Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf ·...

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Probability on Finite GroupsMinkowski functionals and Valuations

ApplicationsAlgebraic and Numeric Programming environments

Aspects of Group Theory in Stochastic Problems

Dr. Marconi BarbosaNICTA/ANU, Canberra, Australia

November 18, 2008

Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems

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SceneFinite Groups crash course outlineMotivational Papers

I Mallow’s Model

I Harmonic analysis on manifolds

I Fourier transforms on groups

I Graph matching: edge info added into node features

I What we do know about Metropolis algorithm? ExactResults?

I Fastest Mixing Markov Chains

I Parallel Coset enumeration

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I Mallow’s Model

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I Mallow’s Model

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I Mallow’s Model

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I Mallow’s Model

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I Mallow’s Model

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I Mallow’s Model

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Outline of what is (would be nice) to come...

I Lagrange Theorem

I Example: Fermat Little theorem and cryptography

I Orbit Counting Theorem

I Example: Cube orbits

I Magic cube group: Scary

I More scary: Baby Monster

I Freaking out: The Monster group

I Group classification: one slide soft crash course

I Group representation: one slide hard crash course

I Invariance, equivalence and symmetry

I Differential invariants, variational problems with symmetry.

I Geometric probability, Minkowski functionals and continuousGroups.

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Lagrange Theorem

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I Verducci & Fligner: Distance Based Ranking Models

I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.

I Fagin: Comparing partial ranks, how efficient?

I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)

I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)

I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)

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Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues

A group G is a set S and a binary operation ∗ satisfying twoproperties:

I A) Closure. For a, b ∈ G , then a ∗ b ∈ G .

I B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) arethe same.

There must be two very special members too:

I C) The identity element e is such that:a ∗ e = e ∗ a = a

I D) The inverse b, for any member a:a ∗ b = b ∗ a = e

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I A subgroup of G , is any subset of elements that still forms agroup.

I Consider a subgroup H of G . Associated with H we createthe following setgH = {g .h : ∀h ∈ H}, This is one of the left cosets of H inG , indexed by g .

Next: An example from integers, the Zn family.

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I A subgroup of G , is any subset of elements that still forms agroup.

I Consider a subgroup H of G . Associated with H we createthe following setgH = {g .h : ∀h ∈ H}, This is one of the left cosets of H inG , indexed by g .

Next: An example from integers, the Zn family.

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Coset example from naturals

Group: G = Z4

Set S : {0, 1, 2, 3}Operation ∗: integer addition mod 4Subgroup: H = {0, 2}0 ∗ H = {0, 2} = H, a trivial coset.1 ∗ H = {1, 3}, first one here2 ∗ H = {2, 4} = {2, 0} = H, trivial again3 ∗ H = {3, 5} = {3, 1}, nothing new...So the subgroup H has only two cosets:H itself and {1, 3}.Note that the cosets form a partition of the group:Z4 = (1 ∗ H)

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The order of every subgroup H divides the order of G .

I Show that all cosets of H have the same order. Definef : aH ⇒ bH by f = ba−1. This is a bijective map with inversef −1 = ab−1.

I Show that cosets from H form a partition of G : Cosets are eitheridentical or disjoint: every element belongs to only one coset.

I Then, the number of elements in G is equal the number of cosets(index) times the number of elements in each coset (which inturn is equal to the order of H)

|G | = |union of its H-cosets|= (number of cosets)*(number of elements in a coset)

= |G : H|.|H|

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= |G : H|.|H|

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= |G : H|.|H|

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= |G : H|.|H|

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Consequences

The order of any element a of a finite group (i.e. the smallest k forwhich ak = e divides G ) divides the order of G . Because the order ofa is the order of a cyclic subgroup generated by a.

I Fermat’s Little theoremap ≡ a (mod p) ;p prime, a integer

I Euler’s Theoremaφ(n) ≡ 1 (mod n); a co-prime nφ(n)(Euler function) counts the number of co-primes from 1 to n.

I Carmichael’s Theorem aλ(n) ≡ 1 (mod n)λ(n)(Carmichael’s function) gives the smallest integer m forwhich a(m) ≡ 1 (mod n)

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Consequences

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Consequences

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Fermat Little theorem proof by group theory

I Basic idea is to recognize G = {1, 2, ..., p − 1}, with theoperation of multiplication mod p as a Group. Some work toprove that every element is invertible

I Assume that a is an element of G and let k be its order. i.e.

I ak ≡ 1 (mod p)

I by Lagrange theorem, k divides the order of G , which is p− 1.So p− 1 = k ∗m then ap−1 = ak∗m = (ak)m = 1m = 1 mod p�

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I ak ≡ 1 (mod p)

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I ak ≡ 1 (mod p)

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I ak ≡ 1 (mod p)

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I Invertibility property.

I Assume b is co-prime (relative prime) to p. Using Bezoutidentity (a linear Diophantine equation)

I bx + py = 1 ( x , y integers)

I bx ≡ 1(modp) x is an inverse for b!

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Orbit counting Theorem

Aliases:Burnside’s LemmaBurnside’s counting theoremThe theorem that is not Burnside’sThe Cauchy-Frobenious lemma, so on.Number of orbits=average number of point fixed by the action ofelements of G.|X/G | = 1

|G |∑

g∈G |X g |

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Coloring

What is the number R of rotationally distinct colorings of a the facesof a cube using 3 colors?Let the X be the set of all 36 = 729 colored cubes. Two elements arein the same orbit precisely when one is a rotation (or composition ofrotations) of the other.

I 1 identity fix all 36 elements of X

I 6 90 degree face rotations fix 33

I 8 120 degree vertex rotation fix 32

I 6 180 edge rotations fix 33