From the Binomial Theorem to Bott periodicity
Basis for dualities?
How does the binomial theorem relate to the triangle of power? (The relationship between a, b, c in {$a=b^c$}?)
Examples
- 2 x 2 square - conceptual factors - "and" and "or"
- eightsome - logical square 2 x 2 vertices and 4 pairs (edges)
- 2 x 2 multiplication
- 3 x 3 factorial experiment - Norman Anderson - multinomial
- girl + boy
- genetics - dominant and recessive
- truth tables
- expand 1.9 to a power = 2 - .01
- |A union B| + |A intersect B| = |A| + |B|
- Russian roulette
- interest rates - and taking the limit - formula for {$e^x$}
- simplexes, cross polytopes, hypercubes, coordinate systems
Infinite Series. Dissecting Hypercubes with Pascal's Triangle.
3 Blue 1 Brown. Derivative formulas through geometry.
Product rule {$u\cdot v + \textrm{d}(u\cdot v) = (u + \textrm{d}u)(v + \textrm{d}v)$}
The power set. The set of all subsets.
- Generalizes to the power object. Related to presheafs and subobject classifiers.
- The inverse image functor of a function between sets has a right adjoint, the universal quantifier between power sets, and likewise a left adjoint, the existential quantifier.
Proof of the binomial theorem
- By bijection
- By induction
- By calculus
Sum of the row is {$2^n$}.
Inclusion-exclusion principle Counting nondivisible integers. Counting derangements. Number of surjective functions. Stirling numbers of the second kind. Rook polynomials. Main proof and algebraic proof. Combinatorial sieve methods.(Brun.)
Directed simplicial complexes
- Binomial distribution
- binomial random variable
- Central limit theorem
- Poisson distribution
- Negative binomial distribution (Meixner polynomials) A discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted r r) occurs. (Russian roulette.) {$P\left(\bigcap_{t\in S} X_t^C\right) = (1-p)^{|S|} = \sum_{n=0}^{|S|} {|S| \choose n} (-p)^n$}. An upper bound for this quantity is {$e^{-p|S|}$}
- Normal distribution
Gaussian binomial coefficient
The coefficient of {$q^r$} in {$\binom{n}{k}_q$} counts the number of inversions, for example, transforming 00011 to 01100 so that 1 is to the left of 0, here in 4 ways.
The coefficient of {$q^r$} in {$\binom{n+m}{m}_q$} counts the number of partitions of r with m or fewer parts each less than or equal to n.
{$\binom{n}{k}_q$} counts the number of k-dimensional vector subspaces of an n-dimensional vector space over {$F_q$} (a Grassmannian).
When k=1 and q=1 we have the Euler characteristic of the complex Grassmannian, and when q=-1, the Euler characteristic of the real Grassmannian.
The number of k-dimensional affine subspaces of {$F_{q^n}$} is {$q^{n-k}\binom{n}{k}_q$}. This allows an interpretation of the identity {$\binom{m}{r}_q=\binom{m-1}{r}_q + q^{m-r}\binom{m-1}{r-1}_q$}.
- Grassmannian Key to Moore's proof of Bott periodicity.
- God's Binomial Identity Expressing a number {$d=d_1*d_2*d_3$} as the sum of signed binomial coefficients {$d=\binom{d_1+d_2+d_3}{3}-\sum_{i<j}\binom{d_i+d_j}{3}+\sum_{i}\binom{d_i}{3}-\binom{0}{3}$}
Finite field with one element
- Michael Attiyah: Classical projective geometry: the twisted cubic, the quadric surface, or the Klein representation of lines in 3-space. These illustrate, respectively, the theory of rational curves, of homogeneous spaces, and of Grassmannians. The classical conic is a rational curve, a quadric and a Grassmannian all in one.
- Any conic lies on some Gr(2,4).
- Richard Borcherds. Algebraic Geometry 20 Grassmannians.
- Dev Sinha. Algebraic Topology From Geometric Perspective
- Andrei Okounkov. Enumerative geometry and geometric representation theory. (Stable envelopes and K-theory.)
- Positive Grassmanian is the subset of the real Grassmannian where all Plücker coordinates are nonnegative
- Amplituhedron By calculating its volume, in effect you’re calculating the amplitude for the given collision.
- Arkani-Hamed and his collaborators had defined the amplituhedron in relation to the positive Grassmannian. They demonstrated that it’s possible to change a positive Grassmannian into the amplituhedron by multiplying it by a type of matrix, effectively providing a mathematical recipe for moving points on the positive Grassmannian over to points on the amplituhedron. As a result, information about the relatively well-studied positive Grassmannian transfers to the relatively unexplored amplituhedron.
- Nima Arkani-Hamed and Thomas Lam in conversation
- A Mathematician’s Unanticipated Journey Through the Physical World
- Lauren K. Williams. The positive Grassmannian, the amplituhedron, and cluster algebras
- Williams wrote a series of papers with the mathematician Sylvie Corteel that explored an unexpected link between the combinatorics of the positive Grassmannian and statistical physics.
- Solitons and the Kadomtsev-Petviashvili equation. {$\partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0$} where {$\lambda=\pm 1$}. A one-to-one relationship between points on the positive Grassmannian and solutions to the KP equation. The large-scale behavior of a wave formation is entirely determined by which cell your point in the positive Grassmannian lies in.
- Planar bicolored graphs.
Bottom-up and top-down duality
Embeddings between the classical groups
Understand concretely the embeddings between the classical groups.
Embeddings over the complex numbers:
{$$U \times U \subset U \subset U \times U$$}
Embeddings over the real numbers or over the quaternions:
{$$O \times O \subset O \subset U\subset \operatorname{Sp} \subset \operatorname{Sp} \times \operatorname{Sp} \subset \operatorname{Sp} \subset U \subset O \subset O \times O$$}
Symmetric spaces
A symmetric space is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. An inversion symmetry is an involution, which in terms of linear algebra is a diagonalizable map whose eigenvalues are all either {$1$} or {$-1$}.
- The isometry group of a Riemannian symmetric space is a Lie group. The isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function.
- The identity component of a topological group G is the connected component {$G_0$} of {$G$} that contains the identity element of the group.
- The action of G on X is called transitive if for any two points {$x, y \in X$} there exists a {$g\in G$} so that {$ g\cdot x=y $}.
- Isotropy is uniformity in all orientations. Given g in G and x in X with {$g\cdot x = x$}, it is said that "x is a fixed point of g" or that "g fixes x". For every x in X, the stabilizer subgroup of G with respect to x (also called the isotropy group) is the set of all elements in G that fix x: {$G_{x}=\{g\in G:g\cdot x=x\}$}
- For a fixed x in X, consider the map {$ f:G\to X $} given by {$ g\mapsto g\cdot x $} By definition the image {$f(G)$} of this map is the orbit {$ G\cdot x $}. {$ f(g)=f(h) $} if and only if g and h lie in the same coset for the stabilizer subgroup {$ G_{x}$}. f induces a bijection between the set {$G/G_{x}$} of cosets for the stabilizer subgroup and the orbit {$ G\cdot x $}, which sends {$ gG_{x}\mapsto g\cdot x $}. This result is known as the orbit-stabilizer theorem.
A symmetric space M is a symmetric space G/K with a compact isotropy group K. G is the identity component of the isometry group of M and is a Lie group acting transitively on M. If we fix some point p of M, then M is diffeomorphic to the quotient G/K, where K denotes the isotropy group of the action of G on M at p.
The irreducible compact Riemannian symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of {$(\mathbf{A}\otimes\mathbf {B})^{n}$}, for normed division algebras A and B. The normed division algebras are {$\mathbb{R},\mathbb{C}, \mathbb{H}, \mathbb{O}$}. A similar construction produces the irreducible non-compact Riemannian symmetric spaces.
There are seven infinite series of compact type where G is a (real) simple Lie group.
Label | G | K | Dimension | Rank | Geometric interpretation |
AI | {$\mathrm{SU}(n)$} | {$\mathrm{SO}(n)$} | {$(n-1)(n+2)/2$} | {$n-1$} | Space of real structures on {$\mathbb{C}^n$} which leave the complex determinant invariant |
AII | {$\mathrm{SU}(2n)$} | {$\mathrm{Sp}(n)$} | {$(n-1)(2n+1) $} | {$n-1$} | Space of quaternionic structures on {$\mathbb{C}^{2n}$} compatible with the Hermitian metric |
AIII | {$\mathrm{SU}(p+q)$} | {$\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(q))$} | {$2pq $} | {$\min(p,q)$} | Grassmannian of complex p-dimensional subspaces of {$\mathbb{C}^{p+q}$} |
BDI | {$\mathrm{SO}(p+q)$} | {$\mathrm{SO}(p) \times \mathrm{SO}(q)$} | {$pq $} | {$\min(p,q)$} | Grassmannian of oriented real p-dimensional subspaces of {$\mathbb{R}^{p+q}$} |
DIII | {$\mathrm{SO}(2n)$} | {$\mathrm{U}(n)$} | {$n(n-1) $} | {$[n/2]$} | Space of orthogonal complex structures on {$\mathbb{R}^{2n}$} |
CI | {$\mathrm{Sp}(n)$} | {$\mathrm{U}(n)$} | {$n(n+1) $} | {$n$} | Space of complex structures on {$\mathbb{H}^n$} compatible with the inner product |
CII | {$\mathrm{Sp}(p+q)$} | {$\mathrm{Sp}(p) \times \mathrm{Sp}(q)$} | {$4pq $} | {$\min(p,q)$} | Grassmannian of quaternionic p-dimensional subspaces of {$\mathbb{H}^{p+q}$} |
Six of these appear in the list of loop spaces. The space AIII does not appear. Also appearing in the list of loop spaces are the Stiefel manifolds. The orthogonal, unitary and symplectic groups are also symmetric spaces, see Eschenburg for these and other examples.
Yongdong Huang. A Uniform Description of Riemannian Symmetric Spaces as Grassmannians Using Magic Square
Suspension functor {$\Sigma$} - Loop space functor {$\Omega$} adjunction
This is a special case of the smash-hom adjunction, which is a special case of the product-hom adjunction.
- Tai-Danae Bradley. What is an adjunction? Part 3.
- Tai-Danae Bradley. One-line proof that the fundamental group of the circle is {$\mathbb{Z}$}.
- Wedge sum If X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints {$x_{0}$} and {$y_{0}$}), then the wedge sum {$X\vee Y$} is the quotient space of the disjoint union of X and Y by the identification {$ x_{0}\sim y_{0}$}.
- Smash product {$ X\wedge Y=(X\times Y)/(X\vee Y)$}
- Product-hom adjunction {$(Y\times\_ ) \vdash \textrm{hom}(Y,\_)$}
- Smash-hom adjunction {$(Y\wedge\_ ) \vdash \textrm{hom}(Y,\_)$}
- Suspension-loop adjunction {$(S^1\wedge\_ ) \vdash \textrm{hom}(S^1,\_)$} where {$S^1$} is the circle.
- Show: The smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere.
- Show: The smash product of two spheres {$S^m$} and {$S^n$} is homeomorphic to the sphere {$S^{m+n}$}.
- The smash product of a space X with a circle is homeomorphic to the reduced suspension of X: {$\Sigma X\cong X\wedge S^1$}
- {$\Sigma^k X\cong X\wedge S^k$}
Loop spaces - symmetric spaces
Loop space | Quotient | Cartan's label | Description |
{$\Omega^0$} | {$\mathbb{Z}\times O/(O \times O)$} | BDI | Real Grassmannian |
{$\Omega^1$} | {$O = (O \times O)/O$} | | Orthogonal group (real Stiefel manifold) |
{$\Omega^2$} | {$O/U$} | DIII | space of complex structures compatible with a given orthogonal structure |
{$\Omega^3$} | {$U/\mathrm{Sp}$} | AII | space of quaternionic structures compatible with a given complex structure |
{$\Omega^4$} | {$\mathbb{Z}\times \mathrm{Sp}/(\mathrm{Sp} \times \mathrm{Sp})$} | CII | Quaternionic Grassmannian |
{$\Omega^5$} | {$\mathrm{Sp} = (\mathrm{Sp} \times \mathrm{Sp})/\mathrm{Sp}$} | | Symplectic group (quaternionic Stiefel manifold) |
{$\Omega^6$} | {$\mathrm{Sp}/U$} | CI | complex Lagrangian Grassmannian |
{$\Omega^7$} | {$U/O$} | AI | Lagrangian Grassmannian |
Sequences in Clifford algebras
Matrices over algebras as follows
{$$\mathbb{C} \oplus \mathbb{C} \subset \mathbb{C} \subset \mathbb{C} \oplus \mathbb{C} $$}
{$$\mathbb{R} \oplus \mathbb{R} \subset \mathbb{R}\subset \mathbb{C}\subset \mathbb{H} \subset \mathbb{H} \oplus \mathbb{H} \subset \mathbb{H} \subset \mathbb{C} \subset \mathbb{R} \subset \mathbb{R} \oplus \mathbb{R} $$}
Dynkin diagrams
Spin
Spin quantum numbers accord with divisions of everything.
- Each type of subatomic particle has fixed spin quantum numbers like 0, 1/2, 1, 3/2, ...
- nullsome : no spin
- onesome: spin 0 : 0
- twosome: spin 1/2 : -1/2, 1/2
- threesome: spin 1 : -1, 0, 1
- foursome: spin 3/2 : -3/2, -1/2, 1/2, 3/2
- fivesome: spin 2 : -2, -1, 0, 1, 2
Spin groups
Spin representations
Spin representations are representations of the spin group {$\textrm{Spin}(n)$}, which is the universal covering space of the special orthogonal group {$\textrm{SO}(n)$}.
Given Lie algebra {$\mathbf{so}(n,\mathbb{C})$} of {$O(n,\mathbb{C}), SO(n,\mathbb{C}), \textrm{Spin}(n,\mathbb{C})$}.
A spin representation S is a real or complex vector space together with a group homomorphism {$\rho$} from {$\textrm{Spin}(n,\mathbb{C})$} or {$\textrm{Spin}(p,q)$} to the general linear group {$\textrm{GL}(S)$} such that the element {$-1$} is not in the kernel of {$\rho$}. This means that the reflection is represented nontrivially.
When n is even, S is not an irreducible representation, but we have {$S_{+}=\wedge ^{\mathrm {even} }W$} and {$S_{-}=\wedge ^{\mathrm {odd} }W$} as invariant subspaces.
Spin representations over complex numbers
Complex representations of the spin group follow a mod-2 Bott periodicity.
- Irreducible complex representations admit a real structure for d=1,2,3 mod 8.
- Irreducible complex representations admit a quaternionic structure for d=5,6,7 mod 8.
Spin representations over real numbers
The type of structure invariant under so(p,q) depends only on the signature p − q modulo 8, and is given by the following table.
p−q mod 8 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Structure | R + R | R | C | H | H + H | H | C | R |
Here R, C and H denote real, hermitian and quaternionic structures respectively, and R + R and H + H indicate that the half-spin representations both admit real or quaternionic structures respectively. The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras.
- A real structure is when there is an invariant complex antilinear map {$r: S → S$} with {$r^2 = id_S$}. The fixed point set of r is then a real vector subspace {$S_\mathbb{R}$} of S with SR ⊗ C = S.
There is an invariant complex antilinear map j: S → S with j2 = −idS. It follows that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure.
There is an invariant complex antilinear map b: S → S∗ that is invertible. This defines a pseudohermitian bilinear form on S and is called a hermitian structure.
Binomial theorem
- The first time we ask yes and no is actually different than the second time we ask it and so on.
- Asking yes and no, repeatedly, has a sort of "octality" where each question takes up positive space and reduces the negative space in the complement. It is as if we have a byte of space (!) to use. And if we fill it up we have to start over with the next byte. And this type of duality of complementarity may be given by an exceptional Lie group/algebra.
- There is another duality of counting forwards and backwards given by the classical Lie groups/algebras.
- Generators of Clifford algebras are an example of the binomial theorem.
- Harley Flanders. Differential Forms with Applications to the Physical Sciences.
https://www.cantorsparadise.com/an-incredibly-useful-generalization-of-the-binomial-theorem-and-its-applications-22d41c1589ce
Idea for Lorentz transformation. Write it out as the generalized binomial theorem (Taylor series) {$(1-x)^{-\frac{1}{2}}=1+\frac{1}{2}x+\frac{3}{8}x^2+... = \sum_{n=0}^{\infty}\frac{(2k)!}{4^k(k!)^2}x^k$} and then we need consider only the initial terms, however many are relevant for the combinatorics, which expresses the generalized binomial theorem. The usual Lorentz transformation only arises in the limit to infinity.
Try to interpret the Gamma function (especially for fractions such as 1/2, pi/sqrt(2), or negative numbers) in terms of the choice function for the binomial theorem.
Pascal's triangle
I am pondering a learning path that would take me and comrades up from the Binomial Theorem through Gaussian binomial coefficients and then Grassmanians to Bott Periodicity. Any thoughts on how to make sense of this? Or any materials to study? I have a new idea that this is about the fact that so-called "independent choices" in are actually not independent cognitively. The first choice provides context for the second choice and so on. This carves out "divisions of everything", or mathematically, chain complexes / exact sequences. I imagine that with the fourth choice we have a short exact sequence and then we start to climb back up the complement so that with the eighth choice we are back to an independent choice. Almost as if we are starting a new byte. :)
Did the second choice take place before or after the first choice? This gives rise to two possibilities: opposites coexist or all is the same.
- How is this related to the Monty Hall problem? Or Bayesian statistics?