Journal of Lie Theory 20 (2010), No. 2, 329--346
Copyright Heldermann Verlag 2010
A Quantum Type Deformation of the Cohomology Ring of Flag Manifolds
Augustin-Liviu Mare
Dept. of Mathematics and Statistics, University of Regina, Regina SK, Canada S4S 0A2
mareal@math.unregina.ca
[Abstract-pdf]
\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.
Keywords: Flag manifolds, cohomology, quantum cohomology, periodic Toda lattice, Schubert polynomials.
MSC: 05E15, 14M15, 57T15
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