Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

三阶张量被广泛用作在固态物理学中对介质的物理特性建模的数学工具。在大多数情况下，它们是基本物理量之间的比例本构张量出现的。本构张量可以认为是介质物理参数的完整集合。本构张量的代数特征可以用作适当识别天然材料（例如晶体）以及设计具有规定特性的人造纳米材料的工具。在本文中，我们研究了相对于其不变分解的一般不对称三阶张量的代数性质。与作用在基本向量空间上的不同组相对应，我们提出了将不同类型的张量分解成不变的次张量的层次结构。特别是，我们讨论了高阶张量分解的非唯一性和可约性问题。对于一般的不对称三阶张量，将明确描述这些特征。在具有规定对称性的特殊张量的情况下，分解被证明是不可约且唯一的。我们给出了两个物理上有趣的模型的显式结果：压电张量作为对对称的例子，霍尔张量作为对偏对称的例子。