ATLAS and CMS

Inclusive & differential cross-section measurements of top-quark pair
production with ATLAS and CMS

Luca Martinelli111Università “La Sapienza” e INFN Roma 1

on behalf the ATLAS and CMS Collaborations222Copyright 2021 CERN for the
benefit of the ATLAS and CMS Collaborations. Reproduction of this article or
parts of it is allowed as specified in the CC-BY-4.0 license.

> Latest results on inclusive top-quark pair production cross-sections are
> presented using collision data collected by ATLAS and CMS experiments at the
> LHC. Inclusive and differential measurements of top-quark pair production
> cross-sections from ATLAS and CMS are presented in the resolved and boosted
> kinematic regions. The cross-sections are measured as a function of various
> kinematic observables of the top quarks, the jets and leptons of the event
> final state.

> PRESENTED AT
>
>
>
>
> $14^{\mathrm{th}}$ International Workshop on Top Quark Physics
> (videoconference), 13–17 September, 2021

##  1 Introduction

The top-quark is the third-generation up-type quark and it is the heaviest
known elementary particle. Good knowledge of the properties of the top-quark
is essential because it introduces sizeable corrections in the electroweak
parameters, with sizeable corrections. A comprehensive set of measurements of
the $t\overline{t}$ production cross-sections ($\sigma_{t\overline{t}}$), both
inclusive and differential, can be used to assess the current level of
understanding of the Standard Model (SM), to perform studies to improve the
Monte Carlo (MC) description of the $t\overline{t}$ final state, thus reducing
the systematic uncertainties of measurements, and also to set limits on the
existence of new physics. ATLAS [1] and CMS [2] experiments at LHC provided a
large number of results during the last years at the $\sqrt{s}=5.02,\ 13$ TeV
in different regions of the phase-space, considering different channels and
topology. In the following, the techniques used by ATLAS and CMS to measure
$\sigma_{t\overline{t}}$ are presented with an overview of the recent results
of the two experiments.

##  2 Inclusive $\sigma_{t\overline{t}}$ measurements at $\sqrt{s}=5.02$ TeV

A measurement of $\sigma_{t\overline{t}}$ at $\sqrt{s}=5.02$ TeV was performed
by both ATLAS [3] and CMS [4].

The ATLAS measurement is made using a sample of $257\ \textrm{pb}^{-1}$ in the
dilepton channel, where the $W$ bosons from both top quarks decay
leptonically. Events are selected requiring an opposite-charge pair of leptons
($ee$, $\mu\mu$ or $e\mu$) and one or two $b$-tagged jets. The
$\sigma_{t\overline{t}}$ measurement is based on counting the number of
$b$-tagged jets and it is extracted simultaneously with a parameter sensitive
to the $b$-tagging efficiency to limit the related uncertainty (0.1%). No
requirements are applied on the number of jets, bringing to very small related
uncertainties (0.03%). The cross-section is $\sigma_{t\overline{t}}=66.0\pm
4.5\textrm{(stat)}\pm 1.6\textrm{(syst)}\pm 1.2\textrm{(lumi)}\pm
0.2\textrm{(beam)}\ \textrm{pb}$, where the first uncertain originates from
statistics, the second from experimental systematics, the third from the
measurement of the integrated luminosity, and the fourth from the beam energy
uncertainty, in perfect agreement with the theoretical NNLO prediction
$\sigma_{t\overline{t}}=68.2\pm 4.8(\textrm{PDF})^{+1.9}_{-2.3}(\alpha_{S})\
\textrm{pb}$.

The CMS measurement is performed with a sample of $302\ \textrm{pb}^{-1}$
using, as ATLAS, the dilepton channel. This analysis is also count-based. It
is combined with the single-lepton channel [5] measurement using the BLUE
combination [6]. The cross-section is measured to be
$\sigma_{t\overline{t}}=63.0\pm 4.1\textrm{(stat)}\pm 3.0(\textrm{syst+lumi})\
\textrm{pb}$, where the first uncertain originates from statistics and the
second from experimental and luminosity systematics, in perfect agreement with
the theoretical NNLO prediction.

##  3 Inclusive $\sigma_{t\overline{t}}$ measurements at $\sqrt{s}=13$ TeV

Using the same technique explained in Section 2, ATLAS uses the dilepton
channel to perform the measurement of $\sigma_{t\overline{t}}$ at the centre-
of-mass energy of 13 TeV [7]. The partial Run-II dataset of $36.1\
\textrm{fb}^{-1}$ collected during the 2015-2016 period is used. This
measurement is limited by the uncertainty on the integrated luminosity (1.9%)
with a small contribution to the uncertainty originating from jets (0.03%). An
impact of 0.67% and 0.52% on the total uncertainty is coming from the
$t\overline{t}$ modelling uncertainties and the normalisation of the tW
background, respectively. The cross-section is measured to be
$\sigma_{t\overline{t}}=826.4\pm 3.6\textrm{(stat)}\pm 11.5\textrm{(syst)}\pm
15.7\textrm{(lumi)}\pm 1.9\textrm{(beam)}\ \textrm{pb}$, where the first
uncertain originates from statistics, the second from experimental
systematics, the third from the measurement of the integrated luminosity, and
the fourth from the beam energy uncertainty, and it is the most precise
inclusive measurement of $\sigma_{t\overline{t}}$ at 13 TeV so far (2.40% of
relative uncertainty). It is in agreement with the theoretical NNLO prediction
$\sigma_{t\overline{t}}=832^{+20}_{-29}(\textrm{scale})\pm
35(\textrm{PDF}+\alpha_{S})\ \textrm{pb}$.

CMS has measured $\sigma_{t\overline{t}}$ in dilepton final states containing
one $\tau$ lepton decaying hadronically and one electron or muon, using $35.9\
\textrm{fb}^{-1}$ of data collected at 13 TeV [8]. This is the first
$\sigma_{t\overline{t}}$ measurement at 13 TeV containing $\tau$ leptons. One
electron or muon, together with at least 3 jets, are required in this
analysis. At least one of the jets must be $b$-tagged and the other two must
be tagged as an hadronic $\tau$ decay. The jet triplet invariant mass is used
to construct the signal-background discriminator:
$D_{jjb}=\sqrt{(m_{W}-m_{jj})^{2}+(m_{t}-m_{jjb})^{2}}$ where $m_{W}$ and
$m_{t}$ are the W boson and top-quark mass respectively, $m_{jj}$ is the
invariant mass of the two non-$b$-tagged jets and $m_{jjb}$ is the invariant
mass of the jet triplet. The cross-section is measured to be
$\sigma_{t\overline{t}}=781\pm 7\textrm{(stat)}\pm 62\textrm{(syst)}\pm
20\textrm{(lumi)}\ \textrm{pb}$, where the first uncertain originates from
statistics, the second from experimental systematics, and the third from the
measurement of the integrated luminosity, and it is dominated by the
systematic uncertainties related to the $\tau$ identification (4.5%) and mis-
identification (2.3%).

A new top-quark pair in the single-lepton channel (lepton+jets) production
measurement has been performed by the ATLAS experiment using the full Run-II
dataset ($139\ \textrm{fb}^{-1}$) at 13 TeV [9]. Exactly one lepton ($e$ or
$\mu$) is required with at least four jets. Three categories are defined
according to the jet and $b$-jet multiplicity in the event: exactly one
$b$-tagged jet plus at least three additional jets (signal region 1, SR1),
exactly two $b$-tagged jets plus exactly two additional jets (SR2) and,
exactly two $b$-tagged jets plus at least three additional jets (SR3). A
profile likelihood approach is used to fit three distributions, one per each
SR: the aplanarity for SR1, the minimum invariant mass of all lepton-jet
combinations in SR2 and the average $\Delta R$ between the three constituent
jets originating from the hadronically decaying top-quark in SR3. The cross-
section is measured to be $\sigma_{t\overline{t}}=830\pm 0.4\textrm{(stat)}\pm
36\textrm{(syst)}\pm 14\textrm{(lumi)}\ \textrm{pb}$, where the first
uncertain originates from statistics, the second from experimental
systematics, and the third from the measurement of the integrated luminosity,
which is systematically limited by the $t\overline{t}$ modelling (2.9%) and by
the jet reconstruction uncertainties (2.6%).

##  4 Differential $\sigma_{t\overline{t}}$ measurements at $\sqrt{s}=13$ TeV

The same analysis technique performed by ATLAS in the dilepton channel
($e\mu$), explained in Section 3, is used to measure single- and double-
differential $\sigma_{t\overline{t}}$ as a function of several lepton
kinematics variables [7], as shown in Figure 1 (left). These measurements show
that generators at NLO plus the parton shower (PS) give a good description of
several observables but Powheg predicts a harder spectrum for the lepton
$\textrm{p}_{T}$ variables.

In the $\sigma_{t\overline{t}}$ measurement in the dilepton channel performed
by CMS [10], in addition to the $e\mu$, the $ee$ and $\mu\mu$ channels are
considered. Double- and triple-differential measurements are performed at
particle-level as shown in Figure 1 (right). The total uncertainties in these
measurements are dominated by the systematic uncertainties and the largest one
is related to the Jet energy scale. The best data-MC agreement is obtained by
the Powheg+Pythia and Powheg+Herwig sample with Powheg+Pythia better in
describing the measurements probing the jet multiplicity and extra radiation
while Powheg+Herwig is better in the ones involving the $\textrm{p}_{T}$
distributions.

Figure 1: left Normalised differential cross-section as a function of lepton
$\textrm{p}_{T}$ measured by ATLAS [7]. right Comparison of the measured
number of jets, invariant mass of the $t\overline{t}$ system and the rapidity
of the $t\overline{t}$ system cross-section to the theoretical predictions
calculated using MC event generators, made by CMS [10].

A new measurement with the full Run-II dataset has been performed by CMS in
the single-lepton channel [11]. The two boosted categories are used for the
background estimation while two resolved categories are used to reconstruct
the system. More than twenty single- or double-differential measurements are
performed with this technique, as shown in Figure 2. Summing all the bins, the
inclusive $\sigma_{t\overline{t}}$ can be extracted: this is the best CMS
measurement of $\sigma_{t\overline{t}}$ so far, with a total relative
uncertainty of 3.2%. The cross-section is measured to be
$\sigma_{t\overline{t}}=791\pm 1\textrm{(stat)}\pm 21\textrm{(syst)}\pm
14\textrm{(lumi)}\ \textrm{pb}$, where the first uncertain originates from
statistics, the second from experimental systematics, and the third from the
measurement of the integrated luminosity. The data-MC agreement for the
differential distributions is good for the Powheg+Pythia and Powheg+Herwig
samples. A softer top $\textrm{p}_{T}$ is observed in data with respect to the
MC predictions. The MC predictions fail to reproduce the jet observables.

Figure 2: Measurements of the $\sigma^{t\overline{t}}_{e/\mu+\textrm{jets}}$
production cross-sections with their total uncertainty obtained as the sum of
the cross-sections in all bins of a distribution as a function of the
kinematic variable used in the determination [11].

The channel in which both top quarks decay hadronically is the most
challenging given the abundant presence of hadrons in the event. The ATLAS
measurement, shown in Figure 3 (left) , is performed requiring at least 6 jets
in the event (with at least 2 $b$-jets) and making some kinematic requirements
on system reconstruction, based on the reconstructed top-quark and W boson
masses [12]. The dominant uncertainties on these measurements are from the PS
and hadronisation modelling and from the multi-jet background estimate. The
differential results show a good agreement with the NLO+PS generator on all
the variables.

Thanks to the high energy provided by the LHC, a new kinematic region can be
investigated, i.e. events with high-$\textrm{p}_{T}$ top quarks. In these
cases, the decays of the top get a large boost and they are all contained in a
cone with a radius of $\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}\sim$1\. The
large-R jets are used to reconstruct the hadronically decays of the top
quarks. Both ATLAS [13] and CMS [14] perform measurements with this type of
events, as shown in Figure 3 (center, right). The ATLAS measurement is
performed in the single-lepton channel and it shows a top $\textrm{p}_{T}$
spectrum softer than MC predictions. If the MC predictions are re-weighted to
the NNLO parton-level, the data-MC agreement improves. The differential
measurements show also that the description of the additional radiation event
is not good for all the tested generators but a good agreement in the shape of
the distribution is visible. The CMS measurements are performed in the single-
lepton channel and in the all-hadronic channel and they show that in both
decay channels the observed absolute cross-sections are significantly lower
than the predictions from theory, while the normalized differential
measurements are well described.

Figure 3: left Particle-level normalised single-differential cross-section as
a function of the azimuthal separation $\Delta\Phi^{t\overline{t}}$ between
the two top-quark candidates, measured by ATLAS [12].(center) Single-
differential cross-section as a function of the hadronic top-quark
$\textrm{p}_{T}$, measured by ATLAS [13].right Differential cross-section
unfolded to the particle-level as a function of the leading top-quark
$\textrm{p}_{T}$, measured by CMS [14].

##  5 Conclusions

Both ATLAS and CMS experiments at LHC have presented a significant number of
inclusive and differential measurements of $\sigma_{t\overline{t}}$, with
unprecedented precision reached in several channels. All the inclusive
measurements are in agreement with the NNLL + NNLO calculations. Important
differential measurements were made in all the channels both at particle- and
parton-level. Future combinations of ATLAS and CMS differential results will
represent a powerful instrument to verify the goodness of several MC
generators and to improve the current knowledge on parton density functions
inside protons.

## References

  * [1] ATLAS Collaboration.  The ATLAS Experiment at the CERN Large Hadron Collider.  JINST, 3:S08003, 2008.
  * [2] CMS Collaboration.  The CMS Experiment at the CERN LHC.  JINST, 3:S08004, 2008.
  * [3] ATLAS Collaboration.  Measurement of the $t\overline{t}$ production cross-section using dilepton events in $pp$ collisions at $\sqrt{s}=5.02$ TeV with the ATLAS detector.  2021\.  https://cds.cern.ch/record/2754223/files/ATLAS-CONF-2021-003.pdf.
  * [4] CMS Collaboration.  Measurement of the inclusive $t\overline{t}$ production cross section in pp collisions at $\sqrt{s}=5.02\,\mathrm{TeV}$.  http://arxiv.org/abs/2112.09114.
  * [5] CMS Collaboration.  Measurement of the inclusive $t\overline{t}$ cross section in pp collisions at $\sqrt{s}=5.02$ TeV using final states with at least one charged lepton.  JHEP, 03:115, 2018.
  * [6] Andrea Valassi and Roberto Chierici.  Information and treatment of unknown correlations in the combination of measurements using the BLUE method.  Eur. Phys. J. C, 74:2717, 2014.
  * [7] ATLAS Collaboration.  Measurement of the $t\overline{t}$ production cross-section and lepton differential distributions in $e\mu$ dilepton events from $pp$ collisions at $\sqrt{s}=13\ TeV$ with the ATLAS detector.  Eur. Phys. J. C, 80(6):528, 2020.
  * [8] CMS Collaboration.  Measurement of the top quark pair production cross section in dilepton final states containing one $\tau$ lepton in pp collisions at $\sqrt{s}=$ 13 TeV.  JHEP, 02:191, 2020.
  * [9] ATLAS Collaboration.  Measurement of the $t\overline{t}$ production cross-section in the lepton+jets channel at $\sqrt{s}=13$ TeV with the ATLAS experiment.  Phys. Lett. B, 810:135797, 2020.
  * [10] CMS Collaboration.  Measurement of $t\overline{t}$ normalised multi-differential cross sections in pp collisions at $\sqrt{s}=13$ TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions.  Eur. Phys. J. C, 80(7):658, 2020.
  * [11] CMS Collaboration.  Measurement of differential $t\overline{t}$ production cross sections in the full kinematic range using lepton+jets events from proton-proton collisions at $\sqrt{s}=$ 13 TeV.  Phys. Rev. D, 104(9):092013, 2021.
  * [12] ATLAS Collaboration.  Measurements of top-quark pair single- and double-differential cross-sections in the all-hadronic channel in $pp$ collisions at $\sqrt{s}=13~{}TeV$ using the ATLAS detector.  JHEP, 01:033, 2021.
  * [13] ATLAS Collaboration.  Measurements of differential cross-sections in top-quark pair events with a high transverse momentum top quark and limits on beyond the Standard Model contributions to top quark pair production with the ATLAS detector.  2021\.  https://cds.cern.ch/record/2777237/files/ATLAS-CONF-2021-031.pdf.
  * [14] CMS Collaboration.  Measurement of differential $t\overline{t}$ production cross sections using top quarks at large transverse momenta in $pp$ collisions at $\sqrt{s}=$ 13 TeV.  Phys. Rev. D, 103(5):052008, 2021.


# Schubert calculus and Intersection theory of Flag manifolds

Haibao Duan and Xuezhi Zhao

###### Abstract

Hilbert’s 15th problem called for a rigorous foundation of Schubert’s
calculus, in which a long standing and challenging part is Schubert’s problem
of characteristics. In the course of securing the foundation of algebraic
geometry, Van der Waerden and André Weil attributed the problem to the
determination of the intersection theory of flag manifolds.

This article surveys the background, content, and resolution of the problem of
characteristics. Our main results are a unified formula for the
characteristics, and a system description for the intersection rings of flag
manifolds. We illustrate the effectiveness of the formula and the algorithm
via explicit examples.

2010 Mathematical Subject: Classification: 14M15; 57T15; 01A65

Key words and phrases: Schubert calculus, Intersection theory, flag manifolds,
Cartan matrix of a Lie group

##  1 Introduction

Hilbert’s 15th problem [43] is an inspiring and far-reaching one. It promotes
the enumerative geometry of the 19th century growing into the algebraic
geometry founded by Van der Waerden and André Weil, and makes Schubert
calculus integrated deeply into many branches of mathematics. However, despite
great many achievements in the 20th century (e.g. [37, 47, 64]), the part of
the problem of finding an effective rule performing the calculus has been
stagnant for a long time, notably, the Schubert’s problem of characteristics
[70, §8], or the Weil’s problem [71, p.331] on the intersection theory of flag
manifolds $G/P$, where $G$ is a compact connected Lie group and $P$ a
parabolic subgroup [4].

In the series of works [18, 19, 21, 27, 29], we have addressed both of the
problem of characteristics and Weil’s problem, implying that the 15th problem
has been solved satisfactorily [30, Remark 6.3]. The purpose of this article
is to give an overview of the background, content and the resolution of
Schubert’s characteristics. In §2 we glimpse the development from Apollonius’s
work ”Tangencies” to Lefschetz’s homology theory, which reflects the evolution
of the basic ideas from enumerative geometry to intersection theory. Section
§3 summarises the pioneer contributions of Van der Waerden, Ehresmann, Weil,
Bernstein-Gel’fand-Gel’fand to Schubert calculus, which led to a great
clarification of Schubert’s characteristics. Our main results are introduced
in §4, where we present a formula expressing the characteristics of a flag
manifold $G/P$ as a polynomial in the Cartan numbers of the Lie group $G$
(Theorem 4.5), develop a system description for the intersection rings of flag
manifolds (Theorem 4.8), and illustrate the effectiveness of our programs in
Examples 4.6, 4.9 and 4.12. In particular, since our approach uses the Cartan
matrices of Lie groups as the main input, the solutions can be implemented
successfully by computer programs, so that the intersection theory of flag
manifolds becomes easily accessible by a broad readers.

##  2 An introduction to intersection theory

In the 2nd century B.C. Apollonius of Perga obtained the following enumerative
result in the paper “Tangencies”.

Apollonius’s Theorem. The number of circles tangent to three general circles
in plane is $8$. $\square$

The original proof of Apollonius was lost, but a record of the theorem by
Pappus dated in the 4th century survived. During the Renaissance different
proofs were founded respectively by Francois Viete, Adriaan van Roomen, Joseph
Diaz Gergonne and Isaac Newton [12, p.159]. For a pictorial illustration of
the theorem see the cover-page story of the book “3264 and all that”[33].

Descartes’s discovery of the Euclidean coordinates makes it possible for
geometers (e.g. Maclaurin, Euler, Bezout) to exploit polynomial system to
characterize geometric figures that satisfy a system of incidence conditions.
Consequently, many enumerative problems admit the following algebraic
formulation.

Problem 2.1. Given a system of $n$ polynomials in $n$ variables with complex
coefficients

  1. (2.1)

$\left\\{\begin{array}[]{c}f_{1}(x_{1},\ldots,x_{n})=0\\\ \vdots\\\
f_{n}(x_{1},\ldots,x_{n})=0\end{array}\right.$

find the number of solutions to the system. $\square$

Problem 2.1 is a fundamental one in algebra. In the case $n=1$ Gauss proved in
the 1820’s that the number of zero’s of a polynomial in a single variable is
the degree of that polynomial, well-known as the fundamental theorem of
algebra.

Letting $g_{i}$ be the homogenization of the polynomial $f_{i}$ in (2.1), we
get in the $n$-dimensional complex projective space $\mathbb{C}P^{n}$ a
hypersurface $N_{i}:=g_{i}^{-1}(0)$. In general, the zero locus of a
homogeneous system on a complex projective space is called a projective
variety. Naturally, the study of problem 2.1 leads to the fundamental problem
of intersection theory.

Problem 2.2. Given $k$ subvarieties $N_{1},\ldots,N_{k}$ in a smooth
projective manifold $M$ that satisfy the dimension constraint $\dim
N_{1}+\cdots+\dim N_{k}=(k-1)\dim M$, find the number
$\left|N_{1}\cap\cdots\cap N_{k}\right|$ of the common intersection points
when the subvarieties $N_{i}$’s are in general position. $\square$

In the course of studying Problem 2.2 S. Lefschetz developed the homology
theory for the cellular complexes [50, 1926]. In the perspective of this
theory let $\alpha_{i}\in H^{\dim M-\dim N_{i}}(M)$ be the Poincaré dual of
the cycle class represented by the subvariety $N_{i}\subset M$.

Problem 2.3. Given $k$ projective subvarieties $N_{1},\ldots,N_{k}$ in a
smooth projective manifold $M$ that satisfy the dimension constraint $\dim
N_{1}+\cdots+\dim N_{k}=(k-1)\dim M$, compute the Kronecker pairing

> $\left\langle\alpha_{1}\cup\cdots\cup\alpha_{k},[M]\right\rangle=?$

where $\cup$ means the cup product on the cohomology ring $H^{\ast}(M)$, and
where $[M]$ denotes the orientation class of $M$. $\square$

Through problems 2.1 to 2.3 we have briefly reviewed three seemingly different
approaches to the problems of enumerative geometry. Given that the effective
computability is the primary task of enumerative geometry, a natural question
is: which approach is the mostly calculable one? The development of the
intersection theory shall tell us the answer.

##  3 Schubert’s problem of characteristics

Hermann Schubert (1848-1911) received his Ph.D. from the University of Halle,
Germany in 1870. His doctoral thesis “The theory of characteristics” [58] is
about enumerative geometry. Prior to this he had shown that there are $16$
spheres tangent to $4$ general spheres in space, a direct extension of the
Apollonius theorem.

In 1879 Schubert published the celebrated book “Calculus of Enumerative
Geometry” [59] that represents the summit of intersection theory in the late
19th century [33, p.2]. While developing M. Chasles’s work on conics [15] he
demonstrated amazing applications of intersection theory to enumerative
geometry, such as

> i) The number of conics tangent to 8 general quadrics in space is 4,407,296;
>
> ii) The number of quadrics tangent to 9 general quadrics in space is
> 666,841,088.
>
> iii) The number of twisted cubic curves tangent to 12 general quadrics in
> space is 5,819,539,783,680.

Nevertheless, in addition to the extensive use of the controversial “principle
of conservation of numbers” [45, 69], Schubert’s exposition was so sketching
that gave “no definition of intersection multiplicity, no way to find it nor
to calculate it” [68]. At the outset of the 20th century Hilbert made finding
rigorous foundations for Schubert calculus one of his celebrated problems,
where he praised also the advantage of the calculus to foresee the final
degree of a polynomial system before carrying out the actual process of
elimination [43].

In order to gain insight into the central part of Schubert’s approach to those
spectacle enumerative numbers, we resort to the table of the characteristics
for the variety of complete conics in space from his book [59, p.95]:

$\mu^{3}\nu^{5}=1$ | $\mu^{2}\nu^{6}=8$ | $\mu\nu^{7}=34$ | $\nu^{8}=92$
---|---|---|---
$\mu^{3}\nu^{4}\rho=2$ | $\mu^{2}\nu^{5}\rho=14$ | $\mu\nu^{6}\rho=52$ | $\nu^{7}\rho=116$
$\mu^{3}\nu^{3}\rho^{2}=4$ | $\mu^{2}\nu^{4}\rho^{2}=24$ | $\mu\nu^{5}\rho^{2}=76$ | $\nu^{6}\rho^{2}=128$
$\mu^{3}\nu^{2}\rho^{3}=4$ | $\mu^{2}\nu^{3}\rho^{3}=24$ | $\mu\nu^{4}\rho^{3}=72$ | $\nu^{5}\rho^{3}=104$
$\mu^{3}\nu\rho^{4}=2$ | $\mu^{2}\nu^{2}\rho^{4}=16$ | $\mu\nu^{3}\rho^{4}=48$ | $\nu^{4}\rho^{4}=64$
$\mu^{3}\rho^{5}=1$ | $\mu^{2}\nu\rho^{5}=8$ | $\mu\nu^{2}\rho^{5}=24$ | $\nu^{3}\rho^{5}=32$
| $\mu^{2}\rho^{6}=4$ | $\mu\nu\rho^{6}=12$ | $\nu^{2}\rho^{6}=16$
|  | $\mu\rho^{7}=6$ | $\nu\rho^{7}=8$
|  |  | $\rho^{8}=4$

Table 1. The characteristics of the space of complete conics on $CP^{3}$.

where the symbols $\mu,\nu$ and $\rho$ stand for the subvarieties of conics
passing through a given point, intersecting a given line, and tangent to a
given plane, respectively. The table consists of the equalities evaluating the
monomials $\mu^{m}\nu^{n}\rho^{8-m-n}$ by integers, which were called
characteristics by Schubert, and the Schubert’s symbolic equations by earlier
researchers. Schubert emphasized that the problem of characteristics is the
fundamental one of enumerative geometry [46, 58, 59, 61]. However, to state
the problem in its natural simplicity and generality, one has to wait until
1950’s for the celebrated “basis theorem of Schubert calculus”. Let us recall
the relevant works on the subject.

The study of the characteristics began with the Italian school headed by
Segre, Enriques and Severi. Two representing papers of the school are “The
principle of conservation of numbers” and “The foundation of enumerative
geometry and the theory of characteristics” due to Severi [62, 63]. Regarding
these works Van de Wareden [69] commented that they “erected an admirable
structure, but its logical foundation was shaky. The notions were not well-
defined, and the proofs were insufficient”.

In the pioneer work “Topological foundation of enumerative geometry” [70,
1930] Van der Waerden interpreted the Schubert’s characteristics in the
perspective of the homology theory developed by Lefschetz [50] (e.g.Problem
2.3). He had the following crucial observations that enlightened the course of
the later studies on the 15th problem:

> 1) Each Schubert’s symbolic equation is a relation on the homology of a
> projective manifold;
>
> 2) The solvability of Schubert’s characteristic problem relies on a finite
> basis of the homology of the relevant manifold;
>
> 3) The determination of the intersection products in homology is the goal of
> all enumerative methods.

C. Ehresmann [34, 1934] went two important steps further: he discovered that

> 4) The parameter spaces of the geometric figures of Schubert are in
> principle certain types of flag manifolds $G/P$ (see Remark 3.6);
>
> 5) For the Grassmannian $G_{n,k}$ of $k$-planes on the $n$-space
> $\mathbb{C}^{n}$ the Schubert symbols form exactly a basis of the homology
> $H_{\ast}(G_{n,k})$.

In what follows we denote by $W(P,G)$ the set of left cosets of the Weyl group
$W(G)$ of $G$ by the Weyl group $W(P)$ of $P$, and let
$l:W(P,G)\rightarrow\mathbb{Z}$ be the associated length function [4]. With
the in-depth research on the structures of Lie groups (e.g. [7]) the vague
term “Schubert symbols” in the early literature was gradually replaced by such
rigorous defined geometric objects as “Schubert cells” or “Schubert
varieties”. In particular, extending Ehresmann’s work [34] on the Grassmannian
$G_{n,k}$, the following result was announced by Chevalley [16] for the
complete flag manifold $G/T$ (where $T\subset G$ is a maximal torus), and
extended to all flag manifolds $G/P$ by Bernstein-Gel’fand-Gel’fand [4,
Proposition 5.1].

Theorem 3.1. Every flag manifold $G/P$ has a canonical decomposition into the
Schubert cells $S_{w}$, parameterized by the elements $w$ of $W(P,G)$,

  1. (3.1)

$G/P=\underset{w\in W(P,G)}{\cup}S_{w},\quad\dim S_{w}=2l(w),$

where the closure $X_{w}$ of each cell $S_{w}$ is a subvariety of $G/P$,
called the Schubert variety on $G/P$ associated to $w\in W(P,G)$. $\square$

Since only even dimensional cells are involved in the partition (3.1), the set
$\\{[X_{w}],w\in W(P,G)\\}$ of fundamental classes forms an additive basis of
the homology $H_{\ast}(G/P)$. The co-cycle classes $s_{w}\in H^{\ast}(G/P)$
Kronecker dual to the basis (i.e. $\left\langle
s_{w},[X_{u}]\right\rangle=\delta_{w,u},~{}w,u\in W(P,G)$) gives rise to the
Schubert class associated to $w\in W(P,G)$. Theorem 3.1 implies the following
result, which is expected by Van der Waerden [70, §8], and is well-known as
the “basis theorem of Schubert calculus”.

Theorem 3.2.([4, Proposition 5.2]) The set $\\{s_{w},w\in W(P,G)\\}$ of
Schubert classes forms a basis of the cohomology $H^{\ast}(G/P)$. $\square$

An immediate consequence of the basis theorem is that any product
$s_{u_{1}}\cdots s_{u_{k}}$ in the Schubert classes can be uniquely expressed
as a linear combination of the basis elements

  1. (3.2)

$s_{u_{1}}\cdots s_{u_{k}}=\sum\limits_{w\in
W(P,G),l(w)=l(u_{1})+\cdots+l(w_{k}),\text{ }}c_{u_{1},\ldots,u_{k}}^{w}\cdot
s_{w}$, $c_{u_{1},\ldots,u_{k}}^{w}\in\mathbb{Z}$

by which Schubert’s problem of characteristics [34, 60, 61, 70] has the
following concise expression 111By Coolidge [13, p.184] “The fundamental
problem which occupies Schubert is to express the product of two of these
symbols in terms of others linearly. He succeeds in part.”:

Problem 3.3. Given any monomial $s_{u_{1}}\cdots s_{u_{k}}$ in the Schubert
classes, determine the characteristics numbers $c_{u_{1},\ldots,u_{k}}^{w}$ in
the linear expansion $(3.2)$. $\square$

In the momentous treatise “Foundations of Algebraic Geometry [71]” A. Weil
completed the definition of intersection multiplicities for the first time,
and summarized the task of Schubert calculus in the context of the modern
intersection theory 222“the classical Schubert calculus amounts to the
determination of the intersection-rings on Grassmann varieties and on the so-
called flag manifolds of projective geometry” [71, p.331], where we note that
for a flag manifold $G/P$ the Chow ring $A^{\ast}(G/P)$ is canonically
isomorphic to the cohomology $H^{\ast}(G/P)$:

Problem 3.4. Determine the intersection rings of flag manifolds $G/P$.
$\square$

Weil commented his problem as “the modern form taken by the topic formerly
known as enumerative geometry” [71, p.331]. We show that

Theorem 3.5. For the flag manifolds $G/P$ the Weil’s problem is equivalent to
the Schubert’s one.

Proof. A ring is an abelian group $R$ that is furnished with a multiplication
$R\times R\rightarrow R$. By the basis theorem the cohomology $H^{\ast}(G/P)$
has a canonical basis consisting of Schubert classes. Therefore, the
multiplication on $H^{\ast}(G/P)$ is uniquely determined by the product among
the basis elements, which is handled by the characteristics
$c_{u_{1},\ldots,u_{k}}^{w}$. $\square$

Remark 3.6. For the case $k=2$ the characteristics
$c_{w_{1},\ldots,w_{k}}^{w}$ admit various interpretations. They are called
the Schubert’s structure constants of the flag manifold $G/P$ in topology; and
the Littlewood-Richardson coefficients in representation theory [51].

In certain cases the parameter spaces of the geometric figures concerned by
Schubert [59, Chapt.IV] fail to be flag manifolds, but can be constructed by
performing finite steps of blow-ups on flag manifolds, see examples in Fulton
[37, Section 10.4], Eisenbud-Harris[33, Chap.13], or in [23] for the
constructions of the parameter spaces of the complete conics and quadrics on
$\mathbb{C}P^{3}$. As results the relevant characteristics can be computed
from those of flag manifolds via strict transformations (e.g [23, Examples
5.11; 5.12]). $\square$

##  4 Intersection theory of flag manifolds

To secure the foundation of a “calculus” it suffices to decide the objects to
be calculated, and to determine accordingly the rules of the calculation (e.g.
[2, Chap.2], [53]). As for the Schubert’s calculus we have seen from §3 that
the objects to be calculated have been clarified to be the Schubert symbols,
or Schubert varieties. In this section we determine the rule of the calculus
by a unified formula computing the characteristics, and apply the formula to
complete the intersection theory of flag manifolds.

###  4.1 Observation and expectation

The major difficulties that one encounters with the problem of characteristics
are fairly transparent:

> i) The simply-connected simple Lie groups $G$ consist of the three infinite
> families of classical Lie groups $Spin(n),Sp(n),SU(n)$, as well as the five
> exceptional ones $G_{2},F_{4},E_{6},E_{7},E_{8}$;
>
> ii) for a simple Lie group $G$ with rank $n$ there are precisely $2^{n}-1$
> parabolic subgroups $P$ on $G$.

That is, there exist plenty of flag manifolds $G/P$ whose geometries and
topologies vary considerably with respect to different choices of $G$ and $P$.
In addition

> iii) the number of Schubert classes of $G/P$ agrees with the Euler
> characteristic $\chi(G/P)$, which is normally very large,

not to mention the number of the relevant characteristics. For instance, for
an exceptional Lie group $G$ with a maximal torus $T$ the Euler characteristic
$\chi(G/T)$ of the flag manifold $G/T$ is given in the table below

> $G$ | $G_{2}$ | $F_{4}$ | $E_{6}$ | $E_{7}$ | $E_{8}$
> ---|---|---|---|---|---
> $\chi(G/T)$ | $12$ | $1152$ | $2^{7}\cdot 3^{4}\cdot 5$ | $2^{10}\cdot 3^{4}\cdot 5\cdot 7$ | $2^{14}\cdot 3^{5}\cdot 5^{2}\cdot 7$
>
> .

Summarizing, studies case by case can never reach a complete solution to the
problem.

On the other hand, according to E. Cartan’s beautiful work on compact Lie
groups, associate to each simple Lie group $G$ there is a Cartan matrix $C$,
which acts the role of “the cosmological constants” to classify all flag
manifolds $G/P$, see discussions in the coming section. As examples, for the
five exceptional Lie groups those matrices are

$G_{2}:\left(\begin{array}[]{cc}2&-1\\\ -3&2\end{array}\right),\ \ \
F_{4}:\left(\begin{array}[]{cccc}2&-1&0&0\\\ -1&2&-2&0\\\ 0&-1&2&-1\\\
0&0&-1&2\end{array}\right),\ \ \
E_{6}:\left(\begin{array}[]{cccccc}2&0&-1&0&0&0\\\ 0&2&0&-1&0&0\\\
-1&0&2&-1&0&0\\\ 0&-1&-1&2&-1&0\\\ 0&0&0&-1&2&-1\\\
0&0&0&0&-1&2\end{array}\right),$
$E_{7}:\left(\begin{array}[]{ccccccc}2&0&-1&0&0&0&0\\\ 0&2&0&-1&0&0&0\\\
-1&0&2&-1&0&0&0\\\ 0&-1&-1&2&-1&0&0\\\ 0&0&0&-1&2&-1&0\\\ 0&0&0&0&-1&2&-1\\\
0&0&0&0&0&-1&2\end{array}\right),\ \ \
E_{8}:\left(\begin{array}[]{cccccccc}2&0&-1&0&0&0&0&0\\\ 0&2&0&-1&0&0&0&0\\\
-1&0&2&-1&0&0&0&0\\\ 0&-1&-1&2&-1&0&0&0\\\ 0&0&0&-1&2&-1&0&0\\\
0&0&0&0&-1&2&-1&0\\\ 0&0&0&0&0&-1&2&-1\\\ 0&0&0&0&0&0&-1&2\end{array}\right).$

This raises the following question: can one express the characteristic
numbers, as well as the intersection ring of a flag manifold $G/P$, merely in
term of the Cartan matrix of the Lie group $G$? In this section we fulfill
this expectation.

###  4.2 Numerical construction of a Weyl group

Therefore, let $C=\left(c_{i,j}\right)_{n\times n}$ be the Cartan matrix of
some compact simple Lie group $G$, and let $\mathbb{R}^{n}$ be the
$n-$dimensional real vector space with basis
$\left\\{\omega_{1},\cdots,\omega_{n}\right\\}$. Define in terms of $C$ the
endomorphisms $\sigma_{i}\in End(\mathbb{R}^{n}),1\leq i\leq n,$ by the
formula

> $\sigma_{i}(\omega_{k})=\left\\{\begin{tabular}[]{l}$\omega_{k}$ if $i\neq
> k$;\\\
> $\omega_{k}-(c_{k,1}\omega_{1}+c_{k,2}\omega_{2}+\cdots+c_{k,n}\omega_{n})$
> if $i\neq k$.\end{tabular}\right.$

By general properties of Cartan matrices we have $\sigma_{i}^{2}=id$, implying
that $\sigma_{i}\in Aut(\mathbb{R}^{n})$. It can be further shown that

Lemma 4.1. The subgroup of $Aut(R^{n})$ generated by the $\sigma_{i}$’s is
isomorphic to the Weyl group $W(G)$ of $G$. $\square$

For each subset $K\subset\\{1,\ldots,n\\}$ there is a parabolic subgroup
$P=P_{K}$, unique up to the conjugations on $G$, whose Weyl group $W(P)$ is
generated by those generators $\sigma_{{j}}\in W(G)$ with $j\notin K$.
Resorting to the length function $l$ on $W(G)$ we can furthermore embed the
set $W(P;G)$ as the subset of the group $W(G)$ [4]

> $W(P;G)=\\{w\in W(G)\mid l(w_{{1}})\geq l(w)$, $w_{{1}}\in wW(P)\\}$,

and put $W^{m}(P;G):=\\{w\in W(P;G)\mid l(w)=m\\}$. By Lemma 4.1 each element
$w\in W^{m}(P;G)$ admits a factorization of the form

> $w=\sigma_{{i}_{{1}}}\circ\cdots\circ\sigma_{{i}_{{m}}}$ with $1\leq
> i_{{1}},\ldots,i_{{m}}\leq n$,

hence can be denoted by $w=\sigma_{I}$, where $I=(i_{{1}},\ldots,i_{{m}})$.
Such expressions of $w$ may not be unique, but the ambiguity can be dispelled
by employing the following notion. Furnish the set
$D(w):=\\{I=(i_{{1}},\ldots,i_{{m}})\mid w=\sigma_{I}\\}$ with the
lexicographical order $\preceq$ on the multi-indices $I$’s. We call a
decomposition $w=\sigma_{I}$ minimized if $I\in D(w)$ is the minimal one.
Clearly we have (e.g. [9])

Lemma 4.2. Every $w\in W(P;G)$ has a unique minimized decomposition. $\square$

It follows that the set $W^{m}(P;G)$ is also ordered by the lexicographical
order $\preceq$ on the multi-index $I$’s, hence can be uniquely presented as

  1. (4.1)

$W^{m}(P;G)=\\{w_{{m,i}}\mid 1\leq
i\leq\beta(m)\\},~{}\beta(m):=\left|W^{m}(P;G)\right|$,

where $w_{{m,i}}$ denotes the $i^{th}$ element in $W^{m}(P;G)$. In [24] the
package “Decomposition” in MATHEMATICA is compiled, whose function is stated
below.

Algorithm I. Decomposition.

> Input: The Cartan matrix $C=(a_{{ij}})_{{n\times n}}$ of $G$, and a subset
> $K\subset\\{1,\ldots,n\\}$ to specify a parabolic subgroup $P$.
>
> Output: The set $W(P;G)$ being presented by the minimized decompositions of
> its elements, together with the index system (4.1) imposed by the order
> $\preceq$. $\square$

Example 4.3. Let $G=SU(n)$ be the special unitary group, and let
$k\in\\{1,\ldots,n-1\\}$. The flag manifold $G/P_{\\{k\\}}$ is the
Grassmannian manifold $G_{n,k}$. Applying the Decomposition to the case
$G_{9,4}$ we obtain the following table of minimized decompositions, as well
as the order imposed by (4.1), for the elements $w\in W(P_{4};SU(9))$ with
$l(w)\leq 8$.

$w_{i,j}$ | decomposition | $w_{i,j}$ | decomposition | $w_{i,j}$ | decomposition
---|---|---|---|---|---
$w_{1,1}$ | $[{4}]$ | $w_{2,1}$ | $[{3,4}]$ | $w_{2,2}$ | $[{5,4}]$
$w_{3,1}$ | $[{2,3,4}]$ | $w_{3,2}$ | $[{3,5,4}]$ | $w_{3,3}$ | $[{6,5,4}]$
$w_{4,1}$ | $[{1,2,3,4}]$ | $w_{4,2}$ | $[{2,3,5,4}]$ | $w_{4,3}$ | $[{3,6,5,4}]$
$w_{4,4}$ | $[{4,3,5,4}]$ | $w_{4,5}$ | $[{7,6,5,4}]$ | $w_{5,1}$ | $[{1,2,3,5,4}]$
$w_{5,2}$ | $[{2,3,6,5,4}]$ | $w_{5,3}$ | $[{2,4,3,5,4}]$ | $w_{5,4}$ | $[{3,7,6,5,4}]$
$w_{5,5}$ | $[{4,3,6,5,4}]$ | $w_{5,6}$ | $[{8,7,6,5,4}]$ | $w_{6,1}$ | $[{1,2,3,6,5,4}]$
$w_{6,2}$ | $[{1,2,4,3,5,4}]$ | $w_{6,3}$ | $[{2,3,7,6,5,4}]$ | $w_{6,4}$ | $[{2,4,3,6,5,4}]$
$w_{6,5}$ | $[{3,2,4,3,5,4}]$ | $w_{6,6}$ | $[{3,8,7,6,5,4}]$ | $w_{6,7}$ | $[{4,3,7,6,5,4}]$
$w_{6,8}$ | $[{5,4,3,6,5,4}]$ | $w_{7,1}$ | $[{1,2,3,7,6,5,4}]$ | $w_{7,2}$ | $[{1,2,4,3,6,5,4}]$
$w_{7,3}$ | $[{1,3,2,4,3,5,4}]$ | $w_{7,4}$ | $[{2,3,8,7,6,5,4}]$ | $w_{7,5}$ | $[{2,4,3,7,6,5,4}]$
$w_{7,6}$ | $[{2,5,4,3,6,5,4}]$ | $w_{7,7}$ | $[{3,2,4,3,6,5,4}]$ | $w_{7,8}$ | $[{4,3,8,7,6,5,4}]$
$w_{7,9}$ | $[{5,4,3,7,6,5,4}]$ | $w_{8,1}$ | $[{1,2,3,8,7,6,5,4}]$ | $w_{8,2}$ | $[{1,2,4,3,7,6,5,4}]$
$w_{8,3}$ | $[{1,2,5,4,3,6,5,4}]$ | $w_{8,4}$ | $[{1,3,2,4,3,6,5,4}]$ | $w_{8,5}$ | $[{2,1,3,2,4,3,5,4}]$
$w_{8,6}$ | $[{2,4,3,8,7,6,5,4}]$ | $w_{8,7}$ | $[{2,5,4,3,7,6,5,4}]$ | $w_{8,8}$ | $[{3,2,4,3,7,6,5,4}]$
$w_{8,9}$ | $[{3,2,5,4,3,6,5,4}]$ | $w_{8,10}$ | $[{5,4,3,8,7,6,5,4}]$ | $w_{8,11}$ | $[{6,5,4,3,7,6,5,4}]$

For more examples of the results produced by Decomposition we refer to [28,
Sections 1.1–7.1].

Geometrically, for any $w\in W(P;G)$ the Schubert variety $X_{w}$ can be
explicitly constructed in terms of its minimized decomposition [4, 19].
$\square$

###  4.3 A unified formula for Schubert’s characteristics

Given an element $w\in W(P_{K};G)$ with minimized decomposition

> $w=\sigma_{i_{1}}\circ\sigma_{i_{2}}\circ\cdots\circ\sigma_{i_{m}}$, $1\leq
> i_{1},i_{2},\cdots,i_{m}\leq n$,

the structure matrix of $w$ is the strictly upper triangular matrix
$A_{w}=\left(a_{s,t}\right)_{m\times m}$ defined by the Cartan matrix
$C=\left(c_{i,j}\right)_{n\times n}$ of $G$ as

> $a_{s,t}=0$ if $s\geq t$, $-c_{i_{s},i_{t}}$ if $s<t$.

As examples recall that the Cartan matrix of the exceptional Lie group $G_{2}$
is

> $C=\left(\begin{array}[]{cc}2&-1\\\ -3&2\end{array}\right)$.

By Lemma 4.1 the Weyl group $W(G_{2})$ has two generators
$\sigma_{1},\sigma_{2}$. Consider the following elements with length $4$:

> $u=\sigma_{1}\circ\sigma_{2}\circ\sigma_{1}\circ\sigma_{2}$ and
> $v=\sigma_{2}\circ\sigma_{1}\circ\sigma_{2}\circ\sigma_{1}$.

From the Cartan matrix $C$ one reads

> $A_{u}=\left(\begin{array}[]{cccc}0&1&-2&1\\\ 0&0&3&-2\\\ 0&0&0&1\\\
> 0&0&0&0\end{array}\right)$ and $A_{v}=\left(\begin{array}[]{cccc}0&3&-2&3\\\
> 0&0&1&-2\\\ 0&0&0&3\\\ 0&0&0&0\end{array}\right)$.

Let $\mathbb{Z}[x_{{1}},\ldots,x_{{m}}]$ be the ring of polynomials in
$x_{{1}},\ldots,x_{{m}}$ that is graded by $\deg x_{i}=1$, and let
$\mathbb{Z}[x_{{1}},\ldots,x_{{m}}]^{(m)}$ be its subgroup spanned by the
monomials with degree $m$. Given a $m\times m$ strictly upper triangular
integer matrix $A=(a_{{i,j}})$ the triangular operator $T_{{A}}$ associated to
$A$ is the linear map

> $T_{A}:\mathbb{Z}[x_{{1}},\ldots,x_{{m}}]^{(m)}\rightarrow\mathbb{Z}$

defined recursively by the following elimination rules:

> i) If $m=1$ (i.e. $A=(0)$) then $T_{{A}}(x_{{1}})=1$;
>
> ii) If $h\in\mathbb{Z}[x_{{1}},\ldots,x_{{m-1}}]^{(m)}$ then $T_{{A}}(h)=0$;
>
> iii) If $h\in\mathbb{Z}[x_{{1}},\ldots,x_{{m-1}}]^{(m-r)}$ with $r\geq 1$
> then
>
> $\qquad T_{{A}}(h\cdot
> x_{{m}}^{r})=T_{{A}_{1}}(h\cdot(a_{{1,m}}x_{{1}}+\cdots+a_{{m-1,m}}x_{{m-1}})^{r-1}),$

where $A_{1}$ is the $(m-1)\times(m-1)$ (strictly upper triangular matrix)
obtained from $A$ by deleting both of the $m^{th}$ column and row. Since every
polynomial $h\in\mathbb{Z}[x_{{1}},\ldots,x_{{m}}]^{(m)}$ admits the unique
expansion

> $h=\sum\limits_{{0\leq r\leq m}}h_{{r}}\cdot x_{{m}}^{r}$ with
> $h_{{r}}\in\mathbb{Z}[x_{{1}},\ldots,x_{{m-1}}]^{(m-r)}$,

the operator $T_{{A}}$ is well-defined by the rules i), ii) and iii). It
follows that

Lemma 4.4. For any polynomial $h\in\mathbb{Z}[x_{{1}},\ldots,x_{{m}}]^{(m)}$,
the number $T_{A}(h)$ is a polynomial in the entries of the matrix $A$ with
degree $m$. $\square$

Extending the main results of [18, 19, 21] we have shown in [30, Theorem 2.4]
the following formula that expresses the Schubert’s characteristics of a flag
manifold $G/P$ as polynomials in the Cartan numbers of the group $G$.

Theorem 4.5. Let $w\in W(P;G)$ be an element with minimized decomposition
$\sigma_{{i}_{{1}}}\circ\cdots\circ\sigma_{{i}_{{m}}}$ and structure matrix
$A_{{w}}$. For any monomial $s_{u_{1}}\cdots s_{u_{k}}$ in the Schubert
classes with total degree $m$ one has

  1. (4.2)

$c_{u_{1},\ldots,u_{k}}^{w}=T_{A_{{w}}}\left(\underset{i=1,\ldots,k}{\prod}\left(\underset{\sigma_{{I}}=u_{i},\left|I\right|=l(u_{i}),I\subseteq\\{1,\ldots,m\\}}{\sum}x_{I}\right)\right)$,

where for a multi-index $I=\\{j_{{1}},\ldots,j_{{t}}\\}$ we have set
$\left|I\right|:=t$ and

>
> $\sigma_{{I}}:=\sigma_{{i}_{{j}_{{1}}}}\circ\cdots\circ\sigma_{{i}_{{j}_{{t}}}}\in
> W(G),~{}~{}x_{I}:=x_{i_{j_{1}}}\cdots
> x_{i_{j_{t}}}\in\mathbb{Z}[x_{{1}},\ldots,x_{{m}}]$. $\square$

Since the matrix $A_{{w}}$ is constructed from the Cartan matrix of the group
$G$ in term of the minimized decomposition of $w$, while the operator
$T_{A_{{w}}}$ is evaluated easily by the elimination rules i)-iii) stated
above, the formula (4.2) indicates an effective algorithm to evaluate the
numbers $c_{u_{1},\ldots,u_{k}}^{w}$. Combining these ideas the package
“Characteristics” in MATHEMATICA has been compiled (e.g. [24]) whose function
is described as follows.

Algorithm II: Characteristics.

> Input: The Cartan matrix $C=(a_{{ij}})_{{n\times n}}$ of $G$, and a subset
> $K\subset\\{1,\ldots,n\\}$ to specify a parabolic subgroup $P$.
>
> Output: The characteristics $c_{u_{1},\ldots,u_{k}}^{w}$ of $G/P$. $\square$

Example 4.6. The characteristics of the Schubert monomials at the top degree.
Let $G/P$ be a flag manifold with $\dim_{\mathbb{C}}G/P=m$. According to the
basis theorem there exists a unique element $w_{0}\in W(P;G)$ so that
$l(w_{0})=m$, and that the Schubert class $s_{w_{0}}$ generates the top degree
cohomology $H^{2m}(G/P)=\mathbb{Z}$. It follows that, for any monomial
$s_{u_{1}}\cdots s_{u_{k}}$ in the Schubert classes with total degree $m$, the
characteristic number $c_{u_{1},\ldots,u_{k}}^{w}$ is given by the equality
(see in Problem 2.3):

$\langle s_{u_{1}}\cdots
s_{u_{k}},[G/P]\rangle=c_{u_{1},\ldots,u_{k}}^{w_{0}},$

which will be abbreviated by $s_{u_{1}}\cdots
s_{u_{k}}=c_{u_{1},\ldots,u_{k}}^{w_{0}}.$ In addition, for an element $w\in
W(P;G)$ with minimized decomposition $w=\sigma_{I}$ we can use the notion
$s_{I}$ to denote the Schubert class $s_{w}$.

The cohomology of the Grassmannians $G_{n,k}$ are the most classical and
archetypal examples of intersection theory [60, 61], [33, p.4]. Traditionally,
the characteristics $c_{u_{1},u_{2}}^{w}$ are given by the combinatorial
Littlewood-Richardson rule [51], rather than a closed formula. In contrast our
formula (4.2) is practical for numerical computation. In term of the
convention above we set

$c_{r}:=s_{\\{k-r+1,k-r+2,\cdots,k\\}}\in H^{2r}(G_{n,k}),r=1,\cdots,k.$

Then $c_{r}$ is also the $r^{th}$ Chern class of the canonical $k$-dimensional
complex vector bundles on $G_{n,k}$ [55]. Applying Characteristics to the case
$G_{9,4}$ we obtain the following table of characteristics for the monomials
in the Chern classes at the top degree $\dim_{\mathbb{C}}G_{9,4}=20$ .

$c_{4}^{5}=1$ | $c_{3}^{4}c_{4}^{2}=1$ | $c_{2}c_{3}^{2}c_{4}^{3}=1$ | $c_{2}c_{3}^{6}=9$
---|---|---|---
$c_{2}^{2}c_{4}^{4}=1$ | $c_{2}^{2}c_{3}^{4}c_{4}=6$ | $c_{2}^{3}c_{3}^{2}c_{4}^{2}=4$ | $c_{2}^{4}c_{4}^{3}=3$
$c_{2}^{4}c_{3}^{4}=45$ | $c_{2}^{5}c_{3}^{2}c_{4}=26$ | $c_{2}^{6}c_{4}^{2}=16$ | $c_{2}^{7}c_{3}^{2}=231$
$c_{2}^{8}c_{4}=126$ | $c_{2}^{10}=1296$ | $c_{1}c_{3}c_{4}^{4}=1$ | $c_{1}c_{3}^{5}c_{4}=4$
$c_{1}c_{2}c_{3}^{3}c_{4}^{2}=3$ | $c_{1}c_{2}^{2}c_{3}c_{4}^{3}=2$ | $c_{1}c_{2}^{2}c_{3}^{5}=29$ | $c_{1}c_{2}^{3}c_{3}^{3}c_{4}=17$
$c_{1}c_{2}^{4}c_{3}c_{4}^{2}=10$ | $c_{1}c_{2}^{5}c_{3}^{3}=141$ | $c_{1}c_{2}^{6}c_{3}c_{4}=76$ | $c_{1}c_{2}^{8}c_{3}=756$
$c_{1}^{2}c_{3}^{2}c_{4}^{3}=2$ | $c_{1}^{2}c_{3}^{6}=19$ | $c_{1}^{2}c_{2}c_{4}^{4}=1$ | $c_{1}^{2}c_{2}c_{3}^{4}c_{4}=12$
$c_{1}^{2}c_{2}^{2}c_{3}^{2}c_{4}^{2}=7$ | $c_{1}^{2}c_{2}^{3}c_{4}^{3}=4$ | $c_{1}^{2}c_{2}^{3}c_{3}^{4}=89$ | $c_{1}^{2}c_{2}^{4}c_{3}^{2}c_{4}=48$
$c_{1}^{2}c_{2}^{5}c_{4}^{2}=26$ | $c_{1}^{2}c_{2}^{6}c_{3}^{2}=451$ | $c_{1}^{2}c_{2}^{7}c_{4}=231$ | $c_{1}^{2}c_{2}^{9}=2556$
$c_{1}^{3}c_{3}^{3}c_{4}^{2}=6$ | $c_{1}^{3}c_{2}c_{3}c_{4}^{3}=3$ | $c_{1}^{3}c_{2}c_{3}^{5}=59$ | $c_{1}^{3}c_{2}^{2}c_{3}^{3}c_{4}=32$
$c_{1}^{3}c_{2}^{3}c_{3}c_{4}^{2}=17$ | $c_{1}^{3}c_{2}^{4}c_{3}^{3}=276$ | $c_{1}^{3}c_{2}^{5}c_{3}c_{4}=141$ | $c_{1}^{3}c_{2}^{7}c_{3}=1491$
$c_{1}^{4}c_{4}^{4}=1$ | $c_{1}^{4}c_{3}^{4}c_{4}=24$ | $c_{1}^{4}c_{2}c_{3}^{2}c_{4}^{2}=12$ | $c_{1}^{4}c_{2}^{2}c_{4}^{3}=6$
$c_{1}^{4}c_{2}^{2}c_{3}^{4}=175$ | $c_{1}^{4}c_{2}^{3}c_{3}^{2}c_{4}=89$ | $c_{1}^{4}c_{2}^{4}c_{4}^{2}=45$ | $c_{1}^{4}c_{2}^{5}c_{3}^{2}=886$
$c_{1}^{4}c_{2}^{6}c_{4}=436$ | $c_{1}^{4}c_{2}^{8}=5112$ | $c_{1}^{5}c_{3}c_{4}^{3}=4$ | $c_{1}^{5}c_{3}^{5}=119$
$c_{1}^{5}c_{2}c_{3}^{3}c_{4}=59$ | $c_{1}^{5}c_{2}^{2}c_{3}c_{4}^{2}=29$ | $c_{1}^{5}c_{2}^{3}c_{3}^{3}=539$ | $c_{1}^{5}c_{2}^{4}c_{3}c_{4}=264$
$c_{1}^{5}c_{2}^{6}c_{3}=2962$ | $c_{1}^{6}c_{3}^{2}c_{4}^{2}=19$ | $c_{1}^{6}c_{2}c_{4}^{3}=9$ | $c_{1}^{6}c_{2}c_{3}^{4}=339$
$c_{1}^{6}c_{2}^{2}c_{3}^{2}c_{4}=164$ | $c_{1}^{6}c_{2}^{3}c_{4}^{2}=79$ | $c_{1}^{6}c_{2}^{4}c_{3}^{2}=1744$ | $c_{1}^{6}c_{2}^{5}c_{4}=832$
$c_{1}^{6}c_{2}^{7}=10302$ | $c_{1}^{7}c_{3}^{3}c_{4}=104$ | $c_{1}^{7}c_{2}c_{3}c_{4}^{2}=49$ | $c_{1}^{7}c_{2}^{2}c_{3}^{3}=1047$
$c_{1}^{7}c_{2}^{3}c_{3}c_{4}=496$ | $c_{1}^{7}c_{2}^{5}c_{3}=5912$ | $c_{1}^{8}c_{4}^{3}=14$ | $c_{1}^{8}c_{3}^{4}=641$
$c_{1}^{8}c_{2}c_{3}^{2}c_{4}=300$ | $c_{1}^{8}c_{2}^{2}c_{4}^{2}=140$ | $c_{1}^{8}c_{2}^{3}c_{3}^{2}=3437$ | $c_{1}^{8}c_{2}^{4}c_{4}=1600$
$c_{1}^{8}c_{2}^{6}=20887$ | $c_{1}^{9}c_{3}c_{4}^{2}=84$ | $c_{1}^{9}c_{2}c_{3}^{3}=2025$ | $c_{1}^{9}c_{2}^{2}c_{3}c_{4}=936$
$c_{1}^{9}c_{2}^{4}c_{3}=11853$ | $c_{1}^{10}c_{3}^{2}c_{4}=552$ | $c_{1}^{10}c_{2}c_{4}^{2}=252$ | $c_{1}^{10}c_{2}^{2}c_{3}^{2}=6792$
$c_{1}^{10}c_{2}^{3}c_{4}=3102$ | $c_{1}^{10}c_{2}^{5}=42597$ | $c_{1}^{11}c_{3}^{3}=3927$ | $c_{1}^{11}c_{2}c_{3}c_{4}=1782$
$c_{1}^{11}c_{2}^{3}c_{3}=23892$ | $c_{1}^{12}c_{4}^{2}=462$ | $c_{1}^{12}c_{2}c_{3}^{2}=13497$ | $c_{1}^{12}c_{2}^{2}c_{4}=6072$
$c_{1}^{12}c_{2}^{4}=87417$ | $c_{1}^{13}c_{3}c_{4}=3432$ | $c_{1}^{13}c_{2}^{2}c_{3}=48477$ | $c_{1}^{14}c_{3}^{2}=27027$
$c_{1}^{14}c_{2}c_{4}=12012$ | $c_{1}^{14}c_{2}^{3}=180609$ | $c_{1}^{15}c_{2}c_{3}=99099$ | $c_{1}^{16}c_{4}=24024$
$c_{1}^{16}c_{2}^{2}=375804$ | $c_{1}^{17}c_{3}=204204$ | $c_{1}^{18}c_{2}=787644$ | $c_{1}^{20}=1662804$

Table 2. The characteristics of the Grassmaniann $G_{9,4}$

The Characteristics works equally well for other types of flag manifolds. For
example consider the flag manifold $E_{6}/P_{\\{2\\}}$, where
$P_{\\{2\\}}=S^{1}\cdot SU(6)$. Following Bourbaki’s numbering of simple roots
[9] let $y_{1},y_{3},y_{4},y_{6}$ be respectively the Schubert classes $s_{I}$
with

$I=\\{2\\},\\{5,4,2\\},\\{6,5,4,2\\},\\{1,3,6,5,4,2\\},$

Then the cohomology $H^{*}(E_{6}/P_{\\{2\\}})$ is generated by
$y_{1},y_{3},y_{4},y_{6}$ by [27, Theorem 3]. Applying Characteristics we
obtain the following table of characteristics for all the monomials in the
Schubert generators $y_{i}^{\prime}s$ at the top degree
$\dim_{\mathbb{C}}E_{6}/P_{\\{2\\}}=21$.

$y_{3}y_{6}^{3}=3$ | $y_{3}y_{4}^{3}y_{6}=3$ | $y_{3}^{3}y_{6}^{2}=21$ | $y_{3}^{3}y_{4}^{3}=21$ | $y_{3}^{5}y_{6}=156$
---|---|---|---|---
$y_{3}^{7}=1158$ | $y_{1}y_{4}^{2}y_{6}^{2}=2$ | $y_{1}y_{4}^{5}=2$ | $y_{1}y_{3}^{2}y_{4}^{2}y_{6}=14$ | $y_{1}y_{3}^{4}y_{4}^{2}=100$
$y_{1}^{2}y_{3}y_{4}y_{6}^{2}=9$ | $y_{1}^{2}y_{3}y_{4}^{4}=9$ | $y_{1}^{2}y_{3}^{3}y_{4}y_{6}=66$ | $y_{1}^{2}y_{3}^{5}y_{4}=483$ | $y_{1}^{3}y_{6}^{3}=6$
$y_{1}^{3}y_{4}^{3}y_{6}=6$ | $y_{1}^{3}y_{3}^{2}y_{6}^{2}=42$ | $y_{1}^{3}y_{3}^{2}y_{4}^{3}=42$ | $y_{1}^{3}y_{3}^{4}y_{6}=312$ | $y_{1}^{3}y_{3}^{6}=2328$
$y_{1}^{4}y_{3}y_{4}^{2}y_{6}=28$ | $y_{1}^{4}y_{3}^{3}y_{4}^{2}=201$ | $y_{1}^{5}y_{4}y_{6}^{2}=18$ | $y_{1}^{5}y_{4}^{4}=18$ | $y_{1}^{5}y_{3}^{2}y_{4}y_{6}=132$
$y_{1}^{5}y_{3}^{4}y_{4}=972$ | $y_{1}^{6}y_{3}y_{6}^{2}=84$ | $y_{1}^{6}y_{3}y_{4}^{3}=84$ | $y_{1}^{6}y_{3}^{3}y_{6}=624$ | $y_{1}^{6}y_{3}^{5}=4677$
$y_{1}^{7}y_{4}^{2}y_{6}=56$ | $y_{1}^{7}y_{3}^{2}y_{4}^{2}=404$ | $y_{1}^{8}y_{3}y_{4}y_{6}=264$ | $y_{1}^{8}y_{3}^{3}y_{4}=1956$ | $y_{1}^{9}y_{6}^{2}=168$
$y_{1}^{9}y_{4}^{3}=168$ | $y_{1}^{9}y_{3}^{2}y_{6}=1248$ | $y_{1}^{9}y_{3}^{4}=9390$ | $y_{1}^{10}y_{3}y_{4}^{2}=813$ | $y_{1}^{11}y_{4}y_{6}=528$
$y_{1}^{11}y_{3}^{2}y_{4}=3936$ | $y_{1}^{12}y_{3}y_{6}=2496$ | $y_{1}^{12}y_{3}^{3}=18837$ | $y_{1}^{13}y_{4}^{2}=1638$ | $y_{1}^{14}y_{3}y_{4}=7917$
$y_{1}^{15}y_{6}=4992$ | $y_{1}^{15}y_{3}^{2}=37752$ | $y_{1}^{17}y_{4}=15912$ | $y_{1}^{18}y_{3}=75582$ | $y_{1}^{21}=151164$

Table 3. The characteristics of the flag manifold $E_{6}/S^{1}\cdot SU(6)$.

The contents of Tables 2 and 3 are compatible with Schubert’s computation in
Table 1. Let $M$ be the variety of complete conics in 3-space. Then
$\dim_{\mathbb{C}}M=8$, while the ring $H^{*}(M)$ is generated by the
Schubert’s symbols $\mu,\rho,\nu\in H^{2}(M)$ [23]. That is, the equalities in
Table 1 consist of the characteristics of the monomials
$\mu^{m}\nu^{n}\rho^{8-m-n}$ in the symbols $\mu,\rho$ and $\nu$ at the top
degree $8$. $\square$

Remark 4.7. Geometrically, the operator $T_{A_{{w}}}$ in (4.2) handles the
integrations along the Schubert cell $X_{w}$ [4, 19].

The formula (4.2) has been extended in [20] to compute the products of the
basis elements of the Grothendieck $K$-theory of flag manifolds, and to
evaluate the Steenrod operations on Schubert classes in [25] . $\square$

###  4.4 The intersection rings of flag manifolds

As in Example 4.6 let $c_{i}\in H^{2i}(G_{{n,k}})$ be the $i^{th}$-Chern
class. Borel [6] has shown that

  1. (4.3)

$H^{\ast}(G_{{n,k}})=\mathbb{Z}[c_{{1}},\ldots,c_{{k}}]/\left\langle
c_{{n-k+1}}^{-1},\ldots,c_{{n}}^{-1}\right\rangle$,

where $c_{{j}}^{-1}$ denotes the component of the formal inverse of
$1+c_{{1}}+\cdots+c_{{k}}$ in degree $j$, and where $\langle\cdots\rangle$
denotes the ideal generated by the enclosed polynomials. Comparing formula
(4.3) with the contents in Table 2 reveals the following phenomena: the
characteristic numbers are essential for enumerative geometry, but fails to be
a concise way to characterize the structure of the ring $H^{\ast}(G_{{n,k}})$.
It is the Weil’s problem that motivates us the following extension of Borel’s
formula (4.3) to all flag manifolds.

Theorem 4.8. For each flag manifold $G/P$ there exist Schubert classes
$y_{1},\cdots,y_{n}$ such that

  1. (4.4)

$H^{\ast}(G/P)=\mathbb{Z}[y_{1},\cdots,y_{n}]/\left\langle
f_{1},\cdots,f_{m}\right\rangle,$

where $f_{i}\in\mathbb{Z}[y_{1},\cdots,y_{n}],1\leq i\leq m$, and where the
numbers $n$ and $m$ are minimum subject to the presentation..

Proof. Let $H^{+}(G/P)$ be the subring of the cohomology $H^{*}(G/P)$ spanned
by the homogeneous elements of positive degrees, and let $D(H^{*}(G/P))$ be
the ideal of the decomposable elements of the ring $H^{+}(G/P)$. Since the
cohomology $H^{*}(G/P)$ is torsion free and has a basis consisting of Schubert
classes, there exist Schubert classes $y_{1},\cdots,y_{n}$ on $G/P$ that
correspond to a basis of the quotient group $H^{+}(G/P)/D(H^{*}(G/P))$. It
follows that the inclusion $\left\\{y_{1},\cdots,y_{n}\right\\}\subset
H^{\ast}(G/P)$ induces a ring epimorphism

$\pi:\mathbb{Z}[y_{1},\cdots,y_{n}]\rightarrow H^{\ast}(G/P).$

By the Hilbert’s basis theorem there exist finitely many polynomials
$f_{1},\cdots,f_{m}\in\mathbb{Z}[y_{1},\cdots,y_{n}]$ such that
$\ker\pi=\left\langle f_{1},\cdots,f_{m}\right\rangle$. We can of course
assume that the number $m$ is minimum with respect to the formula (4.4).

As the cardinality of a basis of the quotient group $H^{+}(G/P)/D(H^{*}(G/P))$
the number $n$ is an invariant of $G/P$. In addition, if one changes the
generators $y_{1},\cdots,y_{n}$ to $y_{1}^{\prime},\cdots,y_{n}^{\prime}$,
then each old generator $y_{i}$ can be expressed as a polynomial $g_{i}$ in
the new ones $y_{1}^{\prime},\cdots,y_{n}^{\prime}$. The invariance of the
number $m$ is shown by the presentation

$H^{\ast}(G/P)=\mathbb{Z}[y_{1}^{\prime},\cdots,y_{n}^{\prime}]/\left\langle
f_{1}^{\prime},\cdots,f_{m}^{\prime}\right\rangle,$

where $f_{j}^{\prime}$ is obtained from $f_{j}$ by substituting $g_{i}$ for
$y_{i},1\leq j\leq m$. $\square$

The proof of Theorem 4.8 singles out two crucial steps in resolving Weil’s
problem:

> i) Find a minimal set of Schubert classes
> $\left\\{y_{1},\cdots,y_{n}\right\\}$ on $G/P$ that generates the quotient
> group $H^{+}(G/P)/D(H^{*}(G/P))$;
>
> ii) Seek a Hilbert basis $\left\\{f_{1},\cdots,f_{m}\right\\}$ of the ideal
> $\ker\pi$.

Since both of the tasks can be implemented by the Characteristics (e.g. [30,
Section 4.4]), we obtain therefore the package “Chow-ring” in MATHEMATICA [27,
30] whose function is stated below.

Algorithm III: Chow-ring.

> Input: The Cartan matrix $C=(a_{{ij}})_{{n\times n}}$ of $G$, and a subset
> $K\subset\\{1,\ldots,n\\}$ to specify a parabolic subgroup $P$.
>
> Output: A presentation (4.4) of the cohomology $H^{\ast}(G/P)$. $\square$

Example 4.9. If $G$ is a simple Lie group of rank $n$ and if
$K=\\{1,\cdots,n\\}$, then the parabolic subgroup $P_{K}$ is a maximal torus
$T$ on $G$, and the flag manifold $G/T$ is called the complete flag manifold
of the group $G$. As applications of the Chow-ring the cohomologies
$H^{\ast}(G/T)$ for the exceptional Lie groups have been determined in terms
of a minimal system of generators and relations in the Schubert classes
(e.g.[29]). We present below the results for $G=F_{4},E_{6},E_{7}$.

  1. (4.5)

$H^{\ast}(F_{4}/T)=\mathbb{Z}[\omega_{1},\cdots,\omega_{4},y_{3},y_{4}]/\left\langle\rho_{2},\rho_{4},r_{3},r_{6},r_{8},r_{12}\right\rangle$,
where

$\rho_{2}=c_{2}-4\omega_{1}^{2}$;

$\rho_{4}=3y_{4}+2\omega_{1}y_{3}-c_{4}$;

$r_{3}=2y_{3}-\omega_{1}^{3}$;

$r_{6}=y_{3}^{2}+2c_{6}-3\omega_{1}^{2}y_{4}$;

$r_{8}=3y_{4}^{2}-\omega_{1}^{2}c_{6}$;

$r_{12}=y_{4}^{3}-c_{6}^{2}$.

  2. (4.6)

$H^{\ast}(E_{6}/T)=\mathbb{Z}[\omega_{1},\ldots,\omega_{6},y_{3},y_{4}]/\left\langle\rho_{2},\rho_{3},\rho_{4},\rho_{5},r_{6},r_{8},r_{9},r_{12}\right\rangle$,
where

$\rho_{2}=4\omega_{2}^{2}-c_{2}$;

$\rho_{3}=2y_{3}+2\omega_{2}^{3}-c_{3}$;

$\rho_{4}=3y_{4}+\omega_{2}^{4}-c_{4}$;

$\rho_{5}=2\omega_{2}^{2}y_{3}-\omega_{2}c_{4}+c_{5}$;

$r_{6}=y_{3}^{2}-\omega_{2}c_{5}+2c_{6}$;

$r_{8}=3y_{4}^{2}-2c_{5}y_{3}-\omega_{2}^{2}c_{6}+\omega_{2}^{3}c_{5};$

$r_{9}=2y_{3}c_{6}-\omega_{2}^{3}c_{6}$;

$r_{12}=y_{4}^{3}-c_{6}^{2}$.

  3. (4.7)

$H^{\ast}(E_{7}/T)=\mathbb{Z}[\omega_{1},\ldots,\omega_{7},y_{3},y_{4},y_{5},y_{9}]/\left\langle\rho_{i},r_{j}\right\rangle$,
where

$\rho_{2}=4\omega_{2}^{2}-c_{2}$;

$\rho_{3}=2y_{3}+2\omega_{2}^{3}-c_{3}$;

$\rho_{4}=3y_{4}+\omega_{2}^{4}-c_{4}$;

$\rho_{5}=2y_{5}-2\omega_{2}^{2}y_{3}+\omega_{2}c_{4}-c_{5}$;

$r_{6}=y_{3}^{2}-\omega_{2}c_{5}+2c_{6}$;

$r_{8}=3y_{4}^{2}+2y_{3}y_{5}-2y_{3}c_{5}+2\omega_{2}c_{7}-\omega_{2}^{2}c_{6}+\omega_{2}^{3}c_{5}$;

$r_{9}=2{y_{9}}+2{y_{4}y_{5}}-2{y_{3}c_{6}}-{\omega_{2}^{2}c_{7}}+{\omega_{2}^{3}c_{6}}$;

$r_{10}=y_{5}^{2}-2y_{3}c_{7}+\omega_{2}^{3}c_{7}$;

$r_{12}=y_{4}^{3}-4y_{5}c_{7}-c_{6}^{2}-2y_{3}y_{9}-2y_{3}y_{4}y_{5}+2\omega_{2}y_{5}c_{6}+3\omega_{2}y_{4}c_{7}+c_{5}c_{7}$;

$r_{14}=c_{7}^{2}-2y_{5}y_{9}+2y_{3}y_{4}c_{7}-\omega_{2}^{3}y_{4}c_{7}$;

$r_{18}=y_{9}^{2}+2y_{5}c_{6}c_{7}-y_{4}c_{7}^{2}-2y_{4}y_{5}y_{9}+2y_{3}y_{5}^{3}-5\omega_{2}y_{5}^{2}c_{7}$,

where the set $\\{\omega_{i},1\leq i\leq rank(G)\\}$ is the Schubert’s basis
of $H^{2}(G/T)$, which is also the set of fundamental dominant weights of the
relevant group $G$ (e.g.[22, Lemma 2.4]); the $c_{r}$’s are certain
polynomials in $\omega_{1},\cdots,\omega_{n}$ defined in [30, (5.17)] whose
geometric implication will be made transparent in Example 4.12 below, and
where in terms of Bourbaki’s numbering of simple roots [9], the $y_{i}$’s are
the Schubert classes $s_{I}$ on $G/T$ specified in the table below:

$y_{i}$ | $y_{3}$ | $y_{4}$ | $y_{5}$ | $y_{9}$
---|---|---|---|---
$F_{4}/T$ | $s_{\\{3,2,1\\}}$ | $s_{\\{4,3,2,1\\}}$ |  |
$E_{6}/T$ | $s_{\\{5,4,2\\}}$ | $s_{\\{6,5,4,2\\}}$ |  |
$E_{7}/T$ | $s_{\\{5,4,2\\}}$ | $s_{\\{6,5,4,2\\}}$ | $s_{\\{7,6,5,4,2\\}}$ | $s_{\\{1,5,4,3,7,6,5,4,2\\}}$

For more examples of the applications of Chow-ring to computing with partial
flag manifolds $G/P$ we refer to [27, Theorems 1-7]. $\square$

###  4.5 Schubert polynomials

In the classical enumerative geometry, the Chern class $c_{i}$ of the
Grassmannian $G_{{n,k}}$ arises firstly as the Poincaré dual of the variety of
the $k$-planes meeting a general $(n-k-i)$-plane on the $n$-space
$\mathbb{C}^{n}$, well-known as the $i^{th}$ special Schubert class of
$G_{{n,k}}$. The celebrated Giambelli formula [40, 1902], expressing an
arbitrary Schubert class $s_{w}$ on $G_{{n,k}}$ as a determinant (i.e. a
polynomial) in the special ones, can be praised as the beginning of the idea
of Schubert polynomials.

In general, suppose that $G/P$ is a flag manifold for which a solution (4.4)
to Weil’s problem has been available. Then, every Schubert class $s_{w}$ can
be expressed as a polynomial in the generators ${y_{1},\cdots,y_{n}}$.
Inspired by the Giambelli formula, we may call the generators
$y_{{1}},\ldots,y_{{n}}$ the special Schubert classes of $G/P$, and ask the
question of expressing an arbitrary Schubert class $s_{{w}}$ on $G/P$ as a
polynomial $\mathcal{G}_{{w}}$ in the special ones, where
$\deg\mathcal{G}_{{w}}=2l(w)$. In particular, starting from Borel’s
presentation of the cohomology ring of the flag manifold $U(n)/T$ [6]

$H^{*}(U(n)/T)=\mathbb{Z}[x_{1},\cdots,x_{n}]/\langle
e_{1},\cdots,e_{n}\rangle,$

where $e_{1},\cdots,e_{n}$ are the elementary symmetric polynomials in
$x_{1},\cdots,x_{n}$, Lascoux and Schüetzenberger [52] proposed a solution to
the problem by the following rules:

> i) Take the generator $\mathcal{G}_{0}=x_{1}^{n-1}x_{2}^{n-2}\cdots
> x_{n-1}\in H^{n(n-1)}(U(n)/T)=\mathbb{Z}$ as the top degree Schubert
> polynomial.
>
> ii) Define $\mathcal{G}_{{w}}:=\mathcal{D}_{ww_{0}}(\mathcal{G}_{0})$, where
> $w_{0}$ denotes the longest element of the Weyl group $W(U(n))$, and where
> $\mathcal{D}_{u}$ is the divided difference operator associated to $u\in
> W(U(n))$ [4].

Strictly speaking, Borel’s generators $x_{1},\cdots,x_{n-1}$ are the simple
roots of the group $SU(n)$, rather than the fundamental dominant weights (i.e.
the special Schubert classes of $U(n)/T$). However, this deficiency is not
serious, because the linear transformation between these two sets of
generators of the ring $H^{*}(U(n)/T)$ is given by the Cartan matrix of $U(n)$
[22, Lemma 2.3]. Extending the work of Lascoux-Schüetzenberger, the Schubert
polynomials of the complete flag manifolds $G/T$ have been defined by Billey-
Haiman [5] for $G=Spin(n),Sp(n)$, and independently by Fomin-Kirillov [35] for
$G=Spin(2n+1).$ In addition, as early as in 1974 Marlin [54] has determined
the ring $H^{*}(G/T)$ for $G=Spin(n)$ in the context of Schubert calculus.

In the traditional approach to Schubert polynomials, there are two necessary
prerequisites (e.g. [5]):

> a) Identify the ring $H^{*}(G/P)$ explicitly in terms of a system of special
> Schubert classes $y_{1},\cdots,y_{n}$ on $G/P$;
>
> b) Specify a polynomial $\mathcal{G}_{0}$ in $y_{1},\cdots,y_{n}$
> representing the top degree Schubert class of $G/P$.

Nevertheless, granted with Characteristics, we have alternatively a linear
algorithm to achieve Schubert polynomials, without resorting to a top degree
Schubert polynomial $\mathcal{G}_{0}$, and to the complicated operators
$\mathcal{D}_{w}$.

> Algorithm IV: Schubert polynomials
>
> Input: A set $\\{y_{{1}},\ldots,y_{{n}}\\}$ of special Schubert classes on
> $G/P$ and an integer $m>0$.
>
> Output: a Schubert polynomials $\mathcal{G}_{{w}}$ for $w\in W^{m}(H;G)$.
> $\square$

We clarify the details of Algorithm IV. Let
$\mathbb{Z}[y_{{1}},\ldots,y_{{n}}]^{(m)}$ be the group of the polynomials of
degree $m$ in the special Schubert classes $y_{{1}},\ldots,y_{{n}}$, and
examine the map

$\pi_{m}:\mathbb{Z}[y_{{1}},\ldots,y_{{n}}]^{(m)}\rightarrow H^{2m}(G/P)$

induced by the inclusion $y_{{1}},\ldots,y_{{n}}\in H^{*}(G/T)$. Let
$B(m):=\\{y^{\alpha_{{1}}},\ldots,y^{\alpha_{{b(m)}}}\\}$ be the monomial
basis of the group $\mathbb{Z}[y_{{1}},\ldots,y_{{n}}]^{(m)}$, and recall by
(4.1) that the Schubert basis of the group $H^{2m}(G/P)$ is
$\\{s_{m,1},\cdots,s_{m,\beta(m)}\\}$. Since each $y^{\alpha}\in B(m)$ is a
monomial in the special Schubert classes, Algorithm II is applicable to expand
it linearly in $\\{s_{m,1},\cdots,s_{m,\beta(m)}\\}$ to get a
$b(m)\times\beta(m)$ matrix $M(\pi_{{m}})$ that satisfies the linear system

$\left(\begin{array}[]{c}y^{\alpha_{{1}}}\\\ \vdots\\\
y^{\alpha_{{b(m)}}}\end{array}\right)=M(\pi_{{m}})\left(\begin{array}[]{c}s_{{m,1}}\\\
\vdots\\\ s_{{m,\beta(m)}}\end{array}\right)$.

Moreover, since the map $\pi_{{m}}$ surjects, the matrix $M(\pi_{{m}})$ has a
$\beta(m)\times\beta(m)$ minor equal to $\pm 1$. Thus, the standard integral
row and column operation diagonalizing $M(\pi_{{m}})$ ([57, p.162-164])
provides us with two invertible matrices $P=P_{{b(m)\times b(m)}}$ and
$Q=Q_{{\beta(m)\times\beta(m)}}$ satisfying the relation

  1. (4.8)

$PM(\pi_{{m}})Q=\left(\begin{array}[]{c}I_{{\beta(m)}}\\\
C\end{array}\right)_{{b(m)\times\beta(m)}},$

where $I_{{\beta(m)}}$ denotes the identity matrix of rank $\beta(m)$.
Summarizing, Algorithm IV can be realized by following procedure.

> Step 1. Compute $M(\pi_{{m}})$ using the Characteristics;
>
> Step 2. Diagonalize $M(\pi_{{m}})$ to get the matrices $P$ and $Q$ in (4.8);
>
> Step 3. Set $\left(\begin{array}[]{c}\mathcal{G}_{{m,1}}\\\ \vdots\\\
> \mathcal{G}_{{m,\beta(m)}}\end{array}\right):=Q\cdot[P]\left(\begin{array}[]{c}y^{\alpha_{{1}}}\\\
> \vdots\\\ y^{\alpha_{{b(m)}}}\end{array}\right)$,

where $[P]$ is the $\beta(m)\times\beta(m)$ matrix formed by the first
$\beta(m)$ rows of $P$. It is clear that

Theorem 4.11. The polynomial $\mathcal{G}_{m,k}(y_{1},\ldots,y_{n})$ is a
Schubert polynomial of the Schubert class $s_{m,k},1\leq k\leq\beta(m)$.
$\square$

Example 4.12. Algorithm IV is suitable for those flag manifolds $G/P$ whose
top degree Schubert polynomials are absent (or in question).

For the exceptional Lie group $G=E_{n}$ with $n=6,7,8$ the parabolic subgroup
$P_{\\{2\\}}$ has a canonical $n$-dimensional complex representation, that
gives rise to the canonical complex $n$-bundle $\xi_{n}$ on the flag manifold
$E_{n}/P_{\\{2\\}}$ [1]. According to Borel-Hirzebruch [8, Section 10] the
Chern classes $c_{i}(\xi_{n})$ can be expressed as a (rather lengthy)
polynomial in positive roots of the group $E_{n}$. However, with respect to
the special Schubert classes of $E_{n}/P_{\\{2\\}}$ specified in the table

$y_{i}$ | $E_{n}/P_{\\{2\\}},\text{ }n=6,7,8$
---|---
$y_{1}$ | $s_{\\{2\\}}\text{, }n=6,7,8$
$y_{3}$ | $s_{\\{5,4,2\\}}\text{, }n=6,7,8$
$y_{4}$ | $s_{\\{6,5,4,2\\}}\text{, }n=6,7,8$
$y_{5}$ | $s_{\\{{7,6,5,4,2}\\}}\text{, }n=7,8$
$y_{6}$ | $s_{\\{1,3,6,5,4,2\\}}\text{, }n=6,7,8$
$y_{7}$ |  $s_{\\{1,3,7,6,5,4,2\\}}$, $n=7,8$
$y_{8}$ |  $s_{\\{1,3,8,7,6,5,4,2\\}}$, $n=8$

we get from Algorithm IV the following concise expressions of the Chern
classes $c_{i}(\xi_{n})$ as polynomials in the special Schubert classes:

| $E_{6}/P_{\\{2\\}}$ | $E_{7}/P_{\\{2\\}}$ | $E_{8}/P_{\\{2\\}}$
---|---|---|---
$c_{1}$ | $3y_{1}$ | $3y_{1}$ | $3y_{1}$
$c_{2}$ | $4y_{1}^{2}$ | $4y_{1}^{2}$ | $4y_{1}^{2}$
$c_{3}$ | $2y_{3}+2y_{1}^{3}$ | $2y_{3}+2y_{1}^{3}$ | $2y_{3}+2y_{1}^{3}$
$c_{4}$ | $3y_{4}+y_{1}^{4}$ | $3y_{4}+y_{1}^{4}$ | $3y_{4}+y_{1}^{4}$
$c_{5}$ | $3y_{1}y_{4}-2y_{1}^{2}y_{3}+y_{1}^{5}$ | $2y_{5}+3y_{1}y_{4}-2y_{1}^{2}y_{3}+y_{1}^{5}$ | $2y_{5}+3y_{1}y_{4}-2y_{1}^{2}y_{3}+y_{1}^{5}$
$c_{6}$ | $y_{6}$ | $y_{6}+2y_{1}y_{5}$ | $5y_{6}+2y_{3}^{2}+6y_{1}y_{5}-6y_{1}^{2}y_{4}$
|  |  | $+4y_{1}^{3}y_{3}-2y_{1}^{6}$
$c_{7}$ | $0$ | $y_{7}$ | $y_{7}+4y_{1}y_{6}+2y_{1}y_{3}^{2}+4y_{1}^{2}y_{5}$
|  |  | $\ -6y_{1}^{3}y_{4}+4y_{1}^{4}y_{3}-2y_{1}^{7}$
$c_{8}$ | $0$ | $0$ | $y_{8}$

By the way, the Schubert polynomials of the flag manifold $E_{6}/P_{\\{2\\}}$
in degrees $m=8,9$ are

$s_{8,1}=y_{4}^{2}-2y_{4}y_{3}y_{1}+y_{4}y_{1}^{4}$ | $s_{9,1}=-y_{6}y_{3}+2y_{4}^{2}y_{1}-2y_{4}y_{3}y_{1}^{2}+y_{4}y_{1}^{5}$
---|---
$s_{8,2}=$ $y_{4}^{2}$ | $s_{9,2}=y_{6}y_{3}-y_{4}^{2}y_{1}$
$s_{8,3}=2y_{4}^{2}-3y_{4}y_{3}y_{1}-y_{4}y_{1}^{4}+3y_{3}^{2}y_{1}^{2}-y_{3}y_{1}^{5}$ | $s_{9,3}=y_{6}y_{3}-y_{4}^{2}y_{1}+2y_{4}y_{3}y_{1}^{2}-y_{4}y_{1}^{5}$
$s_{8,4}=2y_{4}^{2}-5y_{4}y_{3}y_{1}+5y_{3}^{2}y_{1}^{2}-2y_{3}y_{1}^{5}$ | $s_{9,4}=-y_{6}y_{3}-y_{4}y_{1}^{5}+y_{3}^{3}$
$s_{8,5}=-5y_{4}^{2}+8y_{4}y_{3}y_{1}-y_{4}y_{1}^{4}-5y_{3}^{2}y_{1}^{2}+2y_{3}y_{1}^{5}$ | $s_{9,5}=-y_{6}y_{3}-3y_{4}^{2}y_{1}+3y_{4}y_{3}y_{1}^{2}-y_{4}y_{1}^{5}$

$\square$

Remark 4.13. Presumably, Schubert polynomials may be useful to compute the
characteristics numbers. However, in our approach the Schubert’s
characteristics is a preliminary step toward Schubert polynomials (e.g.
Algorithm IV).

Currently, the theory of Schubert polynomials is a powerful tool for
discovering combinatorial structures of the Littlewood-Richardson coefficients
[17, 10, 11], and is essential to the geometric topic of degeneracy loci of
maps between vector bundles [39, 66]. There have also been extensive studies
of Schubert polynomials in quantum cohomology [36], and in the $K$-theory of
flag manifolds. For recent progresses of this branch of contemporary Schubert
calculus, we refer to the articles Kirillov-Narus [48], and Smirnov-Tutubalina
[65]. $\square$

###  4.6 Applications to the topology of homogeneous spaces

For a compact Lie group $G$ with a closed subgroup $H$ the quotient space
$G/H$ is called a homogeneous space of $G$. In contrast to the flag manifolds
the cohomology of a homogeneous space may be nontrivial in odd degrees, and
may contain torsion elements.

A classical problem of topology is to express the cohomology of a Lie group
$G$, or a homogeneous space $G/H$, by a minimal system of explicit generators
and relations. The traditional approaches due to H.Cartan, A.Borel, P.Baum and
H.Toda utilize various spectral sequence techniques [3, 6, 49, 67, 72], and
the calculation encounters the same difficulties when applied to a Lie group
$G$ whose integral cohomology has torsion elements, in particular when $G$ is
one of the exceptional Lie groups.

Schubert calculus makes the cohomology theory of homogeneous spaces appearing
in a new light. For examples, inputting the formulae $(4.5)-(4.7)$ into the
second page of the Serre spectral sequence of the fibration $G\rightarrow G/T$

$E_{2}^{**}(G)=H^{*}(G/T)\otimes H^{*}(T),$

the integral cohomology $H^{*}(G)$, as well as the Hopf algebra structure on
the $\mod p$ cohomology $H^{*}(G;\mathbb{Z}_{p})$, has been determined by
computing with the Schubert classes on $G/T$ [26, 31]. For more examples of
the extension of Schubert calculus to computing with the homogeneous spaces,
see in [27, Sect.5].

##  5 Concluding remarks

Throughout the ages a common hope of geometers is to find calculable
mechanisms among the geometric entities they are caring of (e.g. algebraic
varieties, cellular complexes, vector bundles, or the cobordism classes of
smooth manifolds). The emergence of Schubert’s calculus, or the birth of the
intersection theory, had catered to this demand. Today, Schubert’s calculus
has widely integrated into many branches of mathematics, and has profoundly
affected the trajectories of the development of those fields, such as the
theory of characteristic classes [55], the string theory [44], and algebraic
combinatorics [38]. All of these vigorously witnessed Hilbert’s broad vision
and foresight, and at the same time, put forward the essential request to
explore effective rules performing the computation.

Subject to the plan this article recalled the earlier studies on Schubert
calculus, presented a resolution of the characteristics, and illustrated a
passage from the Cartan matrices of Lie groups to the intersection theory of
flag manifolds, in which the characteristics play a central role. For the
historic significance and rigorous treatments of the enumerative examples of
Schubert mentioned in Section 3, we refer to the survey articles S. Kleiman
[45, 47], or the relevant sections of the books Fulton [37] and Eisenbud-
Harris [33] on intersection theory. For other computer systems that may be
used to perform certain computations in the intersection rings of flag
manifolds, see for example Nikolenko-Semenov [56] (the package ChowMaple06),
Grayson et al. [41, 42] (the package Schubert2 in Macaulay2), and Decker et
al. [32] (the library Schubert in SINGULAR).

ACKNOWLEDGEMENT. The authors would like to thank their referees for
improvements over the earlier version of this paper.

This work is supported by National Science foundation of China: 11771427;
11961131004.

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Haibao Duan<EMAIL_ADDRESS>
Yau Mathematical Science Center, Tsinghua University, Beijing 100084;

Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences,
Beijing 100190.

School of Mathematical Sciences, Dalian University of Technology, Dalian
116024.

Xuezhi Zhao<EMAIL_ADDRESS>
Department of Mathematics, Capital Normal University, Beijing 100048.