Generalized Schubert Calculus

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 11 February 2013

Abstract

In this paper we study the $T\text{-equivariant}$ generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the Bott-Samelson classes. After setup of the general framework we compute, for classes of Schubert varieties of complex dimension $\le$ 3 in rank 2 (including $A_2,$ $B_2,$ $G_2$ and $A_1^{\left(1\right)}\text{),}$ moment graph representatives, Pieri-Chevalley formulas and products of Schubert classes. These computations generalize the computations in equivariant K-theory for rank 2 cases which are given in Griffeth-Ram [GRa0405333].

Introduction

This paper is a study of the generalized equivariant cohomology of flag varieties. We set up a general framework for working with the generalized (equivariant) Schubert calculus which allows for detailed study without the need for knowledge of cobordism or generalized cohomology theories. Working in the context of a complex reductive algebraic group $G\text{,}$ the (generalized) flag variety is $G/B\text{,}$ where $B$ is a Borel subgroup containing the maximal torus $T\text{.}$ The equivariant generalized cohomology theory $h_T$ comes with a (formal) group which is used to combinatorially construct the ring $S=h_T\text{(pt).}$ The Borel model presents $h_T(G/B)$ as a ‘coinvariant ring’ $S{\otimes }_{S^{W_0}}S$ and the moment graph model presents $h_T(G/B)$ via the image of the inclusions of the $T\text{-fixed}$ points of $G/B\text{.}$ Special cases of generalized equivariant cohomology theories are ‘ordinary’ cohomology (corresponding to the additive group) and $K\text{-theory}$ (corresponding to the multiplicative group). The universal formal group law corresponds to complex cobordism.

Our work follows papers of Bressler-Evens [BEv0968883, BEv1044959], Calmés-Petrov-Zainoulline [CPZ0905.1341], Harada-Holm-Henriques [HHH0409305], Hornbostel-Kiritchenko [HKi0903.3936], and Kiritchenko-Krishna [KKr1104.1089], which have laid important foundations. Combining these tools we study the equivariant cohomology of the flag varieties, partial flag varieties, and Schubert varieties via the algebraic and combinatorial study of the rings which appear in the Borel model and the moment graph model. In Sections 2.3 and 3 we review the setup for these models and the connection to the (generalized) nil affine Hecke algebra and the BGG-Demazure operators (see also [HLS1208.4114] and [BEv0968883, BEv1044959]).

One of the main points of our work is to shift the focus from Bott-Samelson classes to Schubert classes. In ordinary equivariant cohomology and equivariant K-theory these agree, but in generalized cohomology the Schubert classes and the Bott-Samelson classes usually differ. Since the Schubert varieties are not, in general, smooth it is not even clear how the Schubert classes (the fundamental classes of the Schubert varieties) should be defined. In Section 5 we give explicit examples of “naive pushforwards” and Bott-Samelson classes and explain why neither of these can possibly be the Schubert classes in general. There are several directions to explore in searching for a good way to define Schubert classes:

1. One can take the lead of Borisov-Libgober [BLi0007108] (see also [Tot2330522]), and define the Schubert class $\left[X_w\right]$ as a ‘corrected’ version of the Bott-Samelson class $\left[Z_{\overrightarrow{w}}\right]$ which, in the end, does not depend on the reduced word $\overrightarrow{w}$ chosen for $w\text{.}$ Borisov-Libgober [BLi0007108, Definition 3.1] obtain a correction factor for the elliptic genus from the discrepancies of the components of the exceptional divisor of a resolution of singularities of a variety with at worst log terminal singularities. Recent papers of Anderson-Stapledon [ASt1203.6678] and Kumar-Schwede [KSc1203.6126] explain that Schubert varieties have Kawamata log terminal singularities and analyze the exceptional divisor in the Bott-Samelson resolution. In Section 5 we compute a possible equivariant algebraic cobordism correction factor for the smallest singular (complex dimension 3) Schubert variety in all rank 2 cases. Though the approach of Borisov-Libgober was a motivation for our computations we have not yet understood how to make our computation of the correction factor for equivariant algebraic cobordism relate to the correction suggested by Borisov-Libgober for the elliptic genus.
2. One can try to define the Schubert classes as classes determined, hopefully uniquely, by positivity properties under multiplication. We have not yet managed to make a definition that is satisfying but our computations of Schubert products do display remarkable positivity features.
3. One can try to use the theory of Soergel bimodules (see [Soe2329762]) to pick out particular generators (as $(S,S)\text{bimodules)}$ of the generalized cohomologies of Schubert varieties which serve as Schubert classes. Though we have not had space to exhibit our computations of the algebraic cobordism case of Soergel bimodules in this paper, our preliminary computations show that generalizing the Soergel bimodule theory to the ring $S$ which appears in Theorem 3.1 is useful for obtaining better understanding of the equivariant generalized cohomology of Schubert varieties.

In Section 7 we provide explicit computations of Schubert classes, and products with Schubert classes in the rank 2 cases. Our computations hold for all rank two cases, but we have only given specific results for Schubert classes of Schubert varieties in $G/B$ of (complex) dimension $\le$ 3. In partiuclar, this provides complete results for types $A_2$ and $B_2$ and partial results for $G_2$ and $A_1^{\left(1\right)}\text{.}$

To some extent this paper is a sequel to [GRa0405333]. That paper considers the case of equivariant $K\text{-theory.}$ In retrospect, [GRa0405333] did not capitalize on the full power of the moment graph model, in particular, that the map $\varphi$ in Theorem 3.1 is a ring homomorphism. This key point is the feature which we exploit in this paper to execute computations similar to those in [GRa0405333], but with greater ease and in greater generality.

Acknowledgements.

We thank the Australian Research Council for continuing support of our research under grants DP0986774, DP120101942 and DP1095815. Many thanks to Geordie Williamson, Omar Ortiz, and Martina Lanini for teaching us the theory of moment graphs and this beautiful way of working with $T\text{-equivariant}$ cohomology theories. We thank Alex Ghitza, Matthew Ando, Megumi Harada, Dave Anderson and Michel Brion for helpful conversations. We also thank Craig Westerland for answering many many questions of all shapes and sizes all along the way. It is a pleasure to dedicate this paper to C.S. Seshadri who, for so many years, has provided so much Schubert calculus support and inspiration.

Notes and References

This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.

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