四阶实张量（d=4）的行列式相关概念解读及入门文献推荐咨询

Hey there! Let's break down your questions step by step since you're coming from a physics background—no overly abstract math jargon where we can avoid it.

First, let's clear up the link between tensors and hypermatrices: in most contexts, they're essentially the same thing, just named differently by different fields. Mathematicians often use "hypermatrix" to talk about multi-dimensional arrays (like how a matrix is 2D, a hypermatrix is 3D+), while physicists default to "tensor" since we frame them as objects that transform in specific ways under coordinate changes. So when you see references to hyperdeterminants, you can directly map that to tensor determinants.

Now, onto the determinant generalization for higher-rank tensors:

• Cayley's rank-3 matrix (d=2) case: Cayley was one of the first to extend the determinant idea beyond 2D matrices. For a 2x2x2 tensor (rank 3, d=2), he defined a polynomial invariant (something that doesn't change under coordinate transformations) that acts like the determinant—it's a single number that encodes key properties of the tensor, like whether it's "degenerate" (similar to a singular matrix).
• Combinatorial hyperdeterminants: These are built using combinatorial rules, focusing on ways to sum products of tensor components with specific sign factors, similar to how the standard determinant uses permutations. They're useful for counting or classifying tensor structures, but might feel a bit disconnected from physical intuition at first.
• Geometrical hyperdeterminants: This is probably more relevant for your physics work. These are defined based on the geometric interpretation of tensors—think of a tensor as defining a set of linear equations or a geometric object (like a hypersurface in higher dimensions). The hyperdeterminant here tells you about the "non-degeneracy" of that geometric object, or whether the associated system of equations has a unique solution (analogous to how a non-zero determinant means a matrix is invertible).

For a rank-4, d=4 tensor, things get more complex because the hyperdeterminant is a much higher-degree polynomial than the standard determinant. Unlike the 2D matrix determinant (degree 2) or Cayley's 2x2x2 hyperdeterminant (degree 4), your case will have a hyperdeterminant of degree 24—so it's a huge polynomial, but its core purpose is still to encode the tensor's invertibility-like properties and transformation invariants.

Now, for references tailored to a physicist:

• Start with "Tensors: Geometry and Applications" by Jeffrey M. Lee. It balances mathematical rigor with physical intuition, and has a section on hyperdeterminants and tensor invariants that's accessible if you know basic linear algebra and tensor calculus.
• If you want something more focused on the physics side, check out chapters on tensor invariants in "Group Theory in Physics" by Wu-Ki Tung. It connects hyperdeterminants to symmetry transformations, which is super useful for physics applications (like in quantum mechanics or general relativity).
• For the historical context and mathematical foundations, "Hyperdeterminants" by Igor Gelfand, Mikhail Kapranov, and Andrei Zelevinsky is the classic text—just note that it's more math-heavy, so you might want to skim the physics-relevant sections first.

Hope this helps you get started! Let me know if you want to dive deeper into any specific part.

备注：内容来源于stack exchange，提问作者César Riquelme