Highest weight vectors of tensors

We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We describe the algebraic relations for classical constructions of corresponding spanning sets of highest weight vectors. The proof is based on important duality that we discover for these highest weight spaces and vectors. As applications of duality, we also give conceptual interpretations to power expansions of Cayley’s first hyperdeterminant and its dual exterior form.

💡 Research Summary

The paper investigates highest weight vectors (HWVs) in the symmetric and alternating tensor algebras of the space V = (ℂⁿ)^{⊗d} under the natural action of the group G = GL(n,ℂ)^d. For a d‑tuple of partitions λ = (λ^{(1)},…,λ^{(d)}) of a total weight m, the authors define the HWV spaces
HWV_λ^{Sym} ⊂ Sym^m(V) and HWV_λ^{Alt} ⊂ ∧^m(V). Their dimensions are given by the generalized Kronecker coefficients g(λ).

The core contribution is an explicit, unified construction of spanning sets for these spaces using d‑dimensional hypermatrices. Two families of vectors are introduced:
• Δ_T indexed by N‑valued hypermatrices T with prescribed marginals, and
• ∇_S indexed by (0,1)‑valued hypermatrices S with the same marginals.
Each vector is obtained by applying a symmetric (for Δ) or alternating (for ∇) projection to a basis element built from the standard basis of ℂⁿ. The formulas involve a stabilizer factor and a sign function defined on words of weight λ.

A striking observation is that the parity of d determines a duality between the index sets: for odd d, HWV_λ^{Sym} is spanned by Δ_T with T ∈ T_{01}(λ′) while HWV_λ^{Alt} is spanned by ∇_S with S ∈ T_N(λ′); for even d the roles of N‑ and (0,1)‑hypermatrices are swapped. Here λ′ denotes the conjugate partitions. This leads to two fundamental theorems:

1. Duality Isomorphism (Theorem 1.2). The linear maps Δ : HWV_{λ′}^{Alt} → HWV_λ^{Sym} and ∇ : HWV_{λ′}^{Sym} → HWV_λ^{Alt} (or their even‑d analogues) are bijections, sending the basis element ∧_eS to ∇_S and the basis element _eT to Δ_T.

2. Coefficient Duality (Theorem 1.3). Define a(T,S)=⟨Δ_T, _eS⟩ and b(S,T)=⟨∇_S, ^eT⟩. For odd d, a(T,S)=±b(S,T); for even d, a(T,S)=±a(S,T) and b(S,T)=±b(T,S). These coefficients are signed sums over intersections of double cosets of Young subgroups, and the matrix {a(T,S)} has rank exactly g(λ). Consequently the coefficients encode not only the dimensions of HWV spaces but also refined combinatorial data such as dimensions of symmetric‑group irreducibles, counts of d‑dimensional Latin hypercubes, and d‑dimensional Alon–Tarsi numbers.

Using the boundary operator ∂^{(ℓ)}_{ij} on hypermatrices, the authors derive a complete set of linear relations among the Δ and ∇ vectors (Theorem 1.1). For each ℓ∈