Canonical summands in the branching of rational Weyl modules

The restriction of the rational Weyl module V_λ from GL (n + 1) ℂ to GL n ℂ has exactly one subrepresentation isomorphic to V_μ for each dominant weight μ interlacing λ. This strengthens an abstract branching equivalence to a uniqueness statement about subspaces inside the given module. Iterating these unique summands is the construction of the canonical lines underlying a Gelfand–Tsetlin basis; choosing vectors and their normalization is additional data.

The multiplicity calculation is made in the space of intertwining maps: its dimension is one for an interlacing weight and zero otherwise. Consequently every nonzero intertwiner from a fixed V_μ has the same image.

References

• W. Fulton and J. Harris, Representation Theory: A First Course (1991), Lecture 15.
• R. Goodman and N. R. Wallach, Symmetry, Representations, and Invariants (2009), Section 8.1.

@[simp]

theorem TauCeti.finrank_intertwiningMap_rationalWeylRep_comp_glBlockSucc {n : ℕ} (l : DominantWeight (n + 1)) (m : DominantWeight n) : Module.finrank ℂ ((rationalWeylRep ℂ n m).IntertwiningMap (MonoidHom.comp (rationalWeylRep ℂ (n + 1) l) (glBlockSucc ℂ n))) = if m ∈ interlacingWeights l then 1 else 0

The multiplicity of V_μ in the restriction of V_λ to GL n ℂ is one exactly when μ interlaces λ, and zero otherwise. This includes arbitrary integer dominant weights.

theorem TauCeti.existsUnique_subrepresentation_rationalWeylRep_comp_glBlockSucc {n : ℕ} (l : DominantWeight (n + 1)) (m : DominantWeight n) (hm : m ∈ interlacingWeights l) : ∃! S : Subrepresentation (MonoidHom.comp (rationalWeylRep ℂ (n + 1) l) (glBlockSucc ℂ n)), Nonempty ((rationalWeylRep ℂ n m).Equiv S.toRepresentation)

An interlacing dominant weight specifies a unique subrepresentation of the restricted Weyl module isomorphic to its Weyl module. The subspace is independent of any branching equivalence.