The Kronecker product and the partition algebra Christopher Bowman - PowerPoint PPT Presentation

The Kronecker product and the partition algebra Christopher Bowman Maud De Visscher Rosa Orellana FPSAC’13 Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 1 / 14

The Kronecker problem Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

The Kronecker problem Complex representations of GL n : simple (Weyl) modules V ( λ ) . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

The Kronecker problem Complex representations of GL n : simple (Weyl) modules V ( λ ) . � c ν V ( λ ) ⊗ V ( µ ) = λ,µ V ( ν ) , ν where c ν λ,µ are the Littlewood-Richardson coefficients. Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

The Kronecker problem Complex representations of GL n : simple (Weyl) modules V ( λ ) . � c ν V ( λ ) ⊗ V ( µ ) = λ,µ V ( ν ) , ν where c ν λ,µ are the Littlewood-Richardson coefficients. Complex representations of S n : simple (Specht) modules S ( λ ) . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

The Kronecker problem Complex representations of GL n : simple (Weyl) modules V ( λ ) . � c ν V ( λ ) ⊗ V ( µ ) = λ,µ V ( ν ) , ν where c ν λ,µ are the Littlewood-Richardson coefficients. Complex representations of S n : simple (Specht) modules S ( λ ) . � g ν S ( λ ) ⊗ S ( µ ) = λ,µ S ( ν ) , ν where g ν λ,µ are the Kronecker coefficients. Combinatorial description of g ν λ,µ ? Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

Schur-Weyl dualities Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

Schur-Weyl dualities Let V n be an n -dimensional C -vector space and r ≥ 1. Then we have the following Schur-Weyl dualities Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

Schur-Weyl dualities Let V n be an n -dimensional C -vector space and r ≥ 1. Then we have the following Schur-Weyl dualities V ⊗ r → ← GL n C S r Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

Schur-Weyl dualities Let V n be an n -dimensional C -vector space and r ≥ 1. Then we have the following Schur-Weyl dualities V ⊗ r → ← GL n C S r ∪ ∩ P r ( n ) Partition algebra S n Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

Schur-Weyl dualities Let V n be an n -dimensional C -vector space and r ≥ 1. Then we have the following Schur-Weyl dualities V ⊗ r → ← GL n C S r ∪ ∩ P r ( n ) Partition algebra S n Idea: Use the partition algebra to study the Kronecker problem. Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

Structure of the talk The partition algebra P r ( n ) : Definition and first properties. 1 Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 4 / 14

Structure of the talk The partition algebra P r ( n ) : Definition and first properties. 1 Combinatorial representation theory of P r ( n ) . 2 Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 4 / 14

Structure of the talk The partition algebra P r ( n ) : Definition and first properties. 1 Combinatorial representation theory of P r ( n ) . 2 Application to the Kronecker problem. 3 Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 4 / 14

1. The partition algebra P r ( n ) Definition and first properties Let r ∈ Z > 0 and n ∈ C . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

1. The partition algebra P r ( n ) Definition and first properties Let r ∈ Z > 0 and n ∈ C . P r ( n ) : C -algebra with basis given by all set partitions of { 1 , 2 , . . . , r , 1 ′ , 2 ′ , . . . , r ′ } . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

1. The partition algebra P r ( n ) Definition and first properties Let r ∈ Z > 0 and n ∈ C . P r ( n ) : C -algebra with basis given by all set partitions of { 1 , 2 , . . . , r , 1 ′ , 2 ′ , . . . , r ′ } . {{ 1 , 2 , 4 , 3 ′ } , { 3 } , { 5 , 1 ′ , 2 ′ } , { 4 ′ } , { 5 ′ }} Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

1. The partition algebra P r ( n ) Definition and first properties Let r ∈ Z > 0 and n ∈ C . P r ( n ) : C -algebra with basis given by all set partitions of { 1 , 2 , . . . , r , 1 ′ , 2 ′ , . . . , r ′ } . 1 2 3 4 5 {{ 1 , 2 , 4 , 3 ′ } , { 3 } , { 5 , 1 ′ , 2 ′ } , { 4 ′ } , { 5 ′ }} ↔ 1 ′ 2 ′ 3 ′ 4 ′ 5 ′ Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

and multiplication given by concatenation and scalar multiplication by n t where t is the number of connected components consisting of middle vertices only. Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

and multiplication given by concatenation and scalar multiplication by n t where t is the number of connected components consisting of middle vertices only. X = Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

and multiplication given by concatenation and scalar multiplication by n t where t is the number of connected components consisting of middle vertices only. X = Y = Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

and multiplication given by concatenation and scalar multiplication by n t where t is the number of connected components consisting of middle vertices only. X = Y = XY = n Remark: The group algebra C S r appears naturally as a subalgebra of P r ( n ) (as the span of all diagrams having precisely r ‘propagating blocks’). Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

Assume throughout this talk that n � = 0. Write P r = P r ( n ) . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

Assume throughout this talk that n � = 0. Write P r = P r ( n ) . e 2 = e . e = 1 . . . n Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

Assume throughout this talk that n � = 0. Write P r = P r ( n ) . e 2 = e . e = 1 . . . n eP r e ∼ P r / P r eP r ∼ = P r − 1 , = C S r . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

Assume throughout this talk that n � = 0. Write P r = P r ( n ) . e 2 = e . e = 1 . . . n eP r e ∼ P r / P r eP r ∼ = P r − 1 , = C S r . Let L be a simple P r -module. Then either eL = 0 and so L is a simple C S r -module, or eL � = 0 and so eL is a simple P r − 1 -module. Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

Assume throughout this talk that n � = 0. Write P r = P r ( n ) . e 2 = e . e = 1 . . . n eP r e ∼ P r / P r eP r ∼ = P r − 1 , = C S r . Let L be a simple P r -module. Then either eL = 0 and so L is a simple C S r -module, or eL � = 0 and so eL is a simple P r − 1 -module. Thus we have that the simple P r -modules are indexed by partitions of degree ≤ r . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

P r is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

P r is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ ≤ r = { λ = ( λ 1 , λ 2 , λ 3 , . . . ) , λ 1 ≥ λ 2 ≥ λ 3 ≥ . . . ≥ 0 , � i λ i ≤ r } . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

P r is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ ≤ r = { λ = ( λ 1 , λ 2 , λ 3 , . . . ) , λ 1 ≥ λ 2 ≥ λ 3 ≥ . . . ≥ 0 , � i λ i ≤ r } . For each λ ∈ Λ ≤ r we have a cell module ∆ r ( λ ) , obtained by ‘inflating’ the corresponding Specht module. Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

P r is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ ≤ r = { λ = ( λ 1 , λ 2 , λ 3 , . . . ) , λ 1 ≥ λ 2 ≥ λ 3 ≥ . . . ≥ 0 , � i λ i ≤ r } . For each λ ∈ Λ ≤ r we have a cell module ∆ r ( λ ) , obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r , ∆ r ( λ ) = S ( λ ) Specht module. Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

P r is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ ≤ r = { λ = ( λ 1 , λ 2 , λ 3 , . . . ) , λ 1 ≥ λ 2 ≥ λ 3 ≥ . . . ≥ 0 , � i λ i ≤ r } . For each λ ∈ Λ ≤ r we have a cell module ∆ r ( λ ) , obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r , ∆ r ( λ ) = S ( λ ) Specht module. λ ⊢ r − 1, ∆ r ( λ ) = P r e ⊗ P r − 1 S ( λ ) . Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14