Groups in action or How to count (mod symmetry) Ross Willard - PowerPoint PPT Presentation

Groups in action or How to count (mod symmetry) Ross Willard University of Waterloo University of Northern British Columbia April 8, 2013

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Motivating problem The newest collectible craze sweeping Northern B.C. is a game played on a 4 × 4 red-and-blue checkerboard. The twist: the colour (red/blue) of each square is random. 3 different boards Jennifer is obsessed with this game! Problem : How many different game boards can she collect?

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge First solution 16 squares. or . Each square can be There are 2 16 = 65 , 536 distinct boards. ∴ What is wrong with this solution?

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge ≡ Different pictures, same board. Problem : Different pictures can represent the same board (by rotating), so 2 16 is too high.

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Second solution ≡ ≡ ≡ Each board is represented by 4 pictures. There are 2 16 distinct pictures. There are 2 16 / 4 = 16 , 384 distinct boards. ∴ What is wrong with this solution?

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Problem : Some boards are represented by fewer than 4 pictures. ≡ Only 2 pictures Only 1 picture So 2 16 / 4 = 16 , 384 is too low. (Correct answer: 16 , 456.)

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge The Big Picture · · · · · · ... . . . . . . Let X = { all 4 × 4 red/blue pictures } X is partitioned into sets (or orbits) of size 4, 2 or 1. We want to count the number of orbits in this partition.

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Generalization: group actions Suppose G and X are sets. Definition An operation of G on X is a function ∗ from G × X to X . Example G = R , X = R n , ∗ = scalar multiplication. Visualization: g ∗ x g g G = x = X

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Now suppose G is a group 1 . Definition An operation ∗ of G on X is a group action if it satisfies the following (natural) conditions: for all g , h ∈ G and x , y ∈ X , (A1) e ∗ x = x. (A2) If g ∗ x = y, then g − 1 ∗ y = x. (A3) h ∗ ( g ∗ x ) = ( hg ) ∗ x. h g h G = g − 1 = X e g hg g − 1 hg e x 1 Elements of G can be composed; G contains an identity element e ; every element has an inverse.

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Our motivating problem is an example of a group action. X = { all 4 × 4 red/blue pictures } e R 3 e e R 3 e R R · · · R 3 R 2 R 3 R 3 R 2 R 3 R R R R R R R 2 e R 3 e e R 3 e R R 2 R 3 R 2 R 2 R R 3 e · · · R e R 3 e R 3 e G = { e , R , R 2 , R 3 } , ... . . . . . . the cyclic group of order 4 R gives rotation by 90 ◦ , R 2 gives rotation by 180 ◦ , etc.

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Orbits and Symmetry sets Similarly, in any group action, the set X is partitioned into orbits. Notation For x ∈ X , we use O x to denote the orbit containing x . Definition If g ∈ G and x ∈ X , we say that g is a symmetry of x , or x is an invariant of g , if g ∗ x = x . Definition Given x ∈ X , the symmetry set (or stabilizer ) of x is G x := { g ∈ G : g ∗ x = x } .

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Orbits and Symmetry sets - Example e e e e R 3 R 3 x 3 R 2 R R R R 3 R R 2 R R 3 R 3 R R 2 R R 3 O x 3 R 3 e R R e R 3 e e R 3 e G x 1 = { e , R 2 } R 2 R 3 R 2 R 2 R O x 2 R ... G x 2 = { e , R , R 2 , R 3 } e R 3 e R 3 e G x 3 = { e } x 1 O x 1 x 2 Note: big O x ≡ small G x . Orbit-Symmetry Set Theorem For any group action, for any x ∈ X , | O x | · | G x | = | G | .

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Invariant sets Definition Given g ∈ G , the invariant set of g is X g := { x ∈ X : g ∗ x = x } . Example : our motivating problem · · · · · · X R 2 = · · · What is X R ? What is X R 2 ?

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Burnside’s Lemma Invariant sets give us a slick way to compute the number of orbits. Burnside’s Lemma Let ∗ be a group action of a finite group G on a set X . Then 1 � # of orbits = | X g | . | G | g ∈ G That is, the number of orbits of the action is the average size of the invariant set X g , as g ranges over the group.

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Example: the Motivating Problem G = { e , R , R 2 , R 3 } . X = { all 4 × 4 red/blue pictures } , To use Burnside, we need to know the sizes of the invariant sets. so | X e | = 2 16 . X e = X , { x ∈ X : x is invariant under 90 ◦ rotation } X R =   a b c a     c u u b   = : a , b , c , u ∈ { r , b } b u u c       a c b a 4 independent choices from { r , b } , so | X R | = 2 4 .

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Similarly, { x ∈ X : x is invariant under 180 ◦ rotation } X R 2 =  a b c d      f u v e   = : a , b , c , d , e , f , u , v ∈ { r , b } e v u f       d c b a so | X R 2 | = 2 8 . { x ∈ X : x is invariant under 270 ◦ rotation } X R 3 = so | X R 3 | = 2 4 . = X R ,

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Summary: | X e | = 2 16 , | X R | = | X R 3 | = 2 4 , | X R 2 | = 2 8 . By Burnside’s Lemma, 1 � # of orbits = | X g | | G | g ∈ G 1 = 4( | X e | + | X R | + | X R 2 | + | X R 3 | ) (2 16 + 2 4 + 2 8 + 2 4 ) / 4 = = 16 , 456 . Thus there are 16,456 different game boards for Jennifer to collect.

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge New problem An internet company from Prince George makes stained-glass windows. They are world-famous for their Random TM line of square, 4 × 4 tiled windows such as the one below: Problem : How many 4 × 4 windows of this kind can the company make, using just red and blue glass?

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Similar to original problem, except there is one new dimension of symmetry: “flipping” (front-to-back). We can model this problem using: • The same set X (of 4 × 4 red/blue pictures). • The dihedral group D 4 = { e , R , R 2 , R 3 , H , V , D , D ′ } . D ′ H V D D 4 acts naturally on X . # of distinct windows = # of orbits under the action of D 4 .

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge 1 We can use Burnside’s Lemma: # of orbits = � g ∈ D 4 | X g | . | D 4 | We have already calculated | X e | , | X R | , | X R 2 | and | X R 3 | . Let’s count the pictures stabilized by the new group operations: a b b a a b c d c d d c b u w e V : D : e f f e c w v f g h h g d e f g | X V | = 2 8 | X D | = 2 10 ∴ ∴ Similarly, | X H | = 2 8 Similarly, | X D ′ | = 2 10

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Summary: | X e | = 2 16 , | X R | = | X R 3 | = 2 4 , | X R 2 | = 2 8 , | X H | = | X V | = 2 8 , | X D | = | X D ′ | = 2 10 . Thus by Burnside’s Lemma, 1 � # of distinct windows = | X g | 8 g ∈ D 4 (2 16 + 2 · 2 10 + 3 · 2 8 + 2 · 2 4 ) / 8 = = 8 , 548 .

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Proof of Burnside’s Lemma Given: G a finite group, X a set, ∗ a group action of G on X . Goal: to count the number of orbits. 1 � First observation: # of orbits = | O x | . x ∈ X Proof by example: if X = then 1 � 1 3 + 1 3 + 1 � + 1 � 1 2 + 1 � � | O x | = 1 + = 1 + 1 + 1 = 3 . 3 2 x ∈ X

Motivating problem Group actions Orbits, Symmetries, Invariants Burnside’s Lemma Applications Proof Challenge Next, create a G -by- X table of all the symmetries. x 1 x 2 x 3 · · · x · · · x n − 1 x n g 1 � � � g 2 � � g 3 � � . . . ? g � . . . g m � � Put � in the ( g , x ) position if g ∗ x = x . (Leave blank otherwise.) Question : How many � s are in the table?