ACTA issues

The Horn inequalities for submodules

H. Bercovici, K. Dykema, W. S. Li

Acta Sci. Math. (Szeged) 79:1-2(2013), 17-30

Abstract. Consider a torsion module $G$ over a discrete valuation ring ${\eufm O}$, and a submodule $G'\subset G$. It is known that the partitions describing the structure of the modules $G,G',$ and $G/G'$ satisfy the Littlewood--Richardson rule. In particular, these partitions must also satisfy all the Horn inequalities. We show that these inequalities can be obtained directly from the intersection theory of Grassmannians. Moreover, when one of these inequalities is saturated, there is a direct summand $H$ of $G$ such that $H\cap G'$ and $(H+G')/G'$ are direct summands of $G'$ and $G/G'$, respectively. The partitions describing these direct summands correspond precisely to the summands appearing in the saturated Horn inequality. These results apply to those Horn inequalities for which the corresponding Littlewood--Richardson coefficient is 1, and these are sufficient to imply all the others.

AMS Subject Classification (1991): 13F10; 14M15,15A23, 20K01

Keyword(s): Horn inequalities, submodules

Received October 9, 2012, and in revised form February 4, 2013. (Registered under 81/2012.)