Dual partially harmonic tensors and Brauer-Schur-Weyl duality
Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $$ \mathfrak{B}_n^{(f)} $$ be the two-sided ideal of the Brauer algebra $$ {\mathfrak{B}_n}\left( { - 2m} \right) $$ over K generated by e1e3⋯e2f-1 where 0 ≤ f ≤ [n/2]. Let $$ \mathcal{H}\mathcal{T}_f^{ \otimes n} $$ be the subspace of partial-harmonic tensors of valence f in V⊗n. In this paper we prove that dim$$ \mathcal{H}\mathcal{T}_f^{ \otimes n} $$ and dim $$ {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) $$ are both independent of K, and the natural homomorphism from $$ {\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{(f)}}}} \right.} {\mathfrak{B}_n^{(f)}}} $$ to $$ {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) $$ is always surjective. We show that $$ \mathcal{H}\mathcal{T}_f^{ \otimes n} $$ has a Weyl filtration and is isomorphic to the dual of $$ {{{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} V}} \right.} V}^{ \otimes n}}\mathfrak{B}_n^{\left( {f + 1} \right)} $$ as an $$ {\text{Sp}}(V) - \left( {{\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right.} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right) $$-bimodule. We obtain an $$ {\text{Sp}}(V) - {\mathfrak{B}_n} $$-bimodules filtration of V⊗n such that each successive quotient is isomorphic to some $$ \nabla \left( \lambda \right) \otimes {z_{g,\lambda }}{\mathfrak{B}_n} $$ with λ ⊢ n 2g, ℓ(λ)≤m and 0 ≤ g ≤ [n/2], where ∇(λ) is the co-Weyl module associated to λ and zg,λ is an explicitly constructed maximal vector of weight λ. As a byproduct, we show that each right $$ {\mathfrak{B}_n} $$-module $$ {z_{g,\lambda }}{\mathfrak{B}_n} $$ is integrally defined and stable under base change.