Young Diagram² Calculator

Young diagrams

diagram λ

diagram μ

⋮⋮ diagram input

Set λ by partition

Set μ by partition

Click a square to set that row length. The page enforces non-increasing row lengths automatically.

⋮⋮basic statistics

⋮⋮littlewood–richardson decomposition

Lie algebra rank fixed
Unchecked: stable n ≫ 0. Checked: tensor product in the fixed-rank Weyl representation ring.

Draw λ and μ, choose stable or fixed-rank mode, then compute.

Fixed-rank mode enumerates dominant candidate highest weights inside the invariant norm bound ‖γ‖ ≤ ‖λ‖ + ‖μ‖, then computes coefficients by Freudenthal characters plus Brauer–Klimyk straightening; the dimension check reports completeness.

⋮⋮lie algebra invariants

Open this card to compute invariants of V^λ at the weight μ.

Uses the selected Dynkin type/rank. The weight μ is interpreted by the same Young-diagram → Dynkin-label convention as the canvas heading.

⋮⋮2d weight slice

Draw λ and μ; this card shows their relative geometry as Lie algebra weights.

λ and μ must generate a 2-dimensional subspace of the weight lattice (i.e. they must be linearly independent) to enable the slice canvas.

⋮⋮slice weights and roots

Show the 2d slice canvas to choose overlay layers.

⋮⋮weight info in 2d slice

Click a lattice point in the 2d slice canvas.

⋮⋮kostka tableaux

Compute all semistandard Young tableaux of shape λ and weight μ. This card is independent of the selected Lie type.

Rows are weakly increasing and columns are strictly increasing; the weight μ = (μ₁, μ₂, …) means entry i appears μᵢ times.

⋮⋮kronecker decomposition

Compute all nonzero Kronecker coefficients g^ν_λ,μ for |λ| = |μ|.

The output lists every nonzero target ν under the current browser cap; no third diagram is required.

⋮⋮plethysm decomposition

Compute all nonzero coefficients in the Schur plethysm s_λ∘s_μ.

The output lists every nonzero target ν under the current browser cap; no third diagram is required.

⋮⋮schur functor

Compute 𝕊^μ(V^λ) for the selected Dynkin type/rank and decompose it as a sum of V^γ.

This card interprets λ as the highest weight of V^λ for the selected Lie algebra, applies the Schur functor with shape μ, then decomposes the resulting character into irreducible representations V^γ.

⋮⋮grassmannian cup product

Presentation:

Compute σ_λ ∪ σ_μ in H*(Gr(r,n)), where r is the number of grid rows and n−r is the number of grid columns.

The Young-diagram presentation uses Littlewood–Richardson coefficients inside the current r × (n−r) rectangle. The Chern presentation uses the Giambelli determinant in Chern classes c_i.

⋮⋮symmetric polynomials

terms

basis

Draw λ and μ to combine their symmetric polynomials.

⋮⋮export

@misc{young_diagram2_calculator,
  title        = {Young Diagram² Calculator},
  author       = {Xiaoxiang Zhou},
  year         = {2026},
  url          = {https://ramified.github.io/web/database/double_young_diagram.html},
  note         = {Browser-side applet for computations with pairs of Young diagrams.}
}