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 ( ) be the
Jiarui Fei
Shanghai Jiao Tong University
June 4, 2018
Let G be the connected, simply connected complex algebraic group
representations of G of highest weight λ. The tensor product of two irreducible representations decomposes as V (µ) ⊗ V (ν) =
cλ
µ,νV (λ).
To compute the multiplicity cλ
µ,ν is not easy.
We consider the algebra of triple-tensor invariants AG := (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,
with the C-dimension of graded component C λ
µ,ν equal to cλ µ,ν.
It turns out that the algebra AG is an upper cluster algebra!
We consider the algebra of triple-tensor invariants AG := (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,
with the C-dimension of graded component C λ
µ,ν equal to cλ µ,ν.
It turns out that the algebra AG is an upper cluster algebra!
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;
◮ The frozen vertices are 0 → Pi, Pi → 0, and Pi Id
− → Pi.
Theorem (Fei)
The algebra AG is an upper cluster algebra C(∆2
Q).
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;
◮ The frozen vertices are 0 → Pi, Pi → 0, and Pi Id
− → Pi.
Theorem (Fei)
The algebra AG is an upper cluster algebra C(∆2
Q).
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;
◮ The frozen vertices are 0 → Pi, Pi → 0, and Pi Id
− → Pi.
Theorem (Fei)
The algebra AG is an upper cluster algebra C(∆2
Q).
1,0
4,4
6,6
◮ There is a basis of AG 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 Tv.
◮ 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 Tv’s.
◮ There is a basis of AG 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 Tv.
◮ 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 Tv’s.
◮ There is a basis of AG 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 Tv.
◮ 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 Tv’s.
◮ There is a basis of AG 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 Tv.
◮ 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 Tv’s.
1,0
1,0
1,0
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) = xg
e
χ
Theorem (Fei)
The generic cluster character maps the lattice points in the cone to a basis of AG.
Corollary
The multiplicity cλ
µ,ν is counted by lattice points in the fibre polytope
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) = xg
e
χ
Theorem (Fei)
The generic cluster character maps the lattice points in the cone to a basis of AG.
Corollary
The multiplicity cλ
µ,ν is counted by lattice points in the fibre polytope
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) = xg
e
χ
Theorem (Fei)
The generic cluster character maps the lattice points in the cone to a basis of AG.
Corollary
The multiplicity cλ
µ,ν is counted by lattice points in the fibre polytope