E0053 definition

Permanent pairing

(1) Definition. The permanent pairing of \(\SS^k(V^*) \times \SS^k(V) \to \IR\) is

\[ (\alpha_1 \cdots \alpha_k, v_1 \cdots v_k) = \sum_{\sigma \in S_k} \prod_i \alpha_i(v_{\sigma(i)}). \]

Distinct degrees pair to zero. The pairing extends to \(\SH(V^*) \times \SS(V) \to \IR\) and to \(\SS(V^*) \times \SH(V) \to \IR\).

(2) Proposition (Non-degeneracy). The permanent pairing is non-degenerate: in a basis \((e_i)\) with dual \((\eps_i)\), \((\eps^\mu, e^\nu) = \nu!\, \delta_{\mu, \nu}\). For finite-dimensional \(V\) it induces \(\SS^k(V^*) \cong \SS^k(V)^*\).

Proof. The pairing matrix is diagonal in the monomial basis, with entry \(\nu!\) at \((\eps^\nu, e^\nu)\). For infinite-dimensional \(V\), the same argument gives injectivity of both adjunction maps \(\SS^k(V^*) \hookrightarrow \SS^k(V)^*\) and \(\SS^k(V) \hookrightarrow \SS^k(V^*)^*\).

(3) Proposition (Filtered duality). For finite-dimensional \(V\): \(\SS(V)^\vee = \SH(V)^\vee = \SS(V^*)\).

Proof. The filtered dual \(\SS(V)^\vee\) consists of functionals vanishing on \(F^k \SS(V)\) for some \(k\), i.e. functionals depending on finitely many graded components. By non-degeneracy, these are exactly \(\SS(V^*)\). The equality \(\SH(V)^\vee = \SS(V)^\vee\) is the general filtered completion result E0050.