Construction of irreducible relative invariant of the prehomogeneous vector space \((\text{SL}_5 {\times} \text{GL}_4,\Lambda^{2}({\mathbb C}^5)\otimes{\mathbb C}^4)\)
The explicit construction of an irreducible relative invariant on a prehomogeneous vector space is a fundamental problem in the study of invariant theory. For almost all of the irreducible prehomogeneous vector spaces, they are already computed explicitly, but some are left as being unsolved problems. The prehomogeneous vector space \((\text{SL}_5\times \text{GL}_4, \Lambda^2\otimes \Lambda^1)\) is known as one of the difficult cases to compute. The authors of this paper construct the irreducible relative invariant of this prehomogeneous vector space. Their method is based on the construction of an equivariant surjection from \((\text{SL}_5\times \text{GL}_4, \Lambda^2\otimes \Lambda^1)\) to \((\text{GL}_4, 2\Lambda_1S^2 (\mathbb{C}^4))\). The irreducible relative invariant of the former prehomogeneous vector space is obtained as the composition of the surjection and the relative invariant of the latter prehomogeneous vector space.
scientific article; zbMATH DE number 1864742