Related papers: Relations in the quantum cohomology ring of G/B

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the…

Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a…

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a…

J. Kock has previously defined a tangency quantum product on formal power series with coefficients in the cohomology ring of any smooth projective variety, and thus a ring that generalizes the quantum cohomology ring. We further generalize…

Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M, T)$ a labeled graph $\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^*(\mG_M)$ of…

We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $\mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible…

Let V be a vector space with a nondegenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(OG) and show that its…

In this paper we consider a class of connected closed $G$-manifolds with a non-empty finite fixed point set, each $M$ of which is totally non-homologous to zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With the help of…

Given a flag variety $Fl(n;r_1, \dots , r_\rho)$, there is natural ring morphism from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of…

Let $T$ be a torus and $B$ a compact $T-$manifold. Goresky, Kottwitz, and MacPherson show in \cite{GKM} that if $B$ is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant…

We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…

Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten…

We consider the Belkale-Kumar cup product $\odot_t$ on $H^*(G/P)$ for a generalized flag variety $G/P$ with parameter $t \in \C^m$, where $m=\dim(H^2(G/P))$. For each $t\in \C^m$, we define an associated parabolic subgroup $P_K \supset P$.…

We prove that the generic quantized coordinate ring $\mathcal{O}_q(G)$ is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group $G$. This answers questions raised by Brown, Lenagan, and the first author.…

For an orbifold X which is the quotient of a manifold Y by a finite group G we construct a noncommutative ring with an action of G such that the orbifold cohomology of X as defined in math.AG/0004129 by Chen and Ruan is the G invariant…

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum…

The quantum cohomology of Grassmannians exhibits two symmetries related to the quantum product, namely a \Bbb {Z}/n action and an involution related to complex conjugation. We construct a new ring by dividing out these symmetries in an…

The real singular cohomology ring of a homogeneous space $G/K$, interpreted as the real Borel equivariant cohomology $H^*_K(G)$, was historically the first computation of equivariant cohomology of any nontrivial connected group action.…

We establish a finiteness property of the quantum K-ring of the complete flag manifold.

Let R denote a commutative Noetherian ring and let I be an ideal of R such that H_i^I(R) = 0, for all integers i greater than or equal to 2. In this paper we shall prove some results concerning the homological properties of I.