[PPT] - Character Polynomials Problem From Stanleys Positivity Problems in PowerPoint Presentation

SLIDE 1

Character Polynomials

SLIDE 2

Problem

From Stanley’s Positivity Problems in

Algebraic Combinatorics

Problem 12: Give a combinatorial

interpretation of the row sums of the character table for Sn (combinatorial proof

f non-negativity)

SLIDE 3

Symmetric Group

Sn = permutations of n things
Contains n! elements
S3=permutations of {1,2,3}

(123, 132, 213, 231, 312, 321)

Permutations can be represented with n × n

matrices

Character: trace of a matrix representation
Character Table: table of all irreducible

characters of a group

SLIDE 4

Representations of S3

vertices of an equilateral triangle

1 1        3 2 2 3 1                3 2 1 2 2              

SLIDE 5

Representations of S3

vertices of an equilateral triangle
pick a permutation: 123  312

3 2 1 2 2               3 2 2 3 1                1 1       

SLIDE 6

Representations of S3

vertices of an equilateral triangle
pick a permutation: 123  312

3 2 2 2 1                3 2 1 2 1               3 1       

SLIDE 7

123  312 is 120° CW rotation
Character = Trace =

1 3 2 2 3 1 2 2               

 1 2  1 2  1

Representations of S3

SLIDE 8

Character Table for S3

1,1,1 2,1 3 3 1 1 1 2,1 2

1

1,1,1 1

1

1

SLIDE 9

Character Table for S4

14 2,12 22 3,1 4 1 1 1 1 1 3 1

1
1

2 2

1

3

1
1

1 1

1

1 1

1

SLIDE 10

Character Polynomials

compute characters without matrices
depend only on small parts of the cycle

type

connections to Murnaghan-Nakayama

rule, Schur functions

SLIDE 11

Character Table for S4

14 2,12 22 3,1 4 1 1 1 1 1 3 1

1
1

2 2

1

3

1
1

1 1

1

1 1

1

Sum

5 2 3 2 1

SLIDE 12

Character Polynomials

Partition Polynomial n 1 n-1,1 n-2,2 n-2,12 n-3,3 n-3,2,1 n-3,13 n-4,14 a1 1

SLIDE 13

Character Table for S4

14 2,12 22 3,1 4 1 1 1 1 1 3 1

1
1

2 2

1

3

1
1

1 1

1

1 1

1

SLIDE 14

Character Polynomials

Partition Polynomial n 1 n-1,1 n-2,2 n-2,12 n-3,3 n-3,2,1 n-3,13 n-4,14 a2  a1 1 a1(a1 1) 2 a1 1 a2  a1  a1(a1 1) 2

SLIDE 15

Character Polynomials

Partition Polynomial n 1 n-1,1 n-2,2 n-2,12 n-3,3 n-3,2,1 n-3,13 n-4,14 a2  a1 1 a1(a1 1) 2 a1 1 a2  a1  a1(a1 1) 2 a3  a1a2  a2  a1(a1 1) 2  a1(a1 1)(a1  2) 6 a3  a1(a1 1)  a1(a1 1)(a1  2) 3  a1 a3  a1a2  a2  a1(a1 1) 2  a1(a1 1)(a1  2) 6  a1 1

SLIDE 16

Character Polynomials

Partition Polynomial n 1 n-1,1 n-2,2 n-2,12 n-3,3 n-3,2,1 n-3,13 n-4,14 a2  a1 1 a1(a1 1) 2 a1 1 a2  a1  a1(a1 1) 2 a3  a1a2  a2  a1(a1 1) 2  a1(a1 1)(a1  2) 6 a3  a1(a1 1)  a1(a1 1)(a1  2) 3  a1 a3  a1a2  a2  a1(a1 1) 2  a1(a1 1)(a1  2) 6  a1 1

2 2 2 2 4 1 3 3 1 2 1 1 1 1 1 1 1 1 1 2 1

( 1) ( 1) 2 2 ( 1)( 2)( 3) ( 1)( 2) ( 1) 1 24 6 2 a a a a a a a a a a a a a a a a a a a a a                   

SLIDE 17

Generating Functions and Row Sums

p(n)xn  1 1 xi

i1



n0



1 1 xi 

i1



(1+x+x2+x3+x4+···)(1+x2+x4+···)(1+x3+x6+···)(1+x4+x8+···)+···

1. x4 · 1 · 1 · 1  1,1,1,1
2. x · 1 · x3 · 1  3,1
3. 1 · x4 · 1 · 1  2,2
4. x2 · x2 · 1 · 1  2,1,1
5. 1 · 1 · 1 · x4  4
p(4)=5

Can get x4 from:

SLIDE 18

Example: n-1,1

a1 1

Character Polynomial:

2 2 3

1 2 3 1 x ux u x u ux        

2 2 3 3

1 1 1 ux u x u x ux      

2 3 1

1 2 3 1

u

x x u u x x



       

counts number of 1s!

SLIDE 19

Example: n-1,1

 u (1 ux)1  x(1 ux)2

1 2 1

(1 (1 ) )

u

u x x u x

 

    

p(n)xn  1 1 xi

i1



n0



1 2 1

1 1 1 1 1 1 1

i i i u i

x x u ux x x

  

      

 

SLIDE 20

Example: n-1,1

( 1) ( 2) ( 3)

n

x p n p n p n           

Row Sum= ( 1) ( 2) ( 3) ( ) p n p n p n p n       

1

1 ( ) 1 1 1

n i n i

x x p n x x x x

 

    



2 3

( ) ( )

n n

x x x p n x



   



xn      a1



SLIDE 21

Rows Sums

Row Row Sum

n n-1,1 n-2,2 n-2,12 n-3,3 n-3,2,1 n-3,13

p(n) ( 1) ( 2) ( 3) ( 4) ( 5) ( ) p n p n p n p n p n p n            ( 2) ( 3) 3 ( 4) 3 ( 5) 5 ( 6) ( 1) p n p n p n p n p n p n             ( ) ( 2) ( 3) ( 4) 3 ( 5) 3 ( 6) ( 1) p n p n p n p n p n p n p n              ( 3) 4 ( 5) 7 ( 6) 12 ( 7) 2 ( 2) p n p n p n p n p n           ( 1) ( 2) ( 4) ( 5) 6 ( 6) ( ) p n p n p n p n p n p n            ( 1) ( 4) 5 ( 5) 10 ( 6) ( 2) 2 ( 3) p n p n p n p n p n p n            

SLIDE 22

Growth of p(n)

p(n-1) ≤ p(n) ≤ p(n-1)+p(n-2)

SLIDE 23

Rows Sums

Row Row Sum Positivity

n n-1,1 n-2,2 n-2,12 n-3,3 n-3,2,1 n-3,13

( ) p n ( 1) ( 2) ( 3) ( 4) ( 5) ( ) p n p n p n p n p n p n            ( 2) ( 3) 3 ( 4) 3 ( 5) 5 ( 6) ( 1) p n p n p n p n p n p n             ( ) ( 2) ( 3) ( 4) 3 ( 5) 3 ( 6) ( 1) p n p n p n p n p n p n p n              ( 3) 4 ( 5) 7 ( 6) 12 ( 7) 2 ( 2) p n p n p n p n p n           ( 1) ( 2) ( 4) ( 5) 6 ( 6) ( ) p n p n p n p n p n p n            ( 1) ( 4) 5 ( 5) 10 ( 6) ( 2) 2 ( 3) p n p n p n p n p n p n            

SLIDE 24

Growth of p(n)

p(n-1) ≤ p(n) ≤ p(n-1)+p(n-2)
super-polynomial, sub-exponential
asymptotics good enough to show that

finitely many subtracted terms guaranteed to cancel out for n sufficiently large

( ) ( )

n n

Q x p n x





SLIDE 25

From the bottom up

The sum of the last row is the number of

self-conjugate partitions of n, call this s(n).

Conjugate row obtained by multiplying by

bottom row

SLIDE 26

Character Table for S4

14 2,12 22 3,1 4 1 1 1 1 1 3 1

1
1

2 2

1

3

1
1

1 1

1

1 1

1

SLIDE 27

From the bottom up

The sum of the last row is the number of

self-conjugate partitions of n, call this s(n).

Conjugate row obtained by multiplying by

bottom row

For every row sum formula in terms of

p(n), the conjugate row has the same formula in terms of s(n).

s(n-1) ≤ s(n) ≤ s(n-1)+s(n-2) for n > 1

1

1 ( ) ( ) ( ) 1 ( 1)i

i n n n n i

s n x Q x s x n x

  

   

  

SLIDE 28