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Representations of the symmetric group, the specialization order, systems and Grassmann manifolds. (English) Zbl 0536.20009

This is an expository article discussing various manifestations of the natural partial ordering \(\leq\) on the set of all partitions of a given natural number: for \(i=(i_ 1,i_ 2,...),\quad j=(j_ 1,j_ 2,...)\) \(i\leq j\) if and only if \(i_ 1+...+i_ p\leq j_ 1+...+j_ p\) for all \(p\geq 1\). In fact the authors deal mainly with the reverse order which they call the specialization order. Besides various classical occurrences of these orderings (permutation representations of the symmetric groups, Gerstenhaber-Hesselink Theorem on orbits of nilpotent matrices, Gale-Ryser Theorem etc.) the authors also discuss their role in control theory (completely reachable systems) and show how some of these manifestations of the orderings are intimately related.
Reviewer: T.Józefiak

MSC:

20C30 Representations of finite symmetric groups
06A06 Partial orders, general
05A17 Combinatorial aspects of partitions of integers
14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
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