Hyperplane arrangements, graphic monoids and moment categories - PowerPoint PPT Presentation

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements, graphic monoids and moment categories Clemens Berger University of Nice-Sophia Antipolis CT 2016 in Halifax August 11, 2016

Hyperplane arrangements, graphic monoids and moment categories Introduction 1 Hyperplane arrangements 2 Graphic monoids 3 Moment categories 4 Unital moment categories 5

Hyperplane arrangements, graphic monoids and moment categories Introduction Purpose of the talk algebraisation (hyperplane arrangements) (graphic monoids) � categorification (graphic monoids) (moment categories) � (unital moment categories) semantics (operads) � Examples (Fox-Neuwirth, Salvetti, McClure-Smith, Berger-Fresse) (braid arrangements) � (symmetric groups) � ( E n -operads)

Hyperplane arrangements, graphic monoids and moment categories Introduction Purpose of the talk algebraisation (hyperplane arrangements) (graphic monoids) � categorification (graphic monoids) (moment categories) � (unital moment categories) semantics (operads) � Examples (Fox-Neuwirth, Salvetti, McClure-Smith, Berger-Fresse) (braid arrangements) � (symmetric groups) � ( E n -operads)

Hyperplane arrangements, graphic monoids and moment categories Introduction Purpose of the talk algebraisation (hyperplane arrangements) (graphic monoids) � categorification (graphic monoids) (moment categories) � (unital moment categories) semantics (operads) � Examples (Fox-Neuwirth, Salvetti, McClure-Smith, Berger-Fresse) (braid arrangements) � (symmetric groups) � ( E n -operads)

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Definition (hyperplane arrangements in R n ) A linear hyperplane arrangement A = { H α ⊂ R n , α ∈ |A|} is essential iff � α ∈|A| H α = (0); Coxeter iff ∀ α, β ∈ |A| : s α ( H β ) ∈ A where s α is the orthogonal reflection with respect to the hyperplane H α . Proposition (Coxeter,Tits) There is a one-to-one correspondence ∼ = (essential Coxeter arrangements) ↔ (finite Coxeter groups) ∼ = A G ↔ G

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Definition (hyperplane arrangements in R n ) A linear hyperplane arrangement A = { H α ⊂ R n , α ∈ |A|} is essential iff � α ∈|A| H α = (0); Coxeter iff ∀ α, β ∈ |A| : s α ( H β ) ∈ A where s α is the orthogonal reflection with respect to the hyperplane H α . Proposition (Coxeter,Tits) There is a one-to-one correspondence ∼ = (essential Coxeter arrangements) ↔ (finite Coxeter groups) ∼ = A G ↔ G

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Definition (hyperplane arrangements in R n ) A linear hyperplane arrangement A = { H α ⊂ R n , α ∈ |A|} is essential iff � α ∈|A| H α = (0); Coxeter iff ∀ α, β ∈ |A| : s α ( H β ) ∈ A where s α is the orthogonal reflection with respect to the hyperplane H α . Proposition (Coxeter,Tits) There is a one-to-one correspondence ∼ = (essential Coxeter arrangements) ↔ (finite Coxeter groups) ∼ = A G ↔ G

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Definition (hyperplane arrangements in R n ) A linear hyperplane arrangement A = { H α ⊂ R n , α ∈ |A|} is essential iff � α ∈|A| H α = (0); Coxeter iff ∀ α, β ∈ |A| : s α ( H β ) ∈ A where s α is the orthogonal reflection with respect to the hyperplane H α . Proposition (Coxeter,Tits) There is a one-to-one correspondence ∼ = (essential Coxeter arrangements) ↔ (finite Coxeter groups) ∼ = A G ↔ G

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Example (symmetric group S 3 and its A S 3 in R 2 ) H + H + [123] 23 12 [132] [213] H − H + · 13 13 [312] [231] H − H − [321] 12 23 Definition (face poset F A ) F A S 3 = { 6 facets of dim 2 , 6 facets of dim 1 , 1 facet of dim 0 }

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Example (symmetric group S 3 and its A S 3 in R 2 ) H + H + [123] 23 12 [132] [213] H − H + · 13 13 [312] [231] H − H − [321] 12 23 Definition (face poset F A ) F A S 3 = { 6 facets of dim 2 , 6 facets of dim 1 , 1 facet of dim 0 }

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Example (symmetric group S 3 and its A S 3 in R 2 ) H + H + [123] 23 12 [132] [213] H − H + · 13 13 [312] [231] H − H − [321] 12 23 Definition (face poset F A ) F A S 3 = { 6 facets of dim 2 , 6 facets of dim 1 , 1 facet of dim 0 }

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).

Hyperplane arrangements, graphic monoids and moment categories Hyperplane arrangements Lemma (face monoid F A with facets x , y , z ) def ⇐ ⇒ ∀ s ∈ x , t ∈ y : s + ǫ ( t − s ) ∈ z for ǫ > 0 small xy = z (0) is neutral element; ∀ x , y ∈ F A ; xyx = xy x ⊂ ¯ y ⇐ ⇒ xy = y ; the univ. comm. quotient of F A is a geometric lattice L A . Definition ( k -th complement of an arrangement) M k ( A ) = R n ⊗ R k − � α ∈|A| H α ⊗ R k Theorem (Orlik-Solomon, Salvetti) L A ( F A ) determines cohomology (homotopy type) of M 2 ( A ).