Tensor Product Multiplicities via Upper Cluster Algebras Jiarui Fei - PowerPoint PPT Presentation

Tensor Product Multiplicities via Upper Cluster Algebras Jiarui Fei Shanghai Jiao Tong University June 4, 2018

Tensor Product Multiplicities Let G be the connected, simply connected complex algebraic group of type Q . Let V ( λ ) be the irreducible (finite-dimensional) representations of G of highest weight λ . The tensor product of two irreducible representations decomposes as � c λ V ( µ ) ⊗ V ( ν ) = µ,ν V ( λ ) . λ ∈ P + To compute the multiplicity c λ µ,ν is not easy.

The Algebra of Triple-tensor Invariants We consider the algebra of triple-tensor invariants A G := ( k [ G ] U − ⊗ k [ G ] U − ⊗ k [ G ] U ) G . The algebra is multigraded by a triple of dominant weights ( µ, ν, λ ) � C λ µ,ν , λ,µ,ν ∈ P + with the C -dimension of graded component C λ µ,ν equal to c λ µ,ν . It turns out that the algebra A G is an upper cluster algebra!

The Algebra of Triple-tensor Invariants We consider the algebra of triple-tensor invariants A G := ( k [ G ] U − ⊗ k [ G ] U − ⊗ k [ G ] U ) G . The algebra is multigraded by a triple of dominant weights ( µ, ν, λ ) � C λ µ,ν , λ,µ,ν ∈ P + with the C -dimension of graded component C λ µ,ν equal to c λ µ,ν . It turns out that the algebra A G is an upper cluster algebra!

The Quiver ∆ 2 Q The data needed to define an upper cluster algebra is a seed (same as the cluster algebra). The ice quiver ∆ 2 Q in the initial seed can be constructed from the certain category related to Q (ADE quiver). ◮ The vertices of ∆ 2 Q are indecomposable projective presentations P + → P − ; ◮ The arrows of ∆ 2 Q are irreducible morphisms and the AR-translations; Id ◮ The frozen vertices are 0 → P i , P i → 0, and P i − → P i . Theorem (Fei) The algebra A G is an upper cluster algebra C (∆ 2 Q ) .

The Quiver ∆ 2 Q The data needed to define an upper cluster algebra is a seed (same as the cluster algebra). The ice quiver ∆ 2 Q in the initial seed can be constructed from the certain category related to Q (ADE quiver). ◮ The vertices of ∆ 2 Q are indecomposable projective presentations P + → P − ; ◮ The arrows of ∆ 2 Q are irreducible morphisms and the AR-translations; Id ◮ The frozen vertices are 0 → P i , P i → 0, and P i − → P i . Theorem (Fei) The algebra A G is an upper cluster algebra C (∆ 2 Q ) .

The Quiver ∆ 2 Q The data needed to define an upper cluster algebra is a seed (same as the cluster algebra). The ice quiver ∆ 2 Q in the initial seed can be constructed from the certain category related to Q (ADE quiver). ◮ The vertices of ∆ 2 Q are indecomposable projective presentations P + → P − ; ◮ The arrows of ∆ 2 Q are irreducible morphisms and the AR-translations; Id ◮ The frozen vertices are 0 → P i , P i → 0, and P i − → P i . Theorem (Fei) The algebra A G is an upper cluster algebra C (∆ 2 Q ) .

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A Basis of A G ◮ There is a basis of A G parametrized by µ -supported g-vectors. ◮ All such g-vectors lies in a cone, which can be explicitly described by the representation theory of ∆ 2 Q . ◮ For each frozen vertex v of ∆ 2 Q , there is an associated boundary representation T v . ◮ The cone has a hyperplane presentation { x ∈ R (∆ 2 Q ) 0 | Hx ≥ 0 } where the rows of the matrix H are given by the dimension vectors of subrepresentations of T v ’s.

A Basis of A G ◮ There is a basis of A G parametrized by µ -supported g-vectors. ◮ All such g-vectors lies in a cone, which can be explicitly described by the representation theory of ∆ 2 Q . ◮ For each frozen vertex v of ∆ 2 Q , there is an associated boundary representation T v . ◮ The cone has a hyperplane presentation { x ∈ R (∆ 2 Q ) 0 | Hx ≥ 0 } where the rows of the matrix H are given by the dimension vectors of subrepresentations of T v ’s.

A Basis of A G ◮ There is a basis of A G parametrized by µ -supported g-vectors. ◮ All such g-vectors lies in a cone, which can be explicitly described by the representation theory of ∆ 2 Q . ◮ For each frozen vertex v of ∆ 2 Q , there is an associated boundary representation T v . ◮ The cone has a hyperplane presentation { x ∈ R (∆ 2 Q ) 0 | Hx ≥ 0 } where the rows of the matrix H are given by the dimension vectors of subrepresentations of T v ’s.

A Basis of A G ◮ There is a basis of A G parametrized by µ -supported g-vectors. ◮ All such g-vectors lies in a cone, which can be explicitly described by the representation theory of ∆ 2 Q . ◮ For each frozen vertex v of ∆ 2 Q , there is an associated boundary representation T v . ◮ The cone has a hyperplane presentation { x ∈ R (∆ 2 Q ) 0 | Hx ≥ 0 } where the rows of the matrix H are given by the dimension vectors of subrepresentations of T v ’s.

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The Generic Cluster Character Given a vector g ∈ Z (∆ 2 Q ) 0 , we can associate a generic representation M := Coker(g) of (∆ 2 Q , W 2 Q ). The generic character CC maps µ -supported g-vectors (lattice points in the cone) to the upper cluster algebra. CC (g) = x g � Gr e ( M ) y e . � � χ e Theorem (Fei) The generic cluster character maps the lattice points in the cone to a basis of A G . Corollary The multiplicity c λ µ,ν is counted by lattice points in the fibre polytope of the cone defined by ( µ, ν, λ ) .

The Generic Cluster Character Given a vector g ∈ Z (∆ 2 Q ) 0 , we can associate a generic representation M := Coker(g) of (∆ 2 Q , W 2 Q ). The generic character CC maps µ -supported g-vectors (lattice points in the cone) to the upper cluster algebra. CC (g) = x g � Gr e ( M ) y e . � � χ e Theorem (Fei) The generic cluster character maps the lattice points in the cone to a basis of A G . Corollary The multiplicity c λ µ,ν is counted by lattice points in the fibre polytope of the cone defined by ( µ, ν, λ ) .

The Generic Cluster Character Given a vector g ∈ Z (∆ 2 Q ) 0 , we can associate a generic representation M := Coker(g) of (∆ 2 Q , W 2 Q ). The generic character CC maps µ -supported g-vectors (lattice points in the cone) to the upper cluster algebra. CC (g) = x g � Gr e ( M ) y e . � � χ e Theorem (Fei) The generic cluster character maps the lattice points in the cone to a basis of A G . Corollary The multiplicity c λ µ,ν is counted by lattice points in the fibre polytope of the cone defined by ( µ, ν, λ ) .

Thank you Happy Birthday to Professor Chari!