Independence Complexes of Finite Groups Casey Pinckney Research - PowerPoint PPT Presentation

Independence Complexes of Finite Groups Casey Pinckney Research Advisors: Dr. Alexander Hulpke, Dr. Chris Peterson Colorado State University August 2017 Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Simplicial Complexes Definition V = { v 1 , . . . , v n } finite set of vertices Simplicial complex ∆ on vertex set V (∆) : A collection of subsets F ⊆ V (∆) (called faces ) with: ◮ F ∈ ∆ and H ⊆ F = ⇒ H ∈ ∆ ◮ { v i } ∈ ∆ for all i . Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Simplicial Complexes x 3 x 5 x 4 x 1 x 2 ∆ = {{ x 1 , x 2 , x 3 } , { x 1 , x 2 } , { x 1 , x 3 } , { x 2 , x 3 } , { x 2 , x 4 } , { x 3 , x 4 } , { x 4 , x 5 } , { x 1 } , { x 2 } , { x 3 } , { x 4 } , { x 5 } , ∅} Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Combinatorial Information Record the number of vertices, edges, triangles, and higher-dimensional faces x 3 x 5 x 4 x 1 x 2 f 0 = 5, f 1 = 6, f 2 = 1 Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Euler Characteristic is a Topological Invariant f (∆) = f 0 − f 1 + f 2 f (∆) = f 0 − f 1 + f 2 − f 3 = 4 − 6 + 4 = 4 − 6 + 4 − 1 = 2 = 1 Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Objects of Study Definition G finite group, non-identity elements G ∗ Independent set: S ⊆ G , no proper subset generates the same subgroup Fact Independent sets of G form a simplicial complex on V (∆) = G ∗ Overarching Goal Study combinatorial properties of independent sets of finite groups via simplicial complexes Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Objects of Study First example C p 1 × C p 2 × · · · × C p n for p i distinct primes Goal Count number of faces of each dimension in the simplicial complex Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Examples G = C 2 × C 3 Independent sets of size 1 : 5 (0 , 1) { (1 , 1) } , { (0 , 2) } , { (1 , 0) } , { (0 , 1) } , { (1 , 2) } (1 , 0) Independent sets of size 2 : 2 (1 , 1) Cannot contain (1 , 1) or (1 , 2) (1 , 2) (each generates whole group) (0 , 2) Must have form { ( ⋆, 0) , (0 , ⋆ ) } ( p 1 − 1)( p 2 − 1) = 2 · 1 = 2 Independent sets of size 3 : 0 { ( ⋆, ) , ( , ⋆ ) , ( , ) } Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Examples G = C p 1 × C p 2 × C p 3 Some Independent sets of size 2: { ( ⋆, 0 , 0) , (0 , ⋆, 0) } , . . . { ( ⋆, ⋆, 0) , (0 , 0 , ⋆ ) } , { ( ⋆, ⋆, 0) , ( ⋆, 0 , ⋆ ) } , { ( ⋆, ⋆, 0) , (0 , ⋆, ⋆ ) } , { ( ⋆, 0 , ⋆ ) , (0 , ⋆, 0) } , { ( ⋆, 0 , ⋆ ) , (0 , ⋆, ⋆ ) } , . . . Each tuple has a unique selling point Counting Technique: Generalize techniques of Hearne and Wagner ( Minimal Covers of Finite Sets ) and Clarke ( Covering a Set by Subsets ) Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Count the Number of Independent Sets n = 5 , k = 3, A i := p i − 1 { ( ⋆, ⋆, 0 , 0 , ⋆ ) , (0 , 0 , ⋆, 0 , ⋆ ) , (0 , 0 , 0 , ⋆, 0) } A 1 A 2 | A 3 | A 4 ↓ A 1 A 3 | A 2 | A 4 A 1 A 2 A 5 | A 3 A 5 | A 4 A 1 A 4 | A 2 | A 3 ↓ A 1 | A 2 A 3 | A 4 A 1 A 2 | A 3 | A 4 A 1 | A 2 A 4 | A 3 ↓ A 1 | A 2 | A 3 A 4 St (4 , 3) = 6 counts the number of ways to partition n = 4 letters into k = 3 parts Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Count the Number of Independent Sets Each remaining non-unique variable A j can appear in exactly ◮ 0 blocks in 1 way � k ◮ 2 blocks in � ways, contributes A 1 A 2 A 3 A 4 A 2 j 2 ◮ 3 blocks in � k ways, contributes A 1 A 2 A 3 A 4 A 3 � 3 j . . . � k ◮ k blocks in � = 1 way, contributes A 1 A 2 A 3 A 4 A k k j Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Number of Independent Sets G = C p 1 × C p 2 × · · · × C p n , p i distinct primes Fix n , k . Let A i = p i − 1. St ( m , k )=number of ways to partition an m -element set into k parts Theorem: The number of independent sets of size k in the simplicial complex for G is: n � k � � k � � � � � � A 2 A k � St ( m , k ) 1 + j + · · · + A i j 2 k m = k S ⊆ [ n ] i ∈ S j / ∈ S | S | = m Independence Complexes of Finite Groups Colorado State University Casey Pinckney

Example counts G = C 2 × C 3 × C 5 × C 7 f (∆ G ) = (1 , 209 , 6232 , 4988 , 48) G = C 11 × C 17 × C 19 × C 557 f (∆ G ) = (1 , 1979020 , 43278735636 , 498994428208 , 1601280) Independence Complexes of Finite Groups Colorado State University Casey Pinckney

The End Thank you! Independence Complexes of Finite Groups Colorado State University Casey Pinckney