We prove that in the algebraic metacomplexity framework, the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. This means that many existing algebraic complexity lower bound proofs can be efficiently converted into isotypic lower bound proofs via highest weight metapolynomials, a notion studied in geometric complexity theory. In the context of algebraic natural proofs, our result means that without loss of generality algebraic natural proofs can be assumed to be isotypic. Our proof is built on the Poincar\'e--Birkhoff--Witt theorem for Lie algebras and on Gelfand--Tsetlin theory, for which we give the necessary comprehensive background.

翻译：我们证明，在代数元复杂度框架下，将元多项式分解为其等型分量可以实现高效计算，即电路规模仅需拟多项式级别的膨胀。这意味着许多现有的代数复杂度下界证明可以通过最高权元多项式（几何复杂度理论中研究的概念）高效转化为等型下界证明。在代数自然证明的语境中，我们的结果表明，在不失一般性的前提下，代数自然证明可被假定为等型证明。我们的证明建立在李代数的庞加莱-伯克霍夫-维特定理以及盖尔范德-采特林理论之上，并为此提供了必要的完整背景知识。