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Peterson varieties are subvarieties of flag varieties. In this note we givean explicit presentation of the (equivariant) cohomology ring of theintersections of Peterson varieties and some Richardson varieties in terms ofgenerators and relations. In particular, our presentation for the intersectionsof Peterson varieties and Schubert varieties generalizes a previous theorem ofHarada-Horiguchi-Masuda, which gives an explicit presentation of the(equivariant) cohomology ring of Peterson varieties.

The cohomology rings of intersections of Peterson varieties and Richardson varieties