Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac-Moody flag varieties $G/P$.
Our main results are as follows. First, we identify a computationally convenient basis of $H^*_{S^1}(Y)$, which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley-Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both $H^*_{S^1}(Y)$ and $H^*(Y)$ are generated in degree 2. Finally, by using our Chevalley-Monk formula we give explicit descriptions (via generators and relations) of both the $S^1$-equivariant cohomology ring $H^*_{S^1}(Y)$ and the ordinary cohomology ring $H^*(Y)$ of the type $A_{n-1}$ Peterson variety. Our methods are both directly from and inspired by those of GKM (Goresky-Kottwitz-MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.
Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of 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 maximal torus T of G. Our description of the ideal J uses the Cartan matrix and is uniform across Lie types. In our …
We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and structure constants. This generalizes Peterson's isomorphism in (co)homology. We prove a formula for the Pontryagin structure constants in the K-homology ring, and we use it to check our conjecture in few situations.
We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}Σ$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on ${\mathrm{w}}Σ$''. For the weighed Schubert classes in ${\mathrm{w}}Σ$, we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley--Monk's formula.
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Qiuzhen College, Tsinghua University（清华大学丘成桐学院）；University of Science and Technology of China（中国科学技术大学）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）；Center for combinatorics, Nankai University（南开大学组合数学中心）
Peterson varieties are special nilpotent Hessenberg varieties that have appeared in the study of quantum cohomology, representation theory, and combinatorics. In type $A$, the Peterson variety $Y$ is a subvariety of the complete flag variety $Fl(n; \mathbb C)$, and is invariant under the action of a subgroup $S\cong \mathbb C^*$ of $T$, where $T$ is the standard (noncompact) torus acting on $Fl(n; \mathbb C)$. Using the Peterson Schubert basis introduced by Harada and Tymoczko obtained by restricting a specific set of Schubert classes from $H_T^*(Fl(n; \mathbb C))$ to $H_S^*(Y)$, we describe t…
We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form o…
We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in $h_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results by Brion, Knutson, Peterson, Tymoczko and others. We then focus on the equivariant oriented cohomology theory corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar's construction of the pa…
A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action …
Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they describe the tangent space to any Schubert variety in G/B at a point xB, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of…
Let $n$ be a fixed positive integer and $h: \{1,2,...,n\} \rightarrow \{1,2,...,n\}$ a Hessenberg function. The main result of this manuscript is to give a systematic method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with $\mathbb{Q}$ coefficients) of any regular nilpotent Hessenberg variety $\mathrm{Hess}(h)$ in type A. Specifically, we give an explicit algorithm, depending only on the Hessenberg function $h$, which produces the $n$ defining relations $\{f_{h(j),j}\}_{j=1}^n$ in the equivariant cohomology ring. Our resu…
Relativistic Toda lattice of type B and quantum $K$-theory of type C flag variety
Faculty of Science and Engineering, Waseda University（早稻田大学理工学府）；Faculty of Business and Commerce, Keio University（庆应义塾大学商学部）；Department of Mathematics, Institute of Science Tokyo（东京科学大学数学系）
Quantum cohomology, shift operators, and Coulomb branches
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所和数学系）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）
Iwahori-Coulomb branches, stable envelopes, and quantum cohomology of cotangent bundles of flag varieties
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所及数学系）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Department of Mathematics, Virginia Tech（弗吉尼亚理工大学数学系）；The Chinese University of Hong Kong（香港中文大学）
