Models and refined models for involutory reflection groups and - PowerPoint PPT Presentation

Models and refined models for involutory reflection groups and classical Weyl groups Models and refined models for involutory reflection groups and classical Weyl groups FABRIZIO CASELLI AND ROBERTA FULCI FPSAC’10 - San Francisco

Models and refined models for involutory reflection groups and classical Weyl groups Plan of the talk 1. Introduce projective reflection groups , a generalization of classical complex reflection groups;

Models and refined models for involutory reflection groups and classical Weyl groups Plan of the talk 1. Introduce projective reflection groups , a generalization of classical complex reflection groups; 2. introduce involutory reflection groups , a family which is bigger than { G ( r , n ) } . Build a Gelfand model for these groups;

Models and refined models for involutory reflection groups and classical Weyl groups Plan of the talk 1. Introduce projective reflection groups , a generalization of classical complex reflection groups; 2. introduce involutory reflection groups , a family which is bigger than { G ( r , n ) } . Build a Gelfand model for these groups; 3. provide a refinement for such model for involutory reflection groups. Two examples: 3a B n ; 3b D n .

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups → 1. Introduce projective reflection groups, a generalization of classical complex reflection groups

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Starting point: complex reflection groups Let V be a ℂ -vector space of finite dimension.

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Starting point: complex reflection groups Let V be a ℂ -vector space of finite dimension. A complex reflection group is a group G < GL ( V ) which is generated by reflections.

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Starting point: complex reflection groups Let V be a ℂ -vector space of finite dimension. A complex reflection group is a group G < GL ( V ) which is generated by reflections. Finite complex reflection groups consist of (Shephard-Todd, 1954; Chevalley, 1955): an infinite family of groups G ( r , p , n ), where p ∣ r ; 34 more sporadic groups.

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Starting point: complex reflection groups Let V be a ℂ -vector space of finite dimension. A complex reflection group is a group G < GL ( V ) which is generated by reflections. Finite complex reflection groups consist of (Shephard-Todd, 1954; Chevalley, 1955): an infinite family of groups G ( r , p , n ), where p ∣ r ; 34 more sporadic groups. Let us focus on G ( r , p , n ).

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , n )? G ( r , p , n ) is the group of monomial matrices such that: the matrix is n × n ; its non-zero entries are r th roots of 1; th root of 1. its permanent is a r p

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , n )? G ( r , p , n ) is the group of monomial matrices such that: the matrix is n × n ; its non-zero entries are r th roots of 1; th root of 1. its permanent is a r p Example ⎛ 0 1 0 0 ⎞ 0 0 0 − 1 ∈ G (2 , 1 , 4) = B 4 ⎜ ⎟ g = ⎜ ⎟ ∈ G (2 , 2 , 4) = D 4 0 0 1 0 / ⎝ ⎠ 1 0 0 0

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , n )? G ( r , p , n ) is the group of monomial matrices such that: the matrix is n × n ; its non-zero entries are r th roots of 1; th root of 1. its permanent is a r p Example ⎛ 0 1 0 0 ⎞ 0 0 0 − 1 ∈ G (2 , 1 , 4) = B 4 ⎜ ⎟ g = ⎜ ⎟ ∈ G (2 , 2 , 4) = D 4 0 0 1 0 / ⎝ ⎠ 1 0 0 0 ⎛ 0 1 0 0 ⎞ − 1 0 0 0 ⎜ ⎟ ⎠ ∈ G (2 , 2 , 4) = D 4 h = ⎜ ⎟ 0 0 − 1 0 ⎝ 1 0 0 0

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Projective reflection groups Let W be a finite complex reflection group.

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Projective reflection groups Let W be a finite complex reflection group. q Id ⟩ , we can consider its quotient W 2 휋 i If W > C q = ⟨ e acting on C q the symmetric power S q ( V ) of V .

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Projective reflection groups Let W be a finite complex reflection group. q Id ⟩ , we can consider its quotient W 2 휋 i If W > C q = ⟨ e acting on C q the symmetric power S q ( V ) of V . Definition (Caselli, 2009) G < GL ( S q ( V )) is a projective reflection group if there exists a reflection group W such that G = W . C q

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups Projective reflection groups Let W be a finite complex reflection group. q Id ⟩ , we can consider its quotient W 2 휋 i If W > C q = ⟨ e acting on C q the symmetric power S q ( V ) of V . Definition (Caselli, 2009) G < GL ( S q ( V )) is a projective reflection group if there exists a reflection group W such that G = W . C q We will focus on the case W = G ( r , p , n ).

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , q , n )? = G ( r , p , n ) G ( r , p , q , n ) def C q

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , q , n )? = G ( r , p , n ) G ( r , p , q , n ) def C q G ( r , p , q , n ) is well defined if:

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , q , n )? = G ( r , p , n ) G ( r , p , q , n ) def C q G ( r , p , q , n ) is well defined if: p ∣ r q ∣ r pq ∣ rn

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , q , n )? = G ( r , p , n ) G ( r , p , q , n ) def C q G ( r , p , q , n ) is well defined if: p ∣ r q ∣ r pq ∣ rn These conditions are symmetric in p and q , so...

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups What is G ( r , p , q , n )? = G ( r , p , n ) G ( r , p , q , n ) def C q G ( r , p , q , n ) is well defined if: p ∣ r q ∣ r pq ∣ rn These conditions are symmetric in p and q , so... G ( r , p , q , n ) exists ⇔ G ( r , q , p , n ) exists .

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups The concept of duality Definition Let G = G ( r , p , q , n ). Its dual group G ∗ is G ( r , q , p , n ).

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups The concept of duality Definition Let G = G ( r , p , q , n ). Its dual group G ∗ is G ( r , q , p , n ). Many algebraic objects related to the group G ( r , p , q , n ) find a natural description via the combinatorics of G ∗ .

Models and refined models for involutory reflection groups and classical Weyl groups 1. Projective reflection groups The concept of duality Definition Let G = G ( r , p , q , n ). Its dual group G ∗ is G ( r , q , p , n ). Many algebraic objects related to the group G ( r , p , q , n ) find a natural description via the combinatorics of G ∗ . For example, its representation theory.

Models and refined models for involutory reflection groups and classical Weyl groups 2. A model for involutory reflection groups → 2. introduce involutory reflection groups. Build a Gelfand model for these groups

Models and refined models for involutory reflection groups and classical Weyl groups 2. A model for involutory reflection groups Gelfand models A Gelfand model of a group G is a G -module containing each irreducible complex representation of G exactly once:

Models and refined models for involutory reflection groups and classical Weyl groups 2. A model for involutory reflection groups Gelfand models A Gelfand model of a group G is a G -module containing each irreducible complex representation of G exactly once: ( M , 휌 ) ∼ ⊕ = ( V 휙 , 휙 ) 휙 ∈ Irr ( G ) Irr ( G ) = { irreducible representations of G } .

Models and refined models for involutory reflection groups and classical Weyl groups 2. A model for involutory reflection groups Gelfand models in recent literature Inglis-Richardson-Saxl, for symmetric groups; Kodiyalam-Verma, for symmetric groups; Aguado-Araujo-Bigeon, for Weyl groups; Baddeley, for wreath products; Adin-Postnikov-Roichman, for the groups G ( r , n )...