In the paper under review, the authors provide a direct argument for the fact that the invariant factors of the terms of a short exact sequence of finitely generated torsion modules over a discrete valuation ring satisfy the Horn inequalities. More generaly, let \(0 \rightarrow B \rightarrow A \rightarrow C \rightarrow 0\) be an exact sequence of such kind of modules over a principal ideal domain \(D\). One has \(A \simeq \bigoplus_{i=1}^n D/(a_i)\) with \(a_n \mid a_{n-1} \mid \cdots \mid a_1\) and, analogously, for \(B\) and \(C\). The authors show that if \(I\), \(J\), \(K\) are subsets of cardinality \(r\) of \(\{1,\dots ,n\}\), verifying certain condition, and with \(c^{\lambda(I)}_{\lambda(J) \lambda(K)} = 1\) then: \[ \prod_{i\in I}a_i \mid\prod_{j\in J}b_j \cdot \prod_{k\in K}c_k\, . \] Moreover, the authors show that if \(D\) is a discrete valuation ring and, for a fixed triple \((I,J,K)\) as above, the corresponding Horn inequality is, actually, an equality then there exists a direct summand \(A^\prime\) of \(A\), with invariant factors \((\lambda_i)_{i\in I}\), such that \(B':= A'\cap B\) and \(C':=\) the image of \(A'\) in \(C\) are direct summands of \(B\) and, respectively, \(C\), and have invariant factors \((\mu_j)_{j\in J}\) and, respectively, \((\nu_k)_{k\in K}\).
The arguments used by the authors are elementary but the starting point is a difficult result (whose proof uses the honeycomb model of A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)]) due to H. Bercovici, B. Collins, K. Dykema, W. S. Li and D. Timotin [J. Funct. Anal. 258, No. 5, 1579–1627 (2010; Zbl 1196.46048)]. The result asserts that if \(\mathbb F\) is an arbitrary field, if \(E_\bullet\), \(E^{\prime}_\bullet\), \(E''_\bullet\) are arbitrary complete flags in \({\mathbb F}^n\) and if \((I,J,K)\) is a triple as above with \(c^{\lambda(I)}_{\lambda(J)\lambda(K)} = 1\) then: \[ \mathfrak{S}(E_\bullet ,I) \cap \mathfrak{S}({\widetilde E}'_\bullet ,{\widetilde J}) \cap \mathfrak{S}({\widetilde E}''_\bullet ,{\widetilde K}) \neq \emptyset \] (the intersection being taken in the set of \(\mathbb F\)-rational points of \(\text{Gr}(r,{\mathbb F}^n)\)). This fact is, in general, not true if \(c^{\lambda(I)}_{\lambda(J)\lambda(K)} > 1\).