A Quantum Type Deformation of the Cohomology Ring of Flag Manifolds

A.-L. Mare

Journal of Lie theory, Tome 20 (2010) no. 2, pp. 329-346.

Résumé

\def\Z{{\Bbb Z}} Let $q_1, \ldots,q_n$ be some variables and consider the ring $K:=\Z[q_1,\ldots,q_n]/( \prod_{i=1}^n q_i)$. We show that there exists a $K$-bilinear product $\star$ on $H^*(F_n;\Z)\otimes K$ which is uniquely determined by some quantum cohomology like properties (most importantly, a degree two relation involving the generators and an analogue of the flatness of the Dubrovin connection). Then we prove that $\star$ satisfies the Frobenius property with respect to the Poincar\'e pairing of $H^*(F_n;\Z)$; this leads immediately to the orthogonality of the corresponding Schubert type polynomials. We also note that if we pick $k\in\{1,\ldots,n\}$ and we formally replace $q_k$ by 0, the ring $(H^*(F_n;\Z)\otimes K,\star)$ becomes isomorphic to the usual small quantum cohomology ring of $F_n$, by an isomorphism which is described precisely.

Détail

Classification : 05E15, 14M15, 57T15
Mots-clés : Flag manifolds, cohomology, quantum cohomology, periodic Toda lattice, Schubert polynomials