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Interlacing inequalities. Matrix groups. (English) Zbl 0745.15010
This overview consists of two parts. The first part is about the attempt to construct a unifying theory on the interlacing inequalities. The Carlson-Sá theory is the only attempt so far in this direction and it does not say any thing for the sufficient part. The second part of this paper is concerned with subgroups of the full linear group defined by polynomial identities. But the only progress made in this direction is on the description of nonsingular matrices \(A\) satisfying \(d_ c(AX)=d_ c(X)\) or \(d_ c(AX)=d_ c(XA)\) for every square matrices \(X\), where \(d_ c\) is the Schur function associated to a function \(c: S_ n\to F\).

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
15A30 Algebraic systems of matrices
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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