Some inequalities in hom sets (Q1181437)

Finite sets, finite Abelian groups, and a few similar categories satisfy a formal Cauchy-Bunjakowkij-Schwarz inequality: \[ |\Hom(X,Y)|\cdot |\Hom(Y,X)|\leq |\Hom(X,X)|\cdot |\Hom(Y,Y)|, \] with equality only if \(X\) and \(Y\)are isomorphic. The author calls this the Cauchy property. Which categories have this property? He describes them in four bunches: i) the categories \(\mathbb{M}_ R\) of finite modules over a commutative ring \(R\) whose ideals are all principal, ii) some categories related to finite sets, iii) finite \(G\)-sets for some groups \(G\) [e.g., \(G=\mathbb{Z}^ 4_ 2\) (this is \(\mathbb{Z}_ 2\oplus\mathbb{Z}_ 2\oplus\mathbb{Z}_ 2\oplus\mathbb{Z}_ 2\)), or \(=\mathbb{Z}_ 8^ 3\), or \(=\mathbb{Z}_ n^ 2\), where \(n\) is the product of the 112 primes between 9000 and 10000, \textit{lack} this property], iv) categories \(\mathbb{M}_ R\), if \(R\) is an algebra over a finite field \(K\), and there are only finitely many indecomposable finite \(R\)-modules \(B_ 1,\dots,B_ n\), and the quadratic form defined by the matrix \((d_{ij})\) of dimensions over \(K\) of \(\Hom(B_ i,B_ j)\) is positive definite. If the Hom-sets are naturally finite-dimensional vector spaces over some field, one can ask if the (extended) Cauchy property holds: \[ d(X,Y)+d(Y,X)\leq d(X,X)+d(Y,Y),\quad\text{strictly less unless }X\approx Y, \] where \(d(X,Y)\) is the dimension of \(\Hom(X,Y)\). All these questions are carefully worked out, becoming at times hard combinatorial problems. There are many examples investigated where the answer is affirmative or negative.