO(2): e^q*u for all real q
O(2): {D00, D01, D10, D11}
{1,1,1,1}
{1,1,-1,-1}
{1,-1,1,-1}
{1,-1,-1,1}
{2*cosh(q*u),0,0,-2*cosh(q*u)} for all positive real q
{2*cosh(q*u),0,0,2*cosh(q*u)} for all positive real q

So instead of a discrete series of irreps, as with O(2) and SO(2), one gets a continuous series.

 

[lpetrich]

I'll now find the conjugacy classes for these groups.

For SO(2), it's trivial. Since it is abelian, every element has its own conjugacy class, a class with only that element in it.

For O(2), the conjugacy classes are {D0(0) = I}, {D0(pi) = -I}, {D0(a),D0(-a)} for all a between 0 and pi, and {D1(a) for all a}

Though both SO(2) and O(2) have uncountable numbers of conjugacy classes, they both have countable numbers of irreps.

-

For SO(1,1), every element is in its own conjugacy class.

I'll consider some in-between groups, then O(1,1)

For the D00-D01 subgroup of O(1,1), the conjugacy classes are {D00(0) = I}, {D00(u), D00(-u)} for all positive u, and {D01(u) for all u}

The D00-D10 subgroup of O(1,1) is analogous, with D10 instead of D00 in the conjugacy classes.

The D00-D11 subgroup of O(1,1) is {I,-I} * SO(1,1), and every element is in its own conjugacy class.

O(1,1) has conjugacy classes {D00(0) = I}, {D00(u), D00(-u)} for all positive u, {D01(u) for all u}, {D10(u) for all u}, {D11(0) = -I}, and {D11(u), D11(-u)} for all positive u.

-

The coset multiplication tables for SO(2) in O(2) and SO(1,1) in O(1,1) are interesting.

SO(2) in O(2):
D0 D1
D1 D0
Group: Z2

SO(1,1) in O(1,1)
D00 D01 D10 D11
D01 D00 D11 D10
D10 D11 D00 D01
D11 D10 D01 D00
Group: Z2 * Z2 -- the 4-group

 

[fast]

If we ever have to take a math test together, I'm gonna cheat off your test.

 

[lpetrich]

Let's see how one might build up to group theory.

One stars with some set S and a binary operation * on it: S*S = S.

If that operation is associative, then we have a "semigroup". I'll now consider some properties of semigroups.

It can have an element that is an identity on one side but not the other, a "left identity" or a "right identity". If it has a left identity, it may also have another one. But if it has both a left identity and a right identity, then they are equal and the only identity.

e1*a = a for all a, a*e2 = a for all a
e1*e2 = e1 from the first definition, e2 from the second one. Thus, e1 = e2.

If there is more than one left identity and a right identity: e11*a = a and e12*a = a, then e11*e2 = e11 = e2 and e12*e2 = e12 = e2
Likewise for a left identity and more than one right identity.

A semigroup can also have a "zero", an element z that makes z as a result of the semigroup operation. A left zero: z1*a = z1, a right zero: a*z1 = z1. If there is both a left zero and a right zero, then they are equal, and the unique zero of the semigroup.

-

A semigroup where the operation has an identity element is called a "monoid". A monoid can have zeros.

A monoid where the operation makes an inverse element for every element is a group. A nontrivial group cannot have zeros.

Let's suppose that a group has one: z*a = z for all a. Since every element has an inverse, we can try a = z^-1, and get e = z. So the only group with a zero is the trivial group, the identity group.

 

[lpetrich]

Let's look at more operations. Consider an analog of addition, +, and an analog of multiplication, *, acting on a set. This is a "ring" if it satisfies these conditions:

• Addition forms an abelian group (identity 0, inverse of a: -a)
• Multiplication is associative
• Multiplication has an identity (1)
• Multiplication is distributive over addition

Some mathematicians define a ring as not necessarily having a multiplicative identity. Relative to a ring with one, such a ring is sometimes called a rng ("rung"; no i) or a pseudoring. But I'll use the multiplicative identity here.

I'll now prove some theorems about rings.

The additive identity is a multiplicative zero: a*0 = 0*a = 0.
Consider a*(b + 0). Use the distributive property: a*b + a*0. But a*(b + 0) = a*b, and thus a*0 = 0. Likewise for 0*a.

If 0 = 1 (both identities equal), then the ring contains only that element, and is thus the "zero ring".
I'll call that element e. For any other element a, e*a = e (additive-identity property), but e*a = a (multiplicative-identity property), thus, a = e.

Something about additive inverses: (-1)*a = a*(-1) = -a.
Use the distributive property: a*(1 + (-1)) = a + a*(-1) = a*0 = 0. Thus a*(-1) = -a, and likewise for (-1)*a.

Every ring contains either the ring of integers Z or the ring of integers modulo some number Z. The modulo value is sometimes called the ring's characteristic. The characteristic of a ring that contains Z is usually set to 0.

Here's the proof. Take 1, and add 1 to itself repeatedly. Do the same with its additive inverse, -1. If one gets zero after adding n 1's or n -1's, one gets Z. If one never gets zero, one gets Z.

Likewise, when a ring contains more elements than its integer subring, those elements all commute with the integer-ring elements under multiplication. Integer rings including modulo-arithmetic integer rings here.

Isn't it fun to prove things?

 

[lpetrich]

Wikipedia lists a chain of ring-like objects starting with commutative rings and ending with algebraic fields.

* A commutative ring is a ring where the multiplication operation is commutative. The integer rings Z and Z are both commutative rings.

* An integral domain has the property that two nonzero elements always have a nonzero product. For a != 0 and b != 0, a*b != 0. It thus has no zero divisors.

Z is an integral domain, but Z is an integral domain only if n is a prime number. The zero ring is not an integral domain.

An integral domain has the "cancellation property": a*b = a*c implies b = c for all a. It's easy to prove that this follows from the lack of zero divisors.

* Integrally closed domains -- I'll pass on them

* A GCD domain havsthe property that any two elements have a nonzero greatest common divisor: a and b have g such that a = g*a1 and b = g*b1

* A unique factorization domain has the property that every element other than 0 can be written as a unique product of a unit element (1, -1, etc.) and some "prime elements".

Here is a ring that is not a UFD: the ring of numbers with the form a + b*sqrt(-5), where a and b are integers. 6 = 2*3 = (1 + sqrt(-5))*(1 - sqrt(-5))

* A principal ideal domain has the property that every ideal is principal, generated by only one element.

A left ideal of a ring: (ideal)*(ring) = (ideal). A right ideal: (ring)*(ideal) = (ideal). If both sides, then a two-sided ideal. An ideal of a commutative ring is always a two-sided ideal.

* A Euclidean domain has a generalization of Euclid's GCD algorithm in it.

* An algebraic field has the property that multiplication over all elements but 0 is an abelian group.

Though the integers do not form a field, the rational numbers, the algebraic numbers, the real numbers, and various other sorts of numbers are all fields.

* A finite field is a field with a finite number of elements. Sometimes called Galois fields, after the famous mathematician Évariste Galois (no, he didn't write down all his mathematical discoveries the night before his fatal duel).