设$E$是任意域$K$上特殊线性群$\mathrm{SL}_2(K)$的自然表示.利用$K$具有素特征时对称幂的两个对偶构造，构造了一个显式同构$\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \mathrm{Sym}_\ell \mathrm{Sym}^m E$。这将厄米互易性推广到任意领域。我们证明了一个类似的明确概括的经典朗斯基同构，即$\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \bigwedge^m \mathrm{Sym}^{\ell+m-1} E$。我们还概括了一个结果首先证明了国王，通过表明，如果$\nabla^λ$是舒尔函子的分区$λ$和$λ^\circ$是补充$λ$在一个矩形与$\ell+1$行，然后$\nabla^λ\mathrm{Sym}^\ell E \cong \nabla^{λ^\circ} \mathrm{Sym}_\ell E$。为了说明这种“plethystic同构”的存在是远远不明显的，我们最终证明，广义$\nabla^λ\mathrm{Sym}^\ell E \cong \nabla^{λ'} \mathrm{Sym}^{\ell + \ell(λ') - \ell(λ)}E$的朗斯基同构，已知持有一个大类的分区在复杂的领域，不概括领域的主要特征，即使在考虑所有可能的对偶。
英文摘要：
Let $E$ be the natural representation of the special linear group $\mathrm{SL}_2(K)$ over an arbitrary field $K$. We use the two dual constructions of the symmetric power when $K$ has prime characteristic to construct an explicit isomorphism $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \mathrm{Sym}_\ell \mathrm{Sym}^m E$. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \bigwedge^m \mathrm{Sym}^{\ell+m-1} E$. We also generalise a result first proved by King, by showing that if $\nabla^λ$ is the Schur functor for the partition $λ$ and $λ^\circ$ is the complement of $λ$ in a rectangle with $\ell+1$ rows, then $\nabla^λ\mathrm{Sym}^\ell E \cong \nabla^{λ^\circ} \mathrm{Sym}_\ell E$. To illustrate that the existence of such `plethystic isomorphisms' is far from obvious, we end by proving that the generalisation $\nabla^λ\mathrm{Sym}^\ell E \cong \nabla^{λ'} \mathrm{Sym}^{\ell + \ell(λ') - \ell(λ)}E$ of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.
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