The cohomology rings of intersections of Peterson varieties and Richardson varieties

Abstract

Peterson varieties are subvarieties of flag varieties. In this note we give an explicit presentation of the (equivariant) cohomology ring of the intersections of Peterson varieties and some Richardson varieties in terms of generators and relations. In particular, our presentation for the intersections of Peterson varieties and Schubert varieties generalizes a previous theorem of Harada-Horiguchi-Masuda, which gives an explicit presentation of the (equivariant) cohomology ring of Peterson varieties. Contents 1. Introduction 1 2. Peterson varieties 4 3. Relations in H * S (Y J I ) 8 4. Proof of Theorem 2.4 12 5. Intersections of Peterson variety and Schubert varieties 15 6. Intersections of Peterson variety and opposite Schubert varieties in type A 17 7. Intersections of Peterson variety and Richardson varieties in type A 25 References 27 2020 Mathematics Subject Classification. 14M15, 55N91. Key words and phrases. flag varieties, Peterson varieties, Richardson varieties, equivariant cohomology. 1 In this manuscript, the cohomology means the singular cohomology with rational coefficients.