Exterior products, elementary symmetric functions, and the Fischer determinant inequality (Q677811)

Determinant inequalities, such as the inequalities of Hadamard and Fisher, have found various applications within mathematical analysis. The author's inequalities improve dramatically upon those of Hadamard and Fischer, and thus offer the prospect of some progress on conjectures such as Redheffer's. Moreover, his results can be constructed to provide both upper and lower bounds for the determinant function. Section 2 contains proofs and theorems which involve multilinear algebra, particularly the Grassman spaces, and some elementary information about group algebras. In section 3 after defining the function \(\psi_t^k\), the author obtains some of their basic properties, including certain expansion formulas that involve another group of functions, namely the functions \(L_t^k\), which he also introduces in this section. The final result is the proof of the identity \(\psi_{t,k}(\cdot)= [\psi_t^k](\cdot)\). The error function \(\varepsilon_k= \text{det } A^{[k]}\text{det } A^{(k)}-\text{det } A\) may be reexpressed as a symmetric polynomial in certain geometric invariants. In section 4 he carefully examines the invariants themselves, relating them to the geometry underlying the matrix \(A\). An explicit formula, and various of the error function's consequences, are presented in section 5.