## Journal of the Mathematical Society of Japan

### The equivariant cohomology rings of Peterson varieties

#### Abstract

The main result of this note gives an efficient presentation of the $S^1$-equivariant cohomology ring of Peterson varieties (in type $A$) as a quotient of a polynomial ring by an ideal $J$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Our result simplifies previous presentations given by Harada-Tymoczko and Bayegan-Harada. In particular, our result gives an affirmative answer to a conjecture of Bayegan and Harada that the defining ideal $J$ is generated by quadratics.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 3 (2015), 1147-1159.

Dates
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1438777443

Digital Object Identifier
doi:10.2969/jmsj/06731147

Mathematical Reviews number (MathSciNet)
MR3376581

Zentralblatt MATH identifier
1339.55006

#### Citation

FUKUKAWA, Yukiko; HARADA, Megumi; MASUDA, Mikiya. The equivariant cohomology rings of Peterson varieties. J. Math. Soc. Japan 67 (2015), no. 3, 1147--1159. doi:10.2969/jmsj/06731147. https://projecteuclid.org/euclid.jmsj/1438777443

#### References

• D. Bayegan and M. Harada, A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson verieties, Involve, 5 (2012), 115–132.
• Y. Fukukawa, H. Ishida and M. Masuda, The cohomology ring of the GKM graph of a flag manifold of classical type, Kyoto J. Math., 54 (2014), 653–677.
• Y. Fukukawa, The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$, arXiv:1207.5229.
• V. Guillemin, T. Holm and C. Zara, A GKM description of the equivariant cohomology ring of a homogeneous space, J. Algebraic Combin., 23 (2006), 21–41.
• M. Harada and J. Tymoczko, Poset pinball, GKM-compatible subspaces, and Hessenberg varieties, arXiv:1007.2750.
• M. Harada and J. Tymoczko, A positive Monk formula in the $S^1$-equivariant cohomology of type A Peterson varieties, Proc. London Math. Soc. (3), 103 (2011), 40–72.
• B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight $\rho$, Selecta Math. (N.S.), 2 (1996), 43–91.
• K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc., 16 (2003), 363–392.
• R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition 1996, Birkhäuser, Boston.
• E. Sommers and J. Tymoczko, Exponents for B-stable ideals, Trans. Amer. Math. Soc., 358 (2006), 3493–3509.