Valiant's conjecture asserts that the circuit complexity classes VP and VNP are distinct, meaning that the permanent does not admit polynomial-size algebraic circuits. As it is the case in many branches of complexity theory, the unconditional separation of these complexity classes seems elusive. In stark contrast, the symmetric analogue of Valiant's conjecture has been proven by Dawar and Wilsenach (2020): the permanent does not admit symmetric algebraic circuits of polynomial size, while the determinant does. Symmetric algebraic circuits are both a powerful computational model and amenable to proving unconditional lower bounds. In this paper, we develop a symmetric algebraic complexity theory by introducing symmetric analogues of the complexity classes VP, VBP, and VF called symVP, symVBP, and symVF. They comprise polynomials that admit symmetric algebraic circuits, skew circuits, and formulas, respectively, of polynomial orbit size. Having defined these classes, we show unconditionally that $\mathsf{symVF} \subsetneq \mathsf{symVBP} \subsetneq \mathsf{symVP}$. To that end, we characterise the polynomials in symVF and symVBP as those that can be written as linear combinations of homomorphism polynomials for patterns of bounded treedepth and pathwidth, respectively. This extends a previous characterisation by Dawar, Pago, and Seppelt (2026) of symVP. Finally, we show that symVBP and symVP contain homomorphism polynomials which are VBP- and VP-complete, respectively. We give general graph-theoretic criteria for homomorphism polynomials and their linear combinations to be VBP-, VP-, or VNP-complete. These conditional lower bounds drastically enlarge the realm of natural polynomials known to be complete for VNP, VP, or VBP. Under the assumption VFPT $\neq$ VW[1], we precisely identify the homomorphism polynomials that lie in VP as those whose patterns have bounded treewidth.

翻译：Valiant猜想断言电路复杂度类VP与VNP是不同的，这意味着永久函数不存在多项式规模的代数电路。正如复杂度理论的许多分支一样，这些复杂度类的无条件分离似乎难以实现。与此形成鲜明对比的是，Dawar与Wilsenach（2020）证明了Valiant猜想的对称类比：永久函数不存在多项式规模的对称代数电路，而行列式函数却存在。对称代数电路既是一种强大的计算模型，又适合用于证明无条件下界。本文通过引入复杂度类VP、VBP与VF的对称类比——分别称为symVP、symVBP与symVF——来发展对称代数复杂度理论。这些类分别包含可由多项式轨道规模的对称代数电路、斜对称电路与公式计算的多元多项式。在定义这些类的基础上，我们无条件地证明了$\mathsf{symVF} \subsetneq \mathsf{symVBP} \subsetneq \mathsf{symVP}$。为此，我们刻画了symVF与symVBP中的多项式，将其分别表征为有界树深与有界路径宽度的模式所对应的同态多项式的线性组合。这扩展了Dawar、Pago与Seppelt（2026）对symVP的先前刻画。最后，我们证明symVBP与symVP分别包含VBP完全与VP完全的同态多项式。我们给出了同态多项式及其线性组合成为VBP完全、VP完全或VNP完全的一般图论判据。这些条件性下界极大地扩展了已知对VNP、VP或VNP完全的自然多项式范畴。在假设VFPT $\neq$ VW[1]下，我们精确地刻画了属于VP的同态多项式，即那些模式具有有界树宽的同态多项式。