The Equivariant Cohomology Rings of Peterson Varieties in All Lie Types

Canadian mathematical bulletin

Megumi Harada¹  Tatsuya Horiguchi²  Mikiya Masuda² 

[1] Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S4K1, Canada;Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan

关键词: equivariant cohomology;    Peterson varieties;    flag varieties;    Monk formula;    Giambelli formula;   

DOI  :  10.4153/CMB-2014-048-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $G$ be a complex semisimple linear algebraic group and let $Pet$ be the associated Peterson variety in the flagvariety $G/B$. The main theorem of this note gives an efficient presentationof the equivariant cohomology ring $H^*_S(Pet)$ of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S cong mathbb{C}^*$ is a certain subgroup of a maximaltorus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda,which was only for Lie type $A$.

【 授权许可】

Unknown