The TIZ-correspondence adjusted for symmetry - Collins, Nathaniel A., author; Wilson, James, advisor; Peterson, Christopher, committee member; Rajopadhye, Sanjay, committee member (Colorado State University. Libraries, 2025)
The TIZ-correspondence ([1], Theorem B) is a ternary Galois correspondence between generalized tensor products, polynomial ideals, and affine schemes of tensor operators. We study the TIZ-correspondence under the presence of symmetry. We provide evidence that this correspondence does not have an internal characterization of symmetry, and we propose three definitions of a generalized symmetric tensor product. For each version, we prove a variant of the TIZ-correspondence in this setting. For the last and most general version, we prove that Lie algebras naturally coordinatize these generalized symmetric tensor products. We prove that every symmetric multilinear map t has a universally smallest generalized tensor product space containing t. We additionally survey the main results in [1]. We give proofs of theorems A-D, demonstrate each theorem with examples, and we provide explicit computations of many of the objects involved.
Keywords: commutative algebra; representation theory; tensors; lie algebras; algebraic geometry; symmetric tensor