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# Multiplicity-Free Schubert Calculus

Published:2009-12-04
Printed: Mar 2010
• Hugh Thomas
• Alexander Yong
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## Abstract

Multiplicity-free algebraic geometry is the study of subvarieties $Y\subseteq X$ with the smallest invariants'' as witnessed by a multiplicity-free Chow ring decomposition of $[Y]\in A^{\star}(X)$ into a predetermined linear basis. This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.
 MSC Classifications: 14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14M05 - Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10] 05E99 - None of the above, but in this section