# The Information Bottleneck Principle in Corporate Hierarchies

Cameron Gordon
Australian Institute for Machine Learning
University of Adelaide
<EMAIL_ADDRESS>

###### Abstract

The hierarchical nature of corporate information processing is a topic of
great interest in economic and management literature. Firms are characterised
by a need to make complex decisions, often aggregating partial and uncertain
information, which greatly exceeds the attention capacity of constituent
individuals. However, the efficient transmission of these signals is still not
fully understood. Recently, the information bottleneck principle has emerged
as a powerful tool for understanding the transmission of relevant information
through intermediate levels in a hierarchical structure. In this paper we note
that the information bottleneck principle may similarly be applied directly to
corporate hierarchies. In doing so we provide a bridge between organisation
theory and that of rapidly expanding work in deep neural networks (DNNs),
including the use of skip connections as a means of more efficient
transmission of information in hierarchical organisations.

##  1 Introduction

The corporate structure of firms has been the subject of economic enquiry
since Coase (1937) [2; 18]. In particular, the structures required for
information processing and communication within firms has received much
interest from economists, organisational theorists, institutional researchers,
cyberneticists, and sociologists [30; 35; 1; 17; 36; 20; 21; 7]. Firms must
acquire, process, and act upon information in order to effect a goal or
strategic commercial objectives, often under conditions of ambiguity, partial
observability, and uncertainty [30; 18]. This requires the coordination of
many individuals whose bounded information capacity is substantially lower
than that of an organisation as a whole [30; 9; 35; 14]. Employees are
typically arranged in a hierarchy with lower levels reporting to higher and so
on up until the board [30; 29; 28]. While decision-making and delegation may
occur at all levels of an organisation, a formal decision-maker (e.g. a board
member or executive) may be separated by several layers of hierarchy from the
original source of information relevant to their decision [14; 30]. Indeed, as
Simon (1997) notes "information most important to top managers comes mainly
from external sources" [30]. Market statistics, supply-chain information,
changes to government policy, media reports, customer interactions, and sales
figures feature among the external information relevant to decision-making.
This aggregated set of information in most cases far exceeds the attention
limit of any one individual to process; hence, the question of how to
efficiently process, transmit, and filter decision-relevant information is of
critical importance for firms [14; 30; 9].

The transfer of relevant information within a firm may be seen principally as
a question of signal transmission. The well-developed theory of the
information bottleneck principle provides a means of analysing the
communication of relevant information that exists between two signals
potentially involving multiple stages of processing [34]. Furthermore, this
theory has recently emerged as a powerful tool for understanding the
information processing that occurs in the similarly hierarchical structure of
deep neural networks (DNNs) [33; 27]. While the hierarchical similarity
between DNNs and corporate hierarchies has been noted sporadically within the
literature (e.g. [19]), the information theoretic and organisational design
consequences of this connection have received little examination.

The primary contribution of this paper is to provide a bridge between
organisational theory and rapidly expanding work in deep neural networks and
information bottleneck theory. We additionally explore a common component of
deep neural networks (skip connections) and consequences for efficient
corporate information processing. We view Helbing et al. [14] as the prior
literature most related to our own. The authors investigate information flow
in hierarchical networks and highlight the importance of side-channels and
shortcut connections for efficient information transmission, however they do
not extend this to the information bottleneck principle or neural
architectures.

##  2 The Information Bottleneck Principle

The information bottleneck principle introduced in [34] describes conditions
for a compressed intermediate representation between two random variable
signals (an input and an output), by reference to the mutual information
shared between these signals and the representation [33; 34]. The principle
states that a compressed intermediate representation should contain the
minimum information with respect to the input while remaining predictive of
the output [34]. Consider two random variables $X$ and $Y$. The entropy of a
random variable $X$ is a measure of its uncertainty [3]:

$H(X)=-\sum\limits_{x\in X}p(x)\log_{2}(p(x)).$ (1)

The mutual information between two random variables $I(X;Y)$ symmetrically
describes the decrease in the uncertainty in $X$ that occurs given that $Y$ is
known (and vice-versa) [3]:

$I(X;Y)=H(X)-H(X|Y)=H(Y)-H(Y|X).$ (2)

Take an intermediate representation $\hat{X}$ between an input $X$ and output
$Y$. Then $I(X;\hat{X})$ describes the information that is preserved between
the input and the representation; and $I(Y;\hat{X})$ that between
representation and the output. Using a control parameter $\beta$, the
information bottleneck principle states that the compressed representation
should minimise the Lagrangian [33; 34; 27]:

$\mathcal{L}[p(\hat{x}|x)]=I(X;\hat{X})-\beta I(Y;\hat{X}),$ (3)

where $p(\hat{x}|x)$ is a mapping for all $x\in X$ satisfying the probability
constraint $\sum\limits_{\hat{x}\in\hat{X}}p(\hat{x})=1$. The parameter
$\beta$ controls the rate of compression and may be viewed as a constraint on
transmission.

The information bottleneck principle has been applied widely to problems of
information processing in decision science [32], psychology [38], and
prediction [10]. Importantly, it has been used to analyse information flows in
deep feedforward neural networks [33; 27; 26; 10]. A deep feedforward neural
network involves a hierarchical processing of layers in which the output of
each layer feeds directly to the next. Typically this is written as a
composition of functions $f(x)=f_{k}(f_{k-1}(f\dots(f_{1}(x))))$ where $x$ is
an input, $f_{i}(x)$ is a function applied at layer $i$, and $k$ is the number
of layers [11].

Tishby and Zaslavsky [33] describe two properties that enable the information
bottleneck principle to be applied to DNNs. The first is the data processing
inequality, a fundamental result in information theory [3]. Briefly stated,
for a Markov chain $X\to Y\to Z$ it must be that $I(X;Y)\geq I(X;Z)$. That is,
that mutual information is non-increasing through processing. This may be seen
visually by observing the sets of information shared between intermediate
representations (see Figure 1). Within a DNN information flow can be
represented as a Markov chain $X\to T_{1}\to T_{2}\to\dots T_{k}\to Y$ where
$T_{i}$ represents the $i^{th}$ hidden layer. The bottleneck representation
implies that at convergence each layer will be the minimal representation
required to predict the next layer subject to network capacity. Successive
neural network layers hence reduce the mutual information shared between the
input and output. The second property (invariance of mutual information to
invertible transformations) enables the principle to hold for different re-
parameterizations and intermediate structural forms [38].

Figure 1: a) The relevant information is the minimal set of information in an
input signal that is predictive of an output signal; without processing this
corresponds to the mutual information $I(X;Y)$. b) Visualizing the data
processing inequality. Taking an intermediate representation forms the Markov
chain $X\to Y\to Z$. The mutual information between input signal $X$ and
output signal $Z$ must be less than or equal to that between the input signal
and the intermediate representation $Y$.

##  3 Hierarchies in Corporations

Corporate organisations are typically arranged in a hierarchy, with reporting
relationships from subordinates to superiors up to the board [30; 2; 6; 17;
18; 8]. Consider the following simplified model of a corporation with $N$
levels of hierarchy. For illustration, we first consider the simple case in
which hierarchies are strict, and information flows only from one hierarchical
layer to the next (i.e. information enters the corporation at layer 1 and is
passed following processing to layer 2). At each layer $L_{i}$ some processing
occurs (e.g. briefing notes, statistical analysis) with active decision taken
by an individual on what information to transmit and what to filter from their
superiors. We can note that we have the Markov chain $L_{1}\to
L_{2}\to\dots\to L_{N}$, with the data processing inequality
$I(L_{1};L_{2})\geq I(L_{1};L_{3})\geq\dots\geq I(L_{1};L_{N})$. Hence, this
structure obeys a data processing inequality. By making the assumption that
transformations within layers are invertible (i.e. prior to transmission the
input information to that layer may be fully recovered), we can begin to apply
the information bottleneck principle to this corporate hierarchy.

Figure 2: Information flow through a hierarchical organisation. Information
entering an organisation at the operational level may have relevance to the
board and is transmitted primarily through reporting lines. Skip connections
within the organisation (informal and formal) enable lower level
representations to be re-transmitted higher in the hierarchy, reducing the
risk of incorrect filtering.

A chief information constraint within firm management is the limited attention
capacity of managers, who each operate within the constraints of bounded
rationality [30; 9; 14]. Hence we may frame an information processing
objective at each level in a hierarchy as transmitting the maximally relevant
information subject to it being less than the attention capacity of managers
at the next level. We can note this is equivalent to filtering or removing
irrelevant information from being transmitted. Assume for the purposes of a
decision that the attention constraint for a relevant decision-maker at level
$k$ is $\alpha_{k}$. With firm external information $X$ we have following
optimisation at level $k-1$ of the organisation to produce a constrained
information representation $L_{k}$ based on the preceding level:

$\underset{L_{k-1}}{\text{argmin
}}I(L_{k-2};L_{k-1})-\alpha_{k}I(L_{k-1};L_{k}),$ (4)

which an be recognised as an information bottleneck. Similar analysis can be
extended to lower levels in the hierarchy where the attention constraint for
the decision-maker is replaced by the sum of attentions of individuals $j$ in
the following layer:

$\underset{L_{k-2}}{\text{argmin
}}I(L_{k-3};L_{k-2})-\sum_{j}\alpha_{j,k-1}I(L_{k-2};L_{k-1}).$ (5)

And finally to the operational level where the signal $X$ enters the
organisation:

$\underset{L_{1}}{\text{argmin
}}I(X;L_{1})-\sum_{j}\alpha_{j,1}I(L_{2};L_{1}).$ (6)

We can view the objective at each level as decoding the information received
and encoding a compressed representation that can be transmitted through
relevant reporting relationships [33; 27].

Aligning the encoding-decoding process at each layer is required to reduce
error in prediction and decision-making. This requires the use of top-down
feedback, a component known to be a critical mechanism for learning across
domains [35; 31]. In deep neural networks feedback usually occurs through
backpropagation of an error signal [25; 11]. In corporate structures feedback
occurs through both formal and informal channels, including performance
reviews, incentives, supervision, hiring decisions, and formal direction in
order to align individual incentives to strategic goals. [30; 2; 36; 18].

##  4 Skip Connections in Corporate Hierarchies

Much of the information relevant to corporate objectives is observed at the
operational level and must be transmitted up the chain [30]. In passing
through several intermediate layers of a strict hierarchy decision-relevant
information may be lost due to the data processing inequality, for example
through misinterpretation or incorrect filtering by a manager. As Simon [30]
writes, "A major communications problem, then, is that much of the information
relevant to the decisions at this level originates at lower levels, and may
not ever reach the higher levels unless the executive is extraordinarily
alert."

One resolution of this issue is that communication networks in corporations
are typically not strict hierarchies, but instead combine a mixture of formal
and informal channels connecting multiple levels in the organisation [30; 14].
It is well-known that different managers selectively attend to different
details, and consultation with multiple groups occurs during making business
decisions in practice [4; 5; 30]. The presence of shortcut, side-channel, and
skip-connections has the effect of combining multiple filtered sources of an
incoming signal, enabling error correction and multiple levels of compression
to be viewed simultaneously [14]. An example of this is the transmission of a
full report alongside a highly compressed executive summary. If relevant
information has been incorrectly filtered by the summary it can be recovered
through the full report. Similarly, lower-level analysts are often made
present while their superiors are reporting to executives. The executive may
rely primarily on the compressed representation presented by the superior as a
matter of efficiency; and seek additional details, confirmation, or
clarification from the subordinate only if required.

The concatenation of a compressed representation with that of an earlier stage
in a hierarchy can be recognised in deep learning as a form of skip
connection, which have wide use in ResNets, ResNeXTs, DenseNets, and related
architectures, enabling extremely deep networks to be trained without the loss
of signal and vanishing gradient known to affect deep strictly sequential
networks [12; 13; 37; 16; 24; 15; 11]. Skip connections can also be viewed as
bridges which reduce path depth when viewed through the lens of network theory
or communication topology [23; 22; 39].

##  5 Conclusion

Corporations need to filter, compress, and transmit information internally in
order effect decisions. In this paper we have argued that the information
bottleneck principle may be used to analyse information processing within
these hierarchical systems. In doing so we provide a bridge between work in
organisational theory and recent advances in the study of neural networks. To
an extent these systems face an analogous problem of making decisions under
constrained resources: computational and financial [2; 30; 18; 8]. While we
have pointed to the example of skip connections as an example of a topological
structure in both systems it is an interesting open question whether other
features known to improve neural network performance may play an analogous
role in organisational design.

## Acknowledgements

The author wishes to thank Dr. Lachlan MacDonald, Dr. Hemanth Saratchandran,
Dr. Patrick O’Callaghan, and Mr. Keaton Jenner for their helpful suggestions
and comments on draft forms of this paper, and Dr. Nan Ye for his help in
understanding the details of information bottleneck theory.

## References

  * Beer [1972] Stafford Beer.  Brain of the Firm: The Managerial Cybernetics of Organization.  1972\.
  * Coase [1937] R. H. Coase.  The Nature of the Firm.  _Economica_ , 4(16):386–405, 1937.  ISSN 1468-0335.
  * Cover and Thomas [2006] Thomas M. Cover and Joy A. Thomas.  _Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)_.  Wiley-Interscience, USA, 2006.  ISBN 0-471-24195-4.
  * Cyert et al. [1956] Richard M. Cyert, Herbert A. Simon, and Donald B. Trow.  Observation of a Business Decision.  _The Journal of Business_ , 29(4):237–248, 1956\.  ISSN 0021-9398.  URL https://www.jstor.org/stable/2350807.  Publisher: University of Chicago Press.
  * Dearborn and Simon [1958] DeWitt C. Dearborn and Herbert A. Simon.  Selective Perception: A Note on the Departmental Identifications of Executives.  _Sociometry_ , 21(2):140–144, 1958.  ISSN 0038-0431.  doi: 10.2307/2785898.  URL https://www.jstor.org/stable/2785898.
  * Dessein and Holden [2022] Wouter Dessein and Richard Holden.  Organizations with Power-Hungry Agents.  _The Journal of Law and Economics_ , 65(S1):S263–S291, February 2022.  ISSN 0022-2186.  doi: 10.1086/718852.  URL https://www.journals.uchicago.edu/doi/full/10.1086/718852.  Publisher: The University of Chicago Press.
  * Durugbo et al. [2013] Christopher Durugbo, Ashutosh Tiwari, and Jeffrey R. Alcock.  Modelling information flow for organisations: A review of approaches and future challenges.  _International Journal of Information Management_ , 33(3):597–610, June 2013.  ISSN 0268-4012.  doi: 10.1016/j.ijinfomgt.2013.01.009.  URL https://www.sciencedirect.com/science/article/pii/S0268401213000121.
  * Foss [1996] Nicolai J. Foss.  Harald B. Malmgren’s Analysis of the Firm: lessons for modern theorists?  _Review of Political Economy_ , 8(4):349–366, October 1996.  ISSN 0953-8259.
  * Gigerenzer and Selten [2001] Gerd Gigerenzer and Reinhard Selten.  _Bounded Rationality: The adaptive toolbox_.  Cambridge: The MIT Press, 2001.
  * Goldfeld and Polyanskiy [2020] Ziv Goldfeld and Yury Polyanskiy.  The Information Bottleneck Problem and Its Applications in Machine Learning.  Technical Report arXiv:2004.14941, arXiv, May 2020.  URL http://arxiv.org/abs/2004.14941.
  * Goodfellow et al. [2016] Ian Goodfellow, Yoshua Bengio, and Aaron Courville.  _Deep Learning_.  MIT Press, 2016.
  * He et al. [2015] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep Residual Learning for Image Recognition.  _arXiv:1512.03385 [cs]_ , December 2015.
  * He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Identity Mappings in Deep Residual Networks.  In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, _Computer Vision – ECCV 2016_ , Lecture Notes in Computer Science, pages 630–645, Cham, 2016. Springer International Publishing.  ISBN 978-3-319-46493-0.  doi: 10.1007/978-3-319-46493-0_38.
  * Helbing et al. [2006] Dirk Helbing, Hendrik Ammoser, and Christian Kühnert.  Information flows in hierarchical networks and the capability of organizations to successfully respond to failures, crises, and disasters.  _Physica A: Statistical Mechanics and its Applications_ , 363(1):141–150, April 2006.  ISSN 0378-4371.  doi: 10.1016/j.physa.2006.01.055.  URL https://www.sciencedirect.com/science/article/pii/S0378437106000999.
  * Hochreiter [1998] Sepp Hochreiter.  The Vanishing Gradient Problem During Learning Recurrent Neural Nets and Problem Solutions.  _International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems_ , 06(02):107–116, April 1998.  ISSN 0218-4885.  doi: 10.1142/S0218488598000094.  URL https://www.worldscientific.com/doi/10.1142/S0218488598000094.  Publisher: World Scientific Publishing Co.
  * Huang et al. [2018] Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger.  Densely Connected Convolutional Networks.  _arXiv:1608.06993 [cs]_ , January 2018.  URL http://arxiv.org/abs/1608.06993.  arXiv: 1608.06993.
  * Knapp [1989] Tim Knapp.  Hierarchies and Control: A New Interpretation and Reevaluation of Oliver Williamson’s "Markets and Hierarchies" Story.  _The Sociological Quarterly_ , 30(3):425–440, 1989.  ISSN 0038-0253.  URL https://www.jstor.org/stable/4120850.
  * Malmgren [1961] H. B. Malmgren.  Information, Expectations and the Theory of the Firm.  _The Quarterly Journal of Economics_ , 75(3):399–421, 1961.  ISSN 0033-5533.  doi: 10.2307/1885131.  URL https://www.jstor.org/stable/1885131.  Publisher: Oxford University Press.
  * Marinescu [2015] Simona-Ioana Marinescu.  Research on Application of Neural Networks in Organizational Management.  _Acta Universitatis Cibiniensis. Technical Series_ , 65(1):64–68, 2015.  doi: doi:10.1515/aucts-2015-0011.
  * Markey-Towler [2017] Brendan Markey-Towler.  I, Roboticus Oeconomicus The philosophy of mind in economics, and why it matters.  _Cambridge Journal of Economics_ , 41(1):203–237, January 2017.  ISSN 0309-166X.  doi: 10.1093/cje/bew014.  URL https://doi.org/10.1093/cje/bew014.
  * Markey-Towler [2019] Brendan Markey-Towler.  Rules, perception and emotion: When do institutions determine behaviour?  _Journal of Institutional Economics_ , 15(3):381–396, June 2019.  ISSN 1744-1374, 1744-1382.  doi: 10.1017/S1744137418000267.  URL https://www.cambridge.org/core/journals/journal-of-institutional-economics/article/rules-perception-and-emotion-when-do-institutions-determine-behaviour/68B3AACDF8CAFB218A5CFFD3F39B5CA5.  Publisher: Cambridge University Press.
  * Momennejad [2022] Ida Momennejad.  Collective minds: social network topology shapes collective cognition.  _Philosophical Transactions of the Royal Society B: Biological Sciences_ , 377(1843):20200315, January 2022.  ISSN 0962-8436, 1471-2970.  doi: 10.1098/rstb.2020.0315.  URL https://royalsocietypublishing.org/doi/10.1098/rstb.2020.0315.
  * Momennejad et al. [2019] Ida Momennejad, Ajua Duker, and Alin Coman.  Bridge ties bind collective memories.  _Nature Communications_ , 10:1578, April 2019.  ISSN 2041-1723.  doi: 10.1038/s41467-019-09452-y.  URL https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6451000/.
  * Pascanu et al. [2013] Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio.  On the difficulty of training Recurrent Neural Networks.  Technical Report arXiv:1211.5063, arXiv, February 2013.  URL http://arxiv.org/abs/1211.5063.
  * Rumelhart et al. [1986] David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams.  Learning representations by back-propagating errors.  _Nature_ , 323(6088):533–536, October 1986.  ISSN 1476-4687.  doi: 10.1038/323533a0.  URL https://doi.org/10.1038/323533a0.
  * Saxe et al. [2019] Andrew M. Saxe, Yamini Bansal, Joel Dapello, Madhu Advani, Artemy Kolchinsky, Brendan D. Tracey, and David D. Cox.  On the information bottleneck theory of deep learning.  _Journal of Statistical Mechanics: Theory and Experiment_ , 2019(12):124020, December 2019.  ISSN 1742-5468.  doi: 10.1088/1742-5468/ab3985.  URL https://doi.org/10.1088/1742-5468/ab3985.  Publisher: IOP Publishing.
  * Shwartz-Ziv and Tishby [2017] Ravid Shwartz-Ziv and Naftali Tishby.  Opening the Black Box of Deep Neural Networks via Information, 2017.
  * Simon [1962] Herbert A. Simon.  The Architecture of Complexity.  _Proceedings of the American Philosophical Society_ , 106(6):467–482, 1962.  ISSN 0003049X.  URL http://www.jstor.org/stable/985254.  Publisher: American Philosophical Society.
  * Simon [1971] Herbert A. Simon.  Designing organizations for an information rich world.  In Martin Greenberger, editor, _Computers, communications, and the public interest_ , pages 37–72. Baltimore, 1971.  ISBN 0-8018-1135-X.  URL http://opacplus.bsb-muenchen.de/search?isbn=0-8018-1135-X.
  * Simon [1997] Herbert A. Simon.  _Administrative Behavior, 4th Edition_.  Free Press, 1997.  ISBN 978-0-684-83582-2.
  * Sutton and Barto [2018] Richard S. Sutton and Andrew G. Barto.  _Reinforcement learning: an introduction_.  Adaptive computation and machine learning series. The MIT Press, Cambridge, Massachusetts, second edition edition, 2018.  ISBN 978-0-262-03924-6.
  * Tishby and Polani [2011] Naftali Tishby and Daniel Polani.  Information Theory of Decisions and Actions.  In Vassilis Cutsuridis, Amir Hussain, and John G. Taylor, editors, _Perception-Action Cycle: Models, Architectures, and Hardware_ , Springer Series in Cognitive and Neural Systems, pages 601–636. Springer, New York, NY, 2011.  ISBN 978-1-4419-1452-1.  doi: 10.1007/978-1-4419-1452-1_19.
  * Tishby and Zaslavsky [2015] Naftali Tishby and Noga Zaslavsky.  Deep Learning and the Information Bottleneck Principle, 2015.  _eprint: 1503.02406.
  * Tishby et al. [1999] Naftali Tishby, Fernando C. Pereira, and William Bialek.  The information bottleneck method.  In _Proc. of the 37-th Annual Allerton Conference on Communication, Control and Computing_ , pages 368–377, 1999.  URL https://arxiv.org/abs/physics/0004057.
  * Wiener [1948] Norbert Wiener.  _Cybernetics: or Control and Communication in the Animal and the Machine_.  MIT Press, Cambridge, MA, 2 edition, 1948.
  * Williamson [1973] Oliver E. Williamson.  Markets and Hierarchies: Some Elementary Considerations.  _The American Economic Review_ , 63(2):316–325, 1973.  ISSN 0002-8282.  URL https://www.jstor.org/stable/1817092.  Publisher: American Economic Association.
  * Xie et al. [2017] Saining Xie, Ross Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He.  Aggregated Residual Transformations for Deep Neural Networks.  In _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 5987–5995, July 2017.  ISBN 978-1-5386-0457-1.  doi: 10.1109/CVPR.2017.634.
  * Zaslavsky et al. [2018] Noga Zaslavsky, Charles Kemp, Terry Regier, and Naftali Tishby.  Efficient human-like semantic representations via the Information Bottleneck principle.  Technical Report arXiv:1808.03353, arXiv, August 2018.  URL http://arxiv.org/abs/1808.03353.
  * Zhou et al. [2021] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun.  Graph Neural Networks: A Review of Methods and Applications.  Technical Report arXiv:1812.08434, arXiv, October 2021.  URL http://arxiv.org/abs/1812.08434.


# ACTION OF WEYL GROUP ON
EQUIVARIANT K-THEORY OF
FLAG VARIETIES

MIESZKO BASZCZAK

###### Abstract

We describe the action of the Weyl group of a semi simple linear group $G$ on
cohomological and K-theoretic invariants of the generalized flag variety
$G/B$. We study the automorphism $s_{i}$, induced by the reflection in the
simple root, on the equivariant $K$-theory ring $K_{T}(G/B)$ using divided
difference operators. Using the localization theorem for torus action and
Borel presentation for the equivariant K-theory ring, we calculate the formula
for this automorphism. Moreover, we expand this formula in the basis
consisting of structure sheaves classes of Schubert varieties. We provide
effective formula (applying properties of Weyl groups) for the approximation
of this expansion, more specifically for the part corresponding to Schubert
varieties with the fixed dimension, which in the case of $G$ being a special
linear group is more exact. Finally, we discuss the above-mentioned formula in
the basis of motivic Chern classes of Schubert varieties.

###### Contents

  1. 1 Preliminaries
    1. 1.1 Equivariant cohomology and equivariant $K$-theory
    2. 1.2 Flag varieties, Schubert varieties and their classes
    3. 1.3 Equivariant cohomology and equivariant $K$-theory of flag varieties
    4. 1.4 Motivic Chern classes
  2. 2 Action of Weyl group
    1. 2.1 Equivariant cohomology
    2. 2.2 Equivariant K-theory
  3. 3 Chevalley formula
  4. 4 Properties of action of Weyl group in $[\mathcal{O}_{\omega}]$ basis
    1. 4.1 Proof of the main theorem
      1. 4.1.1 Part I - preliminaries
      2. 4.1.2 Part II - properties of Weyl group
      3. 4.1.3 Part III - support of Chevalley formula
    2. 4.2 Example of $SL_{n}/B$
  5. 5 Properties of action of Weyl group in $\text{MC}_{y}(X_{\omega}^{\circ})$ basis
  6. 6 Conclusions

## Introduction

Let $G$ be a complex semi-simple Lie group. Let $B$ be a Borel subgroup of $G$
and $G/B$ the associated full flag variety. The fundamental classes of
Schubert varieties in $G/B$ form a basis of (equivariant) cohomology, treated
as an algebra over the (equivariant) cohomology of a point. Similarly, the
equivariant $K$-theory ring $K_{T}(G/B)$, described e.g. in [Uma14], has a
basis consisting of structure sheaves classes of Schubert varieties, treated
as an algebra over $K_{T}(pt)$.

In both of these rings, there are many positivity properties. Such properties
depict different invariants in different bases. From the most fundamental ones
([Bri02] in $K$-theory and [AGM11] in equivariant $K$-theory), through Chern-
Schwartz-MacPherson classes positivity conjecture in ordinary cohomology
([AMSS23]) to the ones that are not proven yet, for example Chern-Schwartz-
MacPherson classes positivity conjecture in equivairant cohomology ([AM16]) as
well as motivic Chern classes positivity conjecture ([FRW21]), both strongly
related to the automorphism $s_{i}$, which is induced by the reflection in a
simple root of the Weyl group. Hence, an interesting question arises: how to
describe this action on the basis of Schubert varieties? In cohomology, the
answer is known and uses the Chevalley formula (as in [AM16]). In $K$-theory,
however, the situation is a bit more complicated.

In this paper, we show that using the Chevalley formula one can calculate a
part of the support of the automorphism $s_{i}$ of the Weyl group. Firstly,
using the Borel presentation (Lemma 1.5 and Lemma 1.6) we derive formulas

$\displaystyle s_{i}^{coh}$
$\displaystyle=\mathrm{id}+c_{1}([\mathcal{L}(\alpha_{i})])\partial_{i}^{coh}$
$\displaystyle s_{i}^{K}$
$\displaystyle=[\mathcal{L}(-\alpha_{i})]\cdot\mathrm{id}-([\mathcal{L}(-\alpha_{i})]-1)\partial_{i}^{K}$

for automorphism $s_{i}$ on both equivariant cohomology ring and equivariant
$K$-theory ring (Lemma 2.6 and Lemma 2.11). Both of these formulas are well
known facts, first of them is examined in [Knu03], but the second one is not
present anywhere in the currently available literature. The main result of
this paper is Theorem 4.1 proving the following formula

$s_{i}^{K}([\mathcal{O}_{\omega}])=\begin{cases*}a_{\omega
s_{i}}[\mathcal{O}_{\omega s_{i}}]+\sum_{\beta\in C_{i}}a_{\omega
s_{i}s_{\beta}}[\mathcal{O}_{\omega s_{i}s_{\beta}}]+l.o.t.&\text{if }
$l(\omega s_{i})>l(\omega)$\\\
[\mathcal{O}_{\omega}]&\text{otherwise}\end{cases*}$

where l.o.t. stands for lower order terms (see the explanation of the notion
in Theorem 4.1). Moreover, for the special case of $SL_{n}/B$ we derive
formula (Equation (4.2))

$\displaystyle s_{i}^{K}([\mathcal{O}_{\omega}])$
$\displaystyle=(1-e^{\omega(\alpha_{i})})[\mathcal{O}_{\omega
s_{i}}]-[\mathcal{O}_{\omega}]$ $\displaystyle-\sum_{\beta\neq\alpha_{i}\text{
positive root such that }l(\omega s_{i}s_{\beta})=l(\omega)\text{ and
}\langle\alpha_{i},\beta\rangle>0}e^{-\omega
s_{i}s_{\beta}(\alpha_{i})}[\mathcal{O}_{\omega s_{i}s_{\beta}}]$
$\displaystyle+\sum_{\beta\neq\alpha_{i}\text{ positive root such that
}l(\omega s_{i}s_{\beta})=l(\omega)\text{ and
}\langle\alpha_{i},\beta\rangle<0}e^{\omega(\alpha_{i})}[\mathcal{O}_{\omega
s_{i}s_{\beta}}]$ $\displaystyle+l.o.t.$

In Chapter 1 we introduce the basic definitions and properties of the objects
we study, such as equivariant cohomology and equivariant $K$-theory of flag
varieties, classes of Schubert varieties in these rings and motivic Chern
classes. In Chapter 2, we describe the automorphism $s_{i}$ on equivariant
cohomology and equivariant $K$-theory. Chapter 3 clarifies the details of the
Chevalley formula. Chapter 4 contains the proof of Theorem 4.1 and Equation
(4.2). In Chapter 5 we consider the automorphism $s_{i}$ in the motivic Chern
classes basis and present the formula for this action in full generality as
well as for some special cases. Chapter 6 points out some problems and
possible ways of further research in this field.

Acknowledgements. I would like to give a special thanks to Andrzej Weber for
his support, guidance and valuable discussions, and Magdalena Zielenkiewicz
for her remarks, comments and overall help in the editing process.

##  1 Preliminaries

###  1.1 Equivariant cohomology and equivariant $K$-theory

Let $X$ be a quasi-projective complex algebraic variety. Let $H_{*}(Y)$ denote
the Borel-Moore homology group and $H^{*}(X)$ the cohomology ring, both with
rational coefficients. Every subvariety $Y\subseteq X$ of complex dimension
$k$ defines _a fundamental class_ $[Y]\in H_{2k}(X)$ ([Ful97, Appendix B.3]).
For $X$ smooth we identify the Borel-Moore homology and cohomology via the
Poincare duality.

Suppose a torus $T=(\mathbb{C}^{*})^{n}$ acts on the variety $X$. _The
$T$-equivariant cohomology ring of $X$_ is defined as the cohomology of the
quotient $H_{T}^{*}(X)=H^{*}(ET\times^{T}X,\mathbb{Q})$ and $ET$ is the total
space of the universal $T$-bundle. This ring is an algebra over the ring
$H^{*}_{T}(pt)$, which is equal to

$H^{*}_{T}(pt)=H^{*}(ET/T,\mathbb{Q})=H^{*}((\mathbb{P}^{\infty})^{n},\mathbb{Q})=\mathbb{Q}[t_{1},\ldots,t_{n}].$

Moreover, this ring can be identified with the polynomial ring
$Sym_{\mathbb{Q}}\Xi(T)$ via the map $\lambda\to c_{1}(\mathcal{K}(\lambda))$,
where $\Xi(T)$ is the group of characters of $T$ and $\mathcal{K}(\lambda)$ is
the linear bundle $ET\times^{T}\mathbb{C}\to ET/T$. Given a $T$-invariant
subvariety $Y$ we define its fundamental class $[Y]_{T}$ as the class of the
subvariety $E_{n}\times^{T}Y$ of $E_{n}\times^{T}X$ for sufficiently large
$n$, where $E_{n}$ is an approximation of the universal $T$-bundle.

We can consider three $K$-theories on $X$ \- the $K$-theory of coherent
equivariant sheaves, the $K$-theory of locally free equivariant sheaves and
the $K$-theory of equivariant topological vector bundles. If $X$ is smooth,
then the first and the second are isomorphic ([Tho87, Theorem 5.7]). Moreover,
if $X$ is a finite union of orbits, the natural map (forgetting the structure
of a vector bundle) is an isomorphism between the second and the third
([FRW21, Theorem 9.1]). In this case, we denote by $K_{T}(X)$ _the equivariant
$K$-theory ring of $X$_. $K_{T}(X)$ is an algebra over $K_{T}(pt)$, which is
equal to $R(T)\simeq\mathbb{Z}[\Xi(T)]$.

###  1.2 Flag varieties, Schubert varieties and their classes

Let $G$ be a connected, complex, semi-simple Lie group, $B_{+}$ a fixed Borel
subgroup, $B_{-}$ the opposite Borel subgroup and $T\subseteq B_{+}$ a maximal
torus. From now on, we denote $B_{+}$ by $B$. Let $W=N(T)/T$ be the Weyl group
and $l\text{: }W\to\mathbb{N}$ the length function. Let
$\alpha_{1},\ldots,\alpha_{n}$ denote the associated set of simple roots. We
denote by $s_{i}$ the reflection in the root $\alpha_{i}$ and by $P_{i}$ the
minimal parabolic subgroup corresponding to the simple root $\alpha_{i}$.

The quotient $X:=G/B$ is called _the full flag variety_ for $G$ and $B$
defined as above. The isomorphism $G/B\simeq G_{comp}/G_{comp}\cap T$, where
$G_{comp}$ is a suitably chosen maximal compact subgroup of $G$, implies that
the Weyl group acts on $G/B$ by right multiplication. The foundation of the
Schubert calculus is a well known partition called the Bruhat decomposition
([Bor91, Section 14.12]),

$G/B=\bigsqcup_{\omega\in W}B\tilde{\omega}B/B$

for $\tilde{\omega}$ being a representative of $\omega\in NT/T$ in $NT$. The
_Schubert cell $X_{\omega}^{\circ}$_ is defined as $B\tilde{\omega}B/B$. The
_Schubert varieties $X_{\omega}$_ are defined as
$\overline{B\tilde{\omega}B/B}$ (closure in Zariski topology). Because these
varieties are $T$-invariant, one can take their fundamental classes
$[X_{\omega}]_{T}$, and these classes form a $H_{T}^{*}(pt)$-basis of
$H_{T}^{*}(G/B)$. The situation is similar when we study $K_{T}(G/B)$. The
structure sheaves $\mathcal{O}_{\omega}$ of $X_{\omega}$ (which are
$T$-equivariant) define classes $[\mathcal{O}_{\omega}]$ in $K_{T}(G/B)$ that
form a $K_{T}(pt)$-basis of $K_{T}(G/B)$.

###  1.3 Equivariant cohomology and equivariant $K$-theory of flag varieties

######  Definition 1.1.

_The Berstain-Gelfand-Gelfand operator $\partial_{i}^{coh}:H^{*}_{T}(G/B)\to
H^{*}_{T}(G/B)$_ is defined as

$\partial_{i}^{coh}(u)=p_{i}^{*}p_{i*}(u),$

where $p_{i}\text{: }G/B\to G/P_{i}$ is the projection and $u$ is an element
of $H^{*}_{T}(G/B)$.

The Demazure operators $\partial_{i}^{K}$ on $K_{T}(G/B)$ were defined in
[Dem74]. They are analogs of the Bernstain-Gelfand-Gelfand operators.

######  Definition 1.2.

_The Demazure operator $\partial_{i}^{K}:K_{T}(G/B)\to K_{T}(G/B)$_ is defined
as

$\partial_{i}^{K}([\mathcal{U}])=p_{i}^{*}p_{i*}([\mathcal{U}]),$

where $p_{i}:G/B\to G/P_{i}$ is the projection and $[\mathcal{U}]$ is an
element of $K_{T}(G/B)$.

The Berstain-Gelfand-Gelfand operator and the Demazure operator satisfy the
following.

######  Proposition 1.3 ([BGG73, Theorem 3.12]).

$\partial_{i}^{coh}([X_{\omega}])=\begin{cases*}[X_{\omega s_{i}}]&\text{if
$l(\omega s_{i})>l(\omega)$},\\\ 0&\text{otherwise.}\end{cases*}$

######  Proposition 1.4 ([Dem74, Theorem 2]).

$\partial_{i}^{K}([\mathcal{O}_{\omega}])=\begin{cases*}[\mathcal{O}_{\omega
s_{i}}]&\text{if $l(\omega s_{i})>l(\omega)$},\\\
[\mathcal{O}_{\omega}]&\text{otherwise.}\end{cases*}$

Let us denote the ring $H^{*}_{T}(pt)$ by $S$ and take an element
$\lambda\in\Xi(T)$. This element can be extended to a character
$\tilde{\lambda}$ of $B$ by composing it with the projection $B\to T$. One can
take a line bundle
$\mathcal{L}(\lambda)=G\times\mathbb{C}/B=G\times\mathbb{C}/(gb,z)\sim(g,\tilde{\lambda}(b)z)$
over $G/B$ which is $G$-equivariant, hence $T$-equivariant. Furthermore, there
is a morphism

$\displaystyle\Xi(T)$ $\displaystyle\to H^{*}_{T}(X)$ $\displaystyle\lambda$
$\displaystyle\to c_{1}^{T}(\mathcal{L}(\lambda)).$

This assignment induces a morphism of rings $c^{T}\text{: }S\to
H_{T}^{*}(G/B)$.

######  Lemma 1.5.

[AS17, Section 1.3] There is an isomorphism called the Borel presentation of
$H^{*}_{T}(G/B)$

$S\otimes_{S^{W}}S\to H^{*}_{T}(G/B)$

defined as $f\otimes g\to f\cdot c^{T}(g)$.

The map $\gamma:R(T)\to K_{T}(G/B)$ induced by the assignment
$\mathbb{Z}[\Xi(T)]\supseteq\Xi(T)\ni\lambda\to[\mathcal{L}(\lambda)]\in
K_{T}(G/B)$, leads to the following lemma called _the Borel’s presentation_ of
$K_{T}(G/B)$.

######  Lemma 1.6.

[HLS10, Theorem 4.4] There is an isomorphism

$R(T)\otimes_{R(T)^{W}}R(T)\to K_{T}(G/B)$

defined as $f\otimes g\to f\cdot\gamma(g)$.

###  1.4 Motivic Chern classes

In this section, we restrict our attention only to equivariant motivic Chern
classes of Schubert cells. For a general definition see e.g. [FRW21]. We
define the equivariant motivic Chern class of Schubert cell
$X_{\omega}^{\circ}$ in equivariant $K$-theory as follows.

######  Definition 1.7.

_The equivariant motivic Chern class of Schubert’s cell $X_{\omega}^{\circ}$_
for the reduced presentation $s_{i_{1}}s_{i_{2}}\ldots s_{i_{n}}$ of $\omega$
is defined as

$\text{MC}_{y}(X_{\omega}^{\circ})=\mathcal{T}_{i_{1}}\mathcal{T}_{i_{2}}\ldots\mathcal{T}_{i_{n}}([\mathcal{O}_{\mathrm{id}}]),$

where _the $K$-theoretic Demazure-Lusztig operator $\mathcal{T}_{i}$_ on
$K_{T}^{*}(G/B)[y]$ is defined as

$\mathcal{T}_{i}=\partial_{i}^{K}+y[\mathcal{L}(\alpha_{i})]\partial_{i}^{K}-\mathrm{id}.$

######  Remark 1.8 ([AMSS22, Proposition 3.4, Remark 5.4]).

The above defintion does not depend on the choice of presentation of element
$\omega$.

The following equality holds ([AMSS22, Corollary 5.2]):

$\mathcal{T}_{i}(\text{MC}_{y}(X_{\omega}^{\circ}))=\begin{dcases*}\text{MC}_{y}(X_{\omega
s_{i}}^{\circ})&$\text{ if }l(\omega s_{i})>l(\omega)$\\\
-(y+1)\text{MC}_{y}(X_{\omega}^{\circ})-y\text{MC}_{y}(X_{\omega
s_{i}}^{\circ})&\text{ otherwise.}\end{dcases*}$

These classes depend on $y$ and after setting $y=const$ form the basis of
$K_{T}^{*}(G/B)$ localized in the ideal generated by the elements of the form
$(1+ye^{-\alpha_{i}})$. Two important special cases are $y=-1$ and $y=0$. In
the first case these classes form a _fixed point basis_ defined as
$[\mathcal{O}_{e_{\omega}}]$ where
$e_{\omega}:=\widetilde{\omega}B\in(G/B)^{T}$ is the fixed point induced by
the element $\omega\in W$ ([AMSS22, Remark 5.3, Lemma 3.8]). In the second
case the class $\text{MC}_{0}(X_{\omega}^{\circ})$ is equal to _the ideal
sheaf of the boundary of the Schubert variety $X_{\omega}$ defined as
$\mathcal{I}_{\omega}=[\mathcal{O}_{\omega}]-[\mathcal{O}_{\partial
X_{\omega}}]$_ ([AMSS22, Lemma 3.8]). We will discuss the automorphism $s_{i}$
in these bases in Chapter 5.

##  2 Action of Weyl group

######  Remark 2.1.

Let us treat $\Xi(T)$ as a lattice in $\mathfrak{h}^{*}$, where $\mathfrak{h}$
is the Cartan subalgebra of the Lie algebra of $G$. By $(-,-)$ we denote an
inner product induced by the Killing form. Moreover, we define _the coroot
$\beta^{\vee}$_ of the root $\beta$ as $2\frac{\beta}{(\beta,\beta)}$. We use
the notation
$\langle\alpha,\beta^{\vee}\rangle:=2\frac{(\alpha,\beta)}{(\beta,\beta)}$.

For every element $\omega\in W$, we choose its representative
$\widetilde{\omega}$ in $N(T)$. We define
$m_{\omega}\in\operatorname{Aut}(G_{comp}/(G_{comp}\cap T))$ as a right hand
side multiplication by $\widetilde{\omega}$. This automorphism, which is
$T$-equivariant, combined with a homeomorphism $G_{comp}/(G_{comp}\cap
T)\approx G/B$, induces (by pullback) automorphism
$\omega^{coh}\in\operatorname{Aut}(H^{*}_{T}(G/B))$ ([MNS22, Section 3.1]) and
$\omega^{K}\in\operatorname{Aut}(K^{*}_{T}(G/B))$ ([MNS22, Section 5.1]). In
next two sections, we will prove formulas for both of these actions for
$\omega=s_{i}$.

######  Remark 2.2.

There are other actions of $W$ on the cohomology ring and $K$-theory ring.
Instead of right hand side multiplication, one can take the left hand side
multiplication. The induced action is completly described in [MNS22]. Further,
instead of taking pullback above, one can take the pushforward map, but we can
easily switch from one case to another, because pullback is adjoint to
pushforward.

###  2.1 Equivariant cohomology

Let $\omega^{S}\text{: }S\to S$ be the automorphism acting on the generators
of $S$ in the same manner as $\omega$ acts on the set of simple roots.

######  Definition 2.3.

_The divided difference operator $\widetilde{\partial}_{i}^{coh}\text{: }S\to
S$_ is defined as

$\widetilde{\partial}_{i}^{coh}(f)=\frac{f-s_{i}^{S}(f)}{-\alpha_{i}}.$

######  Remark 2.4.

This operator is sometimes defined as $\frac{f-s_{i}^{S}(f)}{\alpha_{i}}$, but
for the flag variety $G/B_{-}$. By changing $\alpha_{i}$ for $-\alpha_{i}$, we
translate this definition to our case.

######  Remark 2.5.

The expression $\frac{f-s_{i}^{S}(f)}{-\alpha_{i}}$ formally belongs to a
field of fractions, but we know that $f-s_{i}^{S}(f)$ is divisible by
$\alpha_{i}$.

The Bernstain-Gelfand-Gelfand operator is closely related to the divided
difference operator. The relation between them is given by the isomorphism
from Lemma 1.5

$\Phi:S\otimes_{S^{W}}S\to H_{T}^{*}(G/B),$

which satisfies the following ([Ara89], [BGG73] for non-equivariant case):

$\Phi\circ(\mathrm{id}\otimes\widetilde{\partial}_{i}^{coh})=\partial_{i}^{coh}\circ\Phi.$

Using that fact, we can rewrite the Definition 2.3:

$\displaystyle s_{i}^{S}$
$\displaystyle=\mathrm{id}+\alpha_{i}\widetilde{\partial}_{i}^{coh}$
$\displaystyle\Phi\circ(\mathrm{id}\otimes s_{i}^{S})$
$\displaystyle=\Phi+c_{1}([\mathcal{L}(\alpha_{i})])\partial_{i}^{coh}\circ\Phi$
$\displaystyle\Phi\circ(\mathrm{id}\otimes s_{i}^{S})\circ\Phi^{-1}$
$\displaystyle=\mathrm{id}+c_{1}([\mathcal{L}(\alpha_{i})])\partial_{i}^{coh}$

We give a topological proof of the following key formula (another proof of it
is shown in [Knu03]).

######  Lemma 2.6.

$\Phi\circ(\mathrm{id}\otimes s_{i}^{S})\circ\Phi^{-1}=s_{i}^{coh}$ in
$\operatorname{Aut}(H_{T}^{*}(G/B))$.

###### Proof.

The ring $H_{T}^{*}(G/B)$ can be treated as a subring of the localized ring
$H_{T}^{*}(G/B)_{loc}$, which is defined as the localization of
$H_{T}^{*}(G/B)$ in $\mathfrak{m}/\\{0\\}$ where $\mathfrak{m}$ is a maximal
ideal of $H_{T}(pt)$ generated by the elements $t_{1},t_{2}\ldots t_{n}$. The
fundamental classes $[x]$ of the $T$-fixed points of $G/B$ form a basis of
_the localized cohomology ring_ ([MNS22, Section 2.2]), called _the fixed
point basis_. Let
${\left.\kern-1.2pt\kappa\vphantom{\big{|}}\right|_{x}}=i_{x}^{*}(\kappa)$
denote the restriction of $\kappa$ to the fixed point $x$, where $\kappa\in
H_{T}^{*}(G/B)$ and $i_{x}\text{: }\\{x\\}\to G/B$. We have the following
commutative diagram:

${S\otimes_{S^{W}}S}$${H_{T}^{*}(G/B)}$${H_{T}^{*}(G/B)_{loc}}$${S\otimes_{S^{W}}S}$${H_{T}^{*}(G/B)}$${H_{T}^{*}(G/B)_{loc}}$$\scriptstyle{\mathrm{id}\otimes
s_{i}^{S}}$$\scriptstyle{\Phi}$$\scriptstyle{\Phi\circ(\mathrm{id}\otimes
s_{i}^{S})\circ\Phi^{-1}}$$\scriptstyle{s_{i}}$$\scriptstyle{\Phi}$${1\otimes\lambda}$${c_{1}([\mathcal{L}(\lambda)])}$${\\{\omega(\lambda)\\}_{\omega\in
W}}$${1\otimes
s_{i}(\lambda)}$${c_{1}([\mathcal{L}(s_{i}(\lambda))])}$${\\{(\omega
s_{i})(\lambda)\\}_{\omega\in W}}$$\scriptstyle{1\otimes
s_{i}^{S}}$$\scriptstyle{\Phi}$$\scriptstyle{\Phi\circ(\mathrm{id}\otimes
s_{i}^{S})\circ\Phi^{-1}}$$\scriptstyle{s_{i}}$$\scriptstyle{\Phi}$

The right-most vertical arrow in this diagram is induced by topological action
of Weyl group ($s_{i}$ permutes fixed points, so in cohomology it permutes
summands of the direct sum) and so is the central arrow. This proves the
lemma, so the formula

$s_{i}^{coh}=\mathrm{id}+c_{1}([\mathcal{L}(\alpha_{i})])\partial_{i}^{coh}$

is satisfied. ∎

###  2.2 Equivariant K-theory

Defining the isobaric divided difference operator on the representation ring
$R(T)$ is analogous to equivariant cohomology case. Let $\omega^{R}\text{:
}R(T)\to R(T)$ be the automorphism acting on generators $e^{\lambda}$ of
$R(T)$ by $\omega^{R}(e^{\lambda})=e^{\omega^{S}(\lambda)}$.

######  Definition 2.7 ([HLS10, Chapter 1]).

_The isobaric divided difference operator_
$\widetilde{\partial}_{i}^{K}:R(T)\to R(T)$ is defined as

$\widetilde{\partial}_{i}^{K}(u)=\frac{u-e^{\alpha_{i}}s_{i}^{R}(u)}{1-e^{\alpha_{i}}}.$

######  Remark 2.8.

In [HLS10] this operator is defined as
$\widetilde{\partial}_{i}^{K}(u)=\frac{u-e^{-\alpha_{i}}s_{i}^{R}(u)}{1-e^{-\alpha_{i}}}$,
but for the flag variety $G/B_{-}$. By changing $-\alpha_{i}$ for
$\alpha_{i}$, we translate this definition to our case.

######  Remark 2.9.

As in Remark 2.5, the element
$\frac{u-e^{-\alpha_{i}}s_{i}^{R}(u)}{1-e^{-\alpha_{i}}}$ formally belongs to
the field of fractions of $R(T)$, but since

$\frac{e^{\lambda}-e^{\alpha_{i}}s_{i}^{R}(e^{\lambda})}{1-e^{\alpha_{i}}}=\frac{e^{\lambda}-e^{\alpha_{i}}e^{\lambda-\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}\alpha_{i}}}{1-e^{\alpha_{i}}}=e^{\lambda}\frac{1-e^{\alpha_{i}(-\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}+1)}}{1-e^{\alpha_{i}}}$

it belongs to $R(T)$, because
$\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}$ is an integer.

Analogously to what we have already described in the cohomology case (Section
2.1), the isomorphism (Lemma 1.6)

$\Psi:R(T)\otimes_{R(T)^{W}}R(T)\to K_{T}(G/B)$

results in the following lemma.

######  Lemma 2.10 ([HLS10, Proposition 4.5]).

The following equation is satisfied

$\partial_{i}^{K}\circ\Psi=\Psi\circ(\mathrm{id}\otimes\widetilde{\partial}_{i}^{K}).$

Rewriting Definition 2.7 leads to the equation:

$\displaystyle s_{i}^{R}$
$\displaystyle=e^{-\alpha_{i}}-(e^{-\alpha_{i}}-1)\widetilde{\partial}_{i}^{K}$
$\displaystyle\Psi(\mathrm{id}\otimes s_{i}^{R})$ $\displaystyle=\Psi(1\otimes
e^{-\alpha_{i}})-(\Psi(1\otimes e^{-\alpha_{i}})-\Psi(1\otimes
1))\cdot\partial_{i}^{K}\circ\Psi$ $\displaystyle(\Psi(\mathrm{id}\otimes
s_{i}^{R})\Psi^{-1})([\mathcal{U}])$
$\displaystyle=[\mathcal{L}(-\alpha_{i})][\mathcal{U}]-([\mathcal{L}(-\alpha_{i})]-1)\partial_{i}([\mathcal{U}])$

######  Lemma 2.11.

The equation $\Psi(\mathrm{id}\otimes s_{i}^{R})\Psi^{-1}=s_{i}^{K}$ is
satisfied in $\operatorname{Aut}(K_{T}(G/B))$.

###### Proof.

Analogous to the proof of Lemma 2.6. The formula

$s_{i}^{K}([\mathcal{U}])=[\mathcal{L}(-\alpha_{i})][\mathcal{U}]-([\mathcal{L}(-\alpha_{i})]-1)\partial_{i}^{K}([\mathcal{U}])$

is satisfied, where $[\mathcal{U}]\in K_{T}(G/B)$. ∎

##  3 Chevalley formula

Our next goal is to express the action of the Weyl group in basis of Schubert
varieties, both in cohomology and $K$-theory. In both formulas for these
actions (Lemma 2.6 and Lemma 2.11), multiplication by a linear bundle
(respectively its Chern class) appears, which is described by _the Chevalley
formula_. This knowledge, along with Proposition 1.3 and Proposition 1.4,
allows us to understand the way in which actions $s_{i}^{coh}$ and $s_{i}^{K}$
look like in the bases of Schubert varieties.

First, let us introduce the Chevalley formula in equivariant cohomology.

######  Theorem 3.1 ([Kum02, Theorem 11.1.7, part e]).

Let $[X_{\omega}]$ denote class of the Schubert variety and
$\mathcal{L}(\alpha)$ a linear bundle for $\alpha\in\Xi(T)$. Then

$c_{1}(\mathcal{L}(\alpha))[X_{\omega}]=\omega(\alpha)[X_{\omega}]-\sum_{\beta\text{
positive root such that }l(\omega
s_{\beta})=l(\omega)}\langle\alpha,\beta^{\vee}\rangle[X_{\omega s_{\beta}}].$

The presentation of the Chevalley formula for an equivariant $K$-theory
requires more preparation. To start, we define a family of operators
$T_{\beta}^{0}$ and $T_{\beta}^{1}$ on $R(T)$ for $\beta$ being a root.

######  Definition 3.2.

Let $\beta$ be a root.

$\displaystyle T_{\beta}^{1}(e^{\lambda})$
$\displaystyle=e^{s_{\beta}\lambda};$ $\displaystyle
T_{\beta}^{0}(e^{\lambda})$ $\displaystyle=\begin{dcases*}0&\text{if }
$\langle\lambda,\beta^{\vee}\rangle=0$,\\\
e^{\lambda}+\ldots+e^{\lambda-(\lambda(\beta^{\vee})-1)\beta}&\text{if }
$\langle\lambda,\beta^{\vee}\rangle>0$,\\\
-e^{\lambda+\beta}-\ldots-e^{\lambda-\lambda(\beta^{\vee})\beta}&\text{if }
$\langle\lambda,\beta^{\vee}\rangle<0$.\end{dcases*}$

We define a monoid $W^{\prime}$ with the same set of generators and the same
braid relations as a Weyl group $W$ but instead of relations $s_{i}^{2}=1$,
relations $s_{i}^{2}=s_{i}$ hold. There is an obvious map $\Delta$ from
$W^{\prime}$ to $W$, which for the element $x\in W^{\prime}$ return the
element of $W$ with the same reduced decomposition in $W$ as $x$ have in
$W^{\prime}$. For example $\Delta(s_{i}^{2})=s_{i}$. Let $\omega$ have the
reduced decomposition $s_{i_{1}}s_{i_{2}}\ldots s_{i_{n}}$ and
$\epsilon\in\\{0,1\\}^{n}$. By $x(\epsilon,\omega)$ we denote element
$\Delta(s_{i_{j_{1}}}\ldots s_{i_{j_{k}}})$ for $j_{l}$ such that
$\epsilon_{j_{l}}=1$. For example, if $\omega=s_{1}s_{2}s_{3}$ and
$\epsilon=\\{1,0,1\\}$ we have $x(\epsilon,\omega)=s_{1}s_{3}$. The following
Theorem describes the action of left multiplication by
$[\mathcal{L}(\alpha)]$.

######  Theorem 3.3 ([Wil07, Chapter 1]).

Let $\omega$ have the reduced decomposition $s_{i_{1}}s_{i_{2}}\ldots
s_{i_{n}}$. Then, for every $\alpha\in\Xi(T)$ the formula

$[\mathcal{L}(\alpha)][\mathcal{O}_{\omega}]=\sum_{(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{n})\in\\{0,1\\}^{n}}T_{-\alpha_{i_{1}}}^{\epsilon_{1}}\ldots
T_{-\alpha_{i_{n}}}^{\epsilon_{n}}(e^{\alpha})[\mathcal{O}_{x(\epsilon,\omega)}]$

holds.

######  Remark 3.4.

In [Wil07], the Chevalley formula is considered on flag variety $G/B_{-}$ and
that is the reason why operators $T_{-\alpha_{i}}$ appear in this formula
instead of operators $T_{\alpha_{i}}$.

######  Remark 3.5.

To check the correctness of the above-mentioned remark let us examine an
example of a flag variety $SL_{2}/B$. It is exactly the projective space
$\mathbb{P}^{1}$. It consists of two Schubert cells - a point and a one-
dimensional disc. Both of them correspond to the fixed points of the action of
the Weyl group. Let us name these points $0$ and $\infty$. We know that

$\displaystyle{\left.\kern-1.2pt[\mathcal{O}_{\mathrm{id}}]\vphantom{\big{|}}\right|_{0}}=1-e^{\alpha_{1}}$
$\displaystyle{\left.\kern-1.2pt[\mathcal{O}_{\mathrm{id}}]\vphantom{\big{|}}\right|_{\infty}}=0$
$\displaystyle{\left.\kern-1.2pt[\mathcal{O}_{s_{1}}]\vphantom{\big{|}}\right|_{0}}=1$
$\displaystyle{\left.\kern-1.2pt[\mathcal{O}_{s_{1}}]\vphantom{\big{|}}\right|_{\infty}}=1$
$\displaystyle{\left.\kern-1.2pt[\mathcal{L}(\alpha_{1})][\mathcal{O}_{\mathrm{id}}]\vphantom{\big{|}}\right|_{0}}=e^{\alpha_{1}}-e^{2\alpha_{1}}$
$\displaystyle{\left.\kern-1.2pt[\mathcal{L}(\alpha_{1})][\mathcal{O}_{\mathrm{id}}]\vphantom{\big{|}}\right|_{\infty}}=0$
$\displaystyle{\left.\kern-1.2pt[\mathcal{L}(\alpha_{1})][\mathcal{O}_{s_{1}}]\vphantom{\big{|}}\right|_{0}}=e^{\alpha_{1}}$
$\displaystyle{\left.\kern-1.2pt[\mathcal{L}(\alpha_{1})][\mathcal{O}_{s_{1}}]\vphantom{\big{|}}\right|_{\infty}}=e^{-\alpha_{1}}$

Classes in the equivariant $K$-theory are determined by their restricions to
the fixed point basis. That gives us equalities

$\displaystyle[\mathcal{L}(\alpha_{1})][\mathcal{O}_{\mathrm{id}}]$
$\displaystyle=e^{\alpha_{1}}[\mathcal{O}_{\mathrm{id}}]$
$\displaystyle[\mathcal{L}(\alpha_{1})][\mathcal{O}_{s_{1}}]$
$\displaystyle=e^{-\alpha_{1}}[\mathcal{O}_{s_{1}}]-(1+e^{-\alpha_{1}})[\mathcal{O}_{\mathrm{id}}].$

These calculations coincide with the formulas from Theorem 3.3.

##  4 Properties of action of Weyl group in $[\mathcal{O}_{\omega}]$ basis

Our goal is to study the $s_{i}^{K}$ action in the basis
$[\mathcal{O}_{\omega}]$ in equivariant $K$-theory. From now on, this basis
will be denoted by $\mathcal{B}$.

###  4.1 Proof of the main theorem

In this subsection we prove the following theorem.

######  Theorem 4.1.

Let $\omega\in W$. Then

$s_{i}^{K}([\mathcal{O}_{\omega}])=\begin{cases*}a_{\omega
s_{i}}[\mathcal{O}_{\omega s_{i}}]+\sum_{\beta\in C_{i}}a_{\omega
s_{i}s_{\beta}}[\mathcal{O}_{\omega s_{i}s_{\beta}}]+l.o.t.&\text{if }
$l(\omega s_{i})>l(\omega)$\\\
[\mathcal{O}_{\omega}]&\text{otherwise}\end{cases*}$

where $C_{i}=\\{\gamma\text{: }\gamma\neq\alpha_{i}\text{ is a positive root
such that }l(\omega s_{i}s_{\gamma})=l(\omega)\text{ and
}\langle\alpha_{i},\gamma\rangle\neq 0\\}$, $a_{\omega^{\prime}}$ are non-zero
elements of $R(T)$ and $l.o.t.$ is the sum of terms
$a_{\omega^{\prime}}[\mathcal{O}_{\omega^{\prime}}]$ for
$l(\omega^{\prime})<l(\omega)$.

_The support_ of action for fixed $\omega$ is the set of elements
$[\mathcal{O}_{\omega^{\prime}}]$, that in formula for this action in the
basis $\mathcal{B}$ have non-zero coefficient $a_{\omega^{\prime}}$. Part of
the support, which consist of elements $[\mathcal{O}_{\omega^{\prime}}]$ such
that $l(\omega^{\prime})=n$ for $n\in\mathbb{N}$ is called _the support of
lenght $n$_ of this action.

####  4.1.1 Part I - preliminaries

Lemma 2.11 and Proposition 1.4 imply that

$s_{i}^{K}([\mathcal{O}_{\omega}])=[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega}]-([\mathcal{L}(-\alpha_{i})]-1)[\mathcal{O}_{\omega
s_{i}}]$ (1)

for $l(\omega s_{i})>l(\omega)$ and that

$s_{i}^{K}([\mathcal{O}_{\omega}])=[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega}]-([\mathcal{L}(-\alpha_{i})]-1)[\mathcal{O}_{\omega}]=[\mathcal{O}_{\omega}]$

for $l(\omega s_{i})<l(\omega)$. The second equation proves case $l(\omega
s_{i})\leq l(\omega)$ of Theorem 4.1. From now on we consider the first case
only. The first step in our proof is simplifying Equation 1 using the
Chevalley formula. It is obvious that

$[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega}]=e^{-\omega(\alpha_{i})}[\mathcal{O}_{\omega}]+l.o.t.$

and

$\displaystyle[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega s_{i}}]$
$\displaystyle=e^{\omega s_{i}(-\alpha_{i})}[\mathcal{O}_{\omega
s_{i}}]+\text{ terms of length }l(\omega)+l.o.t.$
$\displaystyle=e^{\omega(\alpha_{i})}[\mathcal{O}_{\omega s_{i}}]+\text{ terms
of length }l(\omega)+l.o.t.$

where "terms of length $l(\omega)$" is a linear combination (over $R(T)$) of
the elements from the support of length $l(\omega)$ of left-multiplication by
$[\mathcal{L}(-\alpha_{i})]$. Applying above equations to Equation (1) we
obtain:

$s_{i}^{K}([\mathcal{O}_{\omega}])=(1-e^{\omega(\alpha_{i})})[\mathcal{O}_{\omega
s_{i}}]+e^{-\omega(\alpha_{i})}[\mathcal{O}_{\omega}]-\text{ terms of length
}l(\omega)+l.o.t.$ (2)

What needs to be understood is which elements belong to the support of length
$l(\omega)$ of left-multiplication by $[\mathcal{L}(-\alpha_{i})]$.

####  4.1.2 Part II - properties of Weyl group

Let us fix a reduced decomposition $s_{i_{1}}s_{i_{2}}\ldots s_{i_{n}}$ of an
element $\omega$ of the Weyl group. The Chevalley formula implies that
elements $[\mathcal{O}_{\omega^{\prime}}]$ in the support of length
$l(\omega)-1$ of left-multiplication by $[\mathcal{L}(-\alpha_{i})]$ are in
correspondence with the subwords
$s_{i_{1}}s_{i_{2}}\ldots\widehat{s_{i_{j}}}\ldots s_{i_{n}}$ of $\omega$.
Moreover, the following lemma holds.

######  Lemma 4.2.

Let $s_{i_{1}}s_{i_{2}}\ldots s_{i_{n}}$ be a fixed representation of the word
$\omega$. Then the words
$\omega_{k}=s_{i_{1}}s_{i_{2}}\ldots\widehat{s_{i_{k}}}\ldots s_{i_{n}}$ and
$\omega_{l}=s_{i_{1}}s_{i_{2}}\ldots\widehat{s_{i_{l}}}\ldots s_{i_{n}}$ are
different.

###### Proof.

Assume that $k=1$ and $l=n$. Otherwise, we can consider the word
$s_{i_{k-1}}\ldots s_{i_{1}}\omega s_{i_{n}}\ldots s_{i_{l+1}}$. The following
equalities hold:

$\displaystyle\omega_{k}$ $\displaystyle=\omega_{l}$ $\displaystyle
s_{i_{1}}\omega_{k}$ $\displaystyle=s_{i_{1}}\omega_{l}$ $\displaystyle\omega$
$\displaystyle=\omega^{\prime}$

where $\omega^{\prime}$ is $\omega$ with the first and the last element
removed. Nevertheless $l(\omega^{\prime})=l(\omega)-2\neq l(\omega)$, which is
a contradiction. ∎

The next step is to prove the following lemma.

######  Lemma 4.3.

Let $\omega=s_{i_{1}}s_{i_{2}}\ldots s_{i_{n}}$. The subwords of $\omega$ of
length $l(\omega)-1$ obtained by removing one element from $\omega$ are in
bijection with the words $\omega s_{\beta}$ such that $l(\omega
s_{\beta})=l(\omega)-1$ for $\beta$ being a positive root.

###### Proof.

  * $(\Leftarrow)$

######  Lemma 4.4.

[Hum90, Theorem 5.8] [Strong Exchange Condition] If
$\omega=s_{i_{1}}s_{i_{2}}\ldots s_{i_{n}}$ is a reduced word and $t$ is a
reflection satisfying $l(\omega t)<l(\omega)$ then there exist $k$ such that
$\omega t=s_{i_{1}}s_{i_{2}}\ldots\widehat{s_{i_{k}}}\ldots s_{i_{n}}$.

  * $(\Rightarrow)$

We must prove that $s_{i_{1}}\ldots\widehat{s_{i_{k}}}\ldots s_{i_{n}}$ is
equal to $\omega t$ for reflection in the positive root $\beta$. Let
$\beta=\omega^{\prime}(\alpha_{k})$ where $\omega^{\prime}$ is defined as
$s_{i_{n}}\ldots s_{i_{k+1}}$.

######  Lemma 4.5.

The root $\beta$ defined as above is a positive root.

###### Proof.

From [Hum90, Theorem 5.4] it follows that if $l(\omega s_{\alpha})>l(\omega)$
for a simple root $\alpha$, then $\omega(\alpha)$ is a positive root. We know
that $\omega^{\prime}s_{i_{k}}$ is a reduced word, because $\omega$ is as
well, so $l(\omega^{\prime}s_{i_{k}})>l(\omega^{\prime})$, so
$\omega^{\prime}(\alpha_{k})$ is a positive root. ∎

Moreover, the element $\omega^{\prime}s_{\alpha_{k}}(\omega^{\prime})^{-1}$ is
a reflection in a positive root $\beta$ ([Hum90, Lemma 5.7]), so when we take
$t=s_{\beta}$, the desired equality holds:

$\omega t=\omega
s_{\beta}=\omega\omega^{\prime}s_{i_{k}}(\omega^{\prime})^{-1}=s_{i_{1}}\ldots\widehat{s_{i_{k}}}\ldots
s_{i_{n}}.$

∎

####  4.1.3 Part III - support of Chevalley formula

Let $\mathrm{sgn}\text{: }\mathbb{Z}\to\\{+,-,0\\}$ denote the function of the
sign of a number. The following lemma holds.

######  Lemma 4.6.

For $\beta$, $\omega^{\prime}$ and $\alpha_{k}$ defined as above and for
$\alpha$ a root, the following equality holds

$\mathrm{sgn}\langle(\omega^{\prime})^{-1}(\alpha),\alpha_{k}^{\vee}\rangle=\mathrm{sgn}(\beta,\alpha).$

###### Proof.

$\displaystyle\mathrm{sgn}\langle(\omega^{\prime})^{-1}(\alpha),\alpha_{k}^{\vee}\rangle$
$\displaystyle=\mathrm{sgn}\frac{2((\omega^{\prime})^{-1}(\alpha),\alpha_{k})}{(\alpha_{k},\alpha_{k})}=\mathrm{sgn}(2((\omega^{\prime})^{-1}(\alpha),\alpha_{k}))=$
$\displaystyle=\mathrm{sgn}((\alpha,\omega^{\prime}(\alpha_{k})))=\mathrm{sgn}((\omega^{\prime}(\alpha_{k}),\alpha))=\mathrm{sgn}(\beta,\alpha)$

∎

From Definition 3.2 we know, that for a fixed $\omega$ and a fixed $i$ the
support of length $l(\omega)$ of left-multiplication $[\mathcal{O}_{\omega
s_{i}}]$ by $[\mathcal{L}(-\alpha_{i})]$ consist of elements
$[\mathcal{O}_{\omega^{\prime\prime}}]$ such that
$\omega^{\prime\prime}=s_{i_{1}}\ldots\widehat{s_{i_{k}}}\ldots
s_{i_{n}}s_{i}$ and
$\langle(\omega^{\prime})^{-1}(\alpha_{i}),\alpha_{k}^{\vee}\rangle\neq 0$ for
$\omega^{\prime}$ defined as in the Subsection 4.1.2. By Lemma 4.2 this is
equivalent to $\omega^{\prime\prime}=\omega s_{i}s_{\beta}$ for positive roots
$\beta$ satisfying $(\beta,\alpha_{i})\neq 0$ and $l(\omega
s_{i}s_{\beta})=l(\omega)$. The only detail demanding explanation now is that
the coefficient of $[\mathcal{O}_{\omega}]$ in
$[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega s_{i}}]$ is not equal to
$e^{-\omega(\alpha_{i})}$. If that was a case, the support of
$s_{i}^{K}([\mathcal{O}_{\omega}])$ would not contain $[\mathcal{O}_{\omega}]$
from Equation (2) (because
$[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega}]=e^{-\omega(\alpha_{i})}[\mathcal{O}_{\omega}]+l.o.t.$).
The Chevalley formula implies that this coefficient is equal to
$e^{-\omega(\alpha_{i})}+1$. Therefore, Theorem 4.1 is proved.

###  4.2 Example of $SL_{n}/B$

In this section, the main goal is to study the automorphism $s_{i}^{K}$ on
$K_{T}(SL_{n}/B)$. In this example, we can calculate coefficients $a_{\omega
s_{i}s_{\beta}}$ from Theorem 4.1.

Every two different roots $\alpha,\beta$ of the Lie algebra
$\mathfrak{sl}_{n}$ satisfy $(\alpha,\beta)\in\\{-1,0,1\\}$, so the operator
$T^{0}_{\beta}$ in the Chevalley formula (Definition 3.2) simplifies to

$\displaystyle T_{\beta}^{0}(e^{\lambda})$
$\displaystyle=\begin{dcases*}0&\text{if }
$\langle\lambda,\beta^{\vee}\rangle=0$,\\\ e^{\lambda}&\text{if }
$\langle\lambda,\beta^{\vee}\rangle=1$,\\\ e^{\lambda}+1&\text{if }
$\lambda=\beta$\\\ -1-e^{-\lambda}&\text{if } $\lambda=-\beta$\\\
-e^{\lambda+\beta}&\text{if }
$\langle\lambda,\beta^{\vee}\rangle=-1$.\end{dcases*}$ (3)

First, we calculate the coefficient of the element $[\mathcal{O}_{\omega}]$ in
$[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega s_{i}}]$. Equation (3) and
Lemma 4.2 show that this coefficient is equal to $1+e^{-\omega(\alpha_{i})}$.
Equation (2) gives

$s_{i}^{K}([\mathcal{O}_{\omega}])=e^{-\omega(\alpha_{i})}[\mathcal{O}_{\omega}]-(1+e^{-\omega(\alpha_{i})})[\mathcal{O}_{\omega}]+\ldots=-[\mathcal{O}_{\omega}]+\ldots.$

Since we know $a_{\omega}=-1$, we calculate the coefficient $a_{\omega
s_{i}s_{\beta}}$ of the element $[\mathcal{O}_{\omega s_{i}s_{\beta}}]$ in
$[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{\omega s_{i}}]$. Let us assume that
$\omega s_{i}s_{\beta}$ is the subword of $\omega s_{i}=s_{i_{1}}\ldots\ldots
s_{i_{n}}s_{i}$ with one element removed as follows:
$s_{i_{1}}\ldots\widehat{s_{i_{k}}}\ldots s_{i_{n}}s_{i}$. Let
$\lambda=s_{i_{k+1}}\ldots s_{i}(-\alpha_{i})$. By the Chevalley formula,
$a_{\omega s_{i}s_{\beta}}$ is equal to

$\displaystyle T_{-\alpha_{i_{1}}}^{1}\ldots T_{-\alpha_{i_{k}}}^{0}\ldots
T_{-\alpha_{i_{n}}}^{1}T_{-\alpha_{i}}^{1}(e^{-\alpha_{i}})$
$\displaystyle=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k}}}^{0}(e^{s_{i_{k+1}}\ldots s_{i}(-\alpha_{i})})=$
$\displaystyle=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k}}}^{0}(e^{\lambda}).$

We will prove, that $\lambda\neq\pm\alpha_{i_{k}}$. Firstly, let us assume
that $i_{k}\neq i$. By [Hum90, Theorem 5.4] if $l(\omega
s_{\alpha})>l(\omega)$ for a simple root $\alpha$, then $\omega(\alpha)$ is a
positive root. Applying this to $\omega^{\prime}=s_{i_{k+1}}\ldots s_{i_{n}}$
and $\omega^{\prime\prime}=s_{i_{k}}\ldots s_{i_{n}}$ implies that both
$\lambda$ and $-\lambda$ are simple roots, which is a contradiction. If
$i_{k}=i$, then Lemma 4.2 implies that
$s_{i_{1}}\ldots\widehat{s_{i_{k}}}\ldots s_{i_{n}}s_{i}=\omega$ and this case
is already calculated. From Equation (3), three options arise.

  * •

$\langle\lambda,-\alpha_{k}^{\vee}\rangle=1$. From Equation (3)

$\displaystyle a_{\omega s_{i}s_{\beta}}$
$\displaystyle=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k}}}^{0}(e^{\lambda})=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k-1}}}^{1}(e^{\lambda})=$
$\displaystyle=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k-1}}}^{1}(e^{s_{i_{k+1}}\ldots s_{i}(-\alpha_{i})})=e^{-\omega
s_{i}s_{\beta}(\alpha_{i})}.$

  * •

$\langle\lambda,-\alpha_{k}^{\vee}\rangle=0$, which implies that
$T_{-\alpha_{i_{k}}}^{0}(e^{\lambda})$ is equal to zero and therefore
$a_{\omega s_{i}s_{\beta}}=0$.

  * •

$\langle\lambda,-\alpha_{k}^{\vee}\rangle=-1$. Equation (3) gives

$T_{-\alpha_{i_{k}}}^{0}(e^{\lambda})=-e^{\lambda-\alpha_{k}}$

Now, let us establish the conditions for $\lambda$ to satisfy the equation
$\langle\lambda,-\alpha_{k}^{\vee}\rangle=-1$. In $\mathfrak{sl_{n}}$,
presenting an element as the sum of simple roots makes every coefficient equal
to either $-1$, $0$ or $1$. Moreover, if coefficients of $\alpha_{i-1}$ and
$\alpha_{i+1}$ are non-zero, then so is the coefficient of $\alpha_{i}$ and
all of them are equal. If we want the equality
$\langle\lambda,-\alpha_{k}^{\vee}\rangle=-1$ to hold, then $\lambda$ written
as a sum of simple roots must look as follows:
$\alpha_{i}+\ldots+\alpha_{k-1}+\alpha_{k}$ or
$\alpha_{k}+\alpha_{k+1}+\ldots+\alpha_{j}$. In both cases
$s_{k}(\lambda)=\lambda-\alpha_{k}$, because

$s_{i}(\alpha_{j})=\begin{dcases*}\alpha_{i}+\alpha_{j}&\text{ if } $i=j\pm
1$\\\ -\alpha_{j}&\text{ if } $i=j$\\\ \alpha_{j}&\text{ otherwise.}\\\
\end{dcases*}$

That gives us the following

$T_{-\alpha_{i_{k}}}^{0}(e^{\lambda})=-e^{\lambda-\alpha_{k}}=-e^{s_{k}(\lambda)}.$

It follows that

$\displaystyle a_{\omega s_{i}s_{\beta}}$
$\displaystyle=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k}}}^{0}(e^{\lambda})=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k-1}}}^{1}(-e^{s_{k}(\lambda)})$
$\displaystyle=T_{-\alpha_{i_{1}}}^{1}\ldots
T_{-\alpha_{i_{k-1}}}^{1}(-e^{s_{k}(s_{i_{k+1}}\ldots
s_{i}(-\alpha_{i}))})=-e^{\omega s_{i}(-\alpha_{i})}=-e^{\omega(\alpha_{i})}.$

Combining these results with the Equation (2) and Lemma 4.6, we obtain:

$\displaystyle s_{i}^{K}([\mathcal{O}_{\omega}])$
$\displaystyle=(1-e^{\omega(\alpha_{i})})[\mathcal{O}_{\omega
s_{i}}]-[\mathcal{O}_{\omega}]$ $\displaystyle-\sum_{\beta\neq\alpha_{i}\text{
positive root such that }l(\omega s_{i}s_{\beta})=l(\omega)\text{ and
}\langle\alpha_{i},\beta\rangle>0}e^{-\omega
s_{i}s_{\beta}(\alpha_{i})}[\mathcal{O}_{\omega s_{i}s_{\beta}}]$
$\displaystyle+\sum_{\beta\neq\alpha_{i}\ \text{ positive root such that
}l(\omega s_{i}s_{\beta})=l(\omega)\text{ and
}\langle\alpha_{i},\beta\rangle<0}e^{\omega(\alpha_{i})}[\mathcal{O}_{\omega
s_{i}s_{\beta}}]$ $\displaystyle+l.o.t.$ (4)

As an example of this formula, we are presenting the full formula for the
$s_{1}$ action below, in the case of the flag variety $SL_{3}/B$.

| $[\mathcal{O}_{\mathrm{id}}]$ | $[\mathcal{O}_{s_{1}}]$ | $[\mathcal{O}_{s_{2}}]$ | $[\mathcal{O}_{s_{1}s_{2}}]$ | $[\mathcal{O}_{s_{2}s_{1}}]$ | $[\mathcal{O}_{s_{1}s_{2}s_{1}}]$
---|---|---|---|---|---|---
$[\mathcal{O}_{\mathrm{id}}]$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
$[\mathcal{O}_{s_{1}}]$ | $1-e^{\alpha_{1}}$ | $1$ | $-e^{\alpha_{1}}$ | $1$ | $0$ | $0$
$[\mathcal{O}_{s_{2}}]$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $0$
$[\mathcal{O}_{s_{1}s_{2}}]$ | $0$ | $0$ | $0$ | $-1$ | $0$ | $0$
$[\mathcal{O}_{s_{2}s_{1}}]$ | $0$ | $0$ | $1-e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{2}}$ | $1$ | $0$
$[\mathcal{O}_{s_{1}s_{2}s_{1}}]$ | $0$ | $0$ | $0$ | $1-e^{\alpha_{2}}$ | $0$ | $1$
Figure 1: Matrix of automorphism $s_{1}$ in the basis $\mathcal{B}$ for the
flag variety $SL_{3}/B$.

##  5 Properties of action of Weyl group in
$\text{MC}_{y}(X_{\omega}^{\circ})$ basis

In this chapter we study the automorphism $s_{i}^{K}$ in the
$mC_{y}(X_{\omega}^{\circ})$ basis of $K_{T}(G/B)_{loc}$. Using Lemma 2.11.

$\displaystyle s_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))$
$\displaystyle=[\mathcal{L}(-\alpha_{i})]mC_{y}(X_{\omega}^{\circ})-([\mathcal{L}(-\alpha_{i})]-1)\partial_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))$

Since $[\mathcal{L}(\alpha_{i})][\mathcal{L}(-\alpha_{i})]=\mathrm{id}$
(because $e^{\alpha_{i}}\cdot e^{-\alpha_{i}}=1$), we get

$\displaystyle s_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))$
$\displaystyle=[\mathcal{L}(-\alpha_{i})]\left(mC_{y}(X_{\omega}^{\circ})-(1-[\mathcal{L}(\alpha_{i})])\partial_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))\right).$
(5)

From Definition 1.7, the following holds for $y\neq 0$:

$[\mathcal{L}(\alpha_{i})]\partial_{i}^{K}=\frac{\mathcal{T}_{i}-\partial_{i}^{K}+\mathrm{id}}{y}.$

That leads to the equation

$\displaystyle s_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))=$
$\displaystyle=[\mathcal{L}(-\alpha_{i})]\left(mC_{y}(X_{\omega}^{\circ})-\partial_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))+\frac{\mathcal{T}_{i}-\partial_{i}^{K}+\mathrm{id}}{y}(mC_{y}(X_{\omega}^{\circ}))\right)=$
$\displaystyle=[\mathcal{L}(-\alpha_{i})]\frac{mC_{y}(X_{\omega
s_{i}}^{\circ})-(y+1)\partial_{i}^{K}(mC_{y}(X_{\omega}^{\circ}))+(y+1)mC_{y}(X_{\omega}^{\circ})}{y}.$

Unfortunately, in this basis no easy description of the action
$\partial_{i}^{K}$ is known, which makes such presentation unuseful. For
$y=-1$ the situation is much easier as we will see further in this chapter.

Another basis worth considering is the basis of motivic Chern classes with
$y=0$, which is the basis of classes
$\mathcal{I}_{\omega}=[\mathcal{O}_{\omega}]-[\mathcal{O}_{\partial
X_{\omega}}]$. They satisfy the following equation

$(\partial_{i}^{K}-\mathrm{id})(\mathcal{I}_{\omega})=\mathcal{T}_{i}(\mathcal{I}_{\omega})=\begin{dcases*}\mathcal{I}_{\omega
s_{i}}&$\text{ if }l(\omega s_{i})>l(\omega)$\\\ -\mathcal{I}_{\omega}&\text{
otherwise.}\end{dcases*}$

Therefore

$s_{i}^{K}(\mathcal{I}_{\omega})=\begin{dcases*}(\mathrm{id}-[\mathcal{L}(-\alpha_{i})])\mathcal{I}_{\omega
s_{i}}+\mathcal{I}_{\omega}&$\text{ if }l(\omega s_{i})>l(\omega)$\\\
[\mathcal{L}(-\alpha_{i})]\mathcal{I}_{\omega}&\text{
otherwise.}\end{dcases*}$

As we have just shown, to describe the automorphism $s_{i}^{K}$ in this basis,
we need to use the Chevalley formula only once, not - as in the basis
$\mathcal{B}$ \- twice. The description of the Chevalley formula in the
$\mathcal{I}_{\omega}$ basis demands further calculations.

| $[\mathcal{I}_{\mathrm{id}}]$ | $[\mathcal{I}_{s_{1}}]$ | $[\mathcal{I}_{s_{2}}]$ | $[\mathcal{I}_{s_{1}s_{2}}]$ | $[\mathcal{I}_{s_{2}s_{1}}]$ | $[\mathcal{I}_{s_{1}s_{2}s_{1}}]$
---|---|---|---|---|---|---
$[\mathcal{I}_{\mathrm{id}}]$ | $-e^{\alpha_{1}}$ | $1+e^{\alpha_{1}}$ | $-e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $-e^{\alpha_{1}+\alpha_{2}}$
$[\mathcal{I}_{s_{1}}]$ | $1-e^{\alpha_{1}}$ | $e^{\alpha_{1}}$ | $-e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $-e^{\alpha_{1}+\alpha_{2}}$
$[\mathcal{I}_{s_{2}}]$ | $0$ | $0$ | $-e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $1+e^{\alpha_{1}+\alpha_{2}}$ | $-e^{\alpha_{1}+\alpha_{2}}$
$[\mathcal{I}_{s_{1}s_{2}}]$ | $0$ | $0$ | $0$ | $-e^{\alpha_{2}}$ | $0$ | $1+e^{\alpha_{2}}$
$[\mathcal{I}_{s_{2}s_{1}}]$ | $0$ | $0$ | $1-e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $e^{\alpha_{1}+\alpha_{2}}$ | $-e^{\alpha_{1}+\alpha_{2}}$
$[\mathcal{I}_{s_{1}s_{2}s_{1}}]$ | $0$ | $0$ | $0$ | $1-e^{\alpha_{2}}$ | $0$ | $e^{\alpha_{2}}$
Figure 2: Matrix of automorphism $s_{1}$ in the $[\mathcal{I}_{\omega}]$ basis
for the flag variety $SL_{3}/B$.

The simplest case is the basis of motivic Chern classes with fixed $y=-1$. In
such basis, the automorphism $s_{i}$ is well described. The following equality
holds ([AMSS22, Lemma 3.7]):

$[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{e_{\omega}}]=e^{-\omega\alpha_{i}}[\mathcal{O}_{e_{\omega}}].$
(6)

Therefore, Definition 1.7 takes the following form

$\mathcal{T}_{i}^{K}=\partial_{i}^{K}-[\mathcal{L}(\alpha_{i})]\partial_{i}^{K}-\mathrm{id}$

and Equation (5) presents as

$s_{i}^{K}([\mathcal{U}])=[\mathcal{L}(-\alpha_{i})](-\mathcal{T}_{i}([\mathcal{U}])).$

Substituting $[\mathcal{O}_{e_{\omega}}]$ in the latter leads to the following
formula

$s_{i}^{K}([\mathcal{O}_{e_{\omega}}])=-[\mathcal{L}(-\alpha_{i})](\mathcal{T}_{i}([\mathcal{O}_{e_{\omega}}]))=-[\mathcal{L}(-\alpha_{i})][\mathcal{O}_{e_{\omega
s_{i}}}]$

which, combined with the Equation (6), results in the final formula

$s_{i}^{K}([\mathcal{O}_{e_{\omega}}])=-e^{\omega\alpha_{i}}[\mathcal{O}_{e_{\omega
s_{i}}}].$

Unfortunately, topological properties of this basis are not interesting,
because elements of it are defined as the classes of the sheaf over point,
which does not provide much information about the topology of the flag
variety. However, we can write the action of the Weyl group in it explicitly,
which in itself is interesting.

| $[\mathcal{O}_{e_{\mathrm{id}}}]$ | $[\mathcal{O}_{e_{s_{1}}}]$ | $[\mathcal{O}_{e_{s_{2}}}]$ | $[\mathcal{O}_{e_{s_{1}s_{2}}}]$ | $[\mathcal{O}_{e_{s_{2}s_{1}}}]$ | $[\mathcal{O}_{e_{s_{1}s_{2}s_{1}}}]$
---|---|---|---|---|---|---
$[\mathcal{O}_{e_{\mathrm{id}}}]$ | $0$ | $-e^{-\alpha_{1}}$ | $0$ | $0$ | $0$ | $0$
$[\mathcal{O}_{e_{s_{1}}}]$ | $-e^{\alpha_{1}}$ | $0$ | $0$ | $0$ | $0$ | $0$
$[\mathcal{O}_{e_{s_{2}}}]$ | $0$ | $0$ | $0$ | $0$ | $-e^{-\alpha_{1}-\alpha_{2}}$ | $0$
$[\mathcal{O}_{e_{s_{1}s_{2}}}]$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-e^{-\alpha_{2}}$
$[\mathcal{O}_{e_{s_{2}s_{1}}}]$ | $0$ | $0$ | $-e^{\alpha_{1}+\alpha_{2}}$ | $0$ | $0$ | $0$
$[\mathcal{O}_{e_{s_{1}s_{2}s_{1}}}]$ | $0$ | $0$ | $0$ | $-e^{\alpha_{2}}$ | $0$ | $0$
Figure 3: Matrix of automorphism $s_{1}$ in the $[\mathcal{O}_{e_{\omega}}]$
basis for the flag variety $SL_{3}/B$.

##  6 Conclusions

In order to calculate the coefficients of the automorphism $s_{i}^{K}$, we
must multiply the class of Schubert variety with the class of a linear bundle
twice, which means using the Chevalley formula twice, which is computationally
challenging. Switching to the motivic Chern classes basis changes the
Chevalley formula (shown in [MNS24]), but the formula is still
combinatorically too complicated to simplify these calculations. As we saw in
Chapter 3, the problem with the Chevalley formula is that the support of it is
not described in terms of the elements of the Weyl group, but the monoid
$W^{\prime}$. We only know that the support is subset of
$\\{[\mathcal{O}_{\omega^{\prime}}]\\}$ for $\omega^{\prime}$ being the
subword of $\omega s_{i}$. That is because, for example, a certain subword can
be constructed in two ways in the Weyl group, but in three ways in the monoid
$W^{\prime}$, as we explain below.

Let us take a word $s_{1}s_{2}s_{1}$ in the Weyl group of $SL_{n}$. It has a
subword $s_{1}$. It can be constructed by removing $s_{2}$ and the last
$s_{1}$ or by removing $s_{2}$ and the first $s_{1}$. However, in this monoid
a relation $s_{1}s_{1}=s_{1}$ holds, so there are three constructions possible
- the two mentioned above and one additional, executed by only removing
$s_{2}$. These additional combinations can transform element from being in the
support to not being in the support and the other way around. For example the
coefficient of $[\mathcal{O}_{\mathrm{id}}]$ in
$s_{1}^{K}[\mathcal{O}_{s_{1}s_{2}}]$ is equal to zero (Figure 1), but would
be non-zero if we would consider only subwords in the Weyl group because one
non-zero term of $[\mathcal{L}(-\alpha_{1})][\mathcal{O}_{s_{1}s_{2}s_{1}}]$
is equal to

$T_{\alpha_{1}}^{1}T_{\alpha_{2}}^{0}T_{\alpha_{1}}^{0}(e^{-\alpha_{1}}).$

A possible solution to this problem might be a different choice of bases or
indexing elements of basis $\mathcal{B}$ not by elements of the Weyl group
$W$, but by elements of the monoid (or, in fact, the distinguished basis of
Hecke algebra) $W^{\prime}$. These two objects have the same elements, but the
fact that the relations holding in the monoid are different from the Weyl
group could help to describe the support of the automorphism $s_{i}^{K}$
explicitely.

## References

  * [AGM11] Dave Anderson, Stephen Griffeth and Ezra Miller  “Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces”  In _J. Eur. Math. Soc. (JEMS)_ 13.1, 2011, pp. 57–84  DOI: 10.4171/JEMS/244
  * [AM16] Paolo Aluffi and Leonardo C. Mihalcea  “Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds”  In _Compos. Math._ 152.12, 2016, pp. 2603–2625  DOI: 10.1112/S0010437X16007685
  * [AMSS22] Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann and Changjian Su  “Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem”, 2022  arXiv:1902.10101 [math.AG]
  * [AMSS23] Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann and Changjian Su  “Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells”  In _Duke Math. J._ 172.17, 2023, pp. 3257–3320  DOI: 10.1215/00127094-2022-0101
  * [Ara89] Alberto Arabia  “Cohomologie $T$-équivariante de la variété de drapeaux d’un groupe de Kac-Moody”  In _Bull. Soc. Math. France_ 117.2, 1989, pp. 129–165  URL: http://www.numdam.org/item?id=BSMF_1989__117_2_129_0
  * [AS17] Praise Adeyemo and Frank Sottile  “Equivariant cohomology theories and the pattern map”  In _Houston J. Math._ 43.2, 2017, pp. 375–393  DOI: 10.1162/coli\\_a\\_00286
  * [BGG73] I. N. Bernšteĭn, I. M. Gelfand and S. I. Gelfand  “Schubert cells, and the cohomology of the spaces $G/P$”  In _Uspehi Mat. Nauk_ 28.3(171), 1973, pp. 3–26
  * [Bor91] Armand Borel  “Linear algebraic groups” 126, Graduate Texts in Mathematics  Springer-Verlag, New York, 1991, pp. xii+288  DOI: 10.1007/978-1-4612-0941-6
  * [Bri02] Michel Brion  “Positivity in the Grothendieck group of complex flag varieties” Special issue in celebration of Claudio Procesi’s 60th birthday  In _J. Algebra_ 258.1, 2002, pp. 137–159  DOI: 10.1016/S0021-8693(02)00505-7
  * [Dem74] Michel Demazure  “Désingularisation des variétés de Schubert généralisées”  In _Ann. Sci. École Norm. Sup. (4)_ 7, 1974, pp. 53–88  URL: http://www.numdam.org/item?id=ASENS_1974_4_7_1_53_0
  * [FRW21] László M. Fehér, Richárd Rimányi and Andrzej Weber  “Motivic Chern classes and K-theoretic stable envelopes”  In _Proc. Lond. Math. Soc. (3)_ 122.1, 2021, pp. 153–189  DOI: 10.1112/plms.12374
  * [Ful97] William Fulton  “Young tableaux” With applications to representation theory and geometry 35, London Mathematical Society Student Texts  Cambridge University Press, Cambridge, 1997, pp. x+260
  * [HLS10] Megumi Harada, Gregory D. Landweber and Reyer Sjamaar  “Divided differences and the Weyl character formula in equivariant $K$-theory”  In _Math. Res. Lett._ 17.3, 2010, pp. 507–527  DOI: 10.4310/MRL.2010.v17.n3.a10
  * [Hum90] James E. Humphreys  “Reflection groups and Coxeter groups” 29, Cambridge Studies in Advanced Mathematics  Cambridge University Press, Cambridge, 1990, pp. xii+204  DOI: 10.1017/CBO9780511623646
  * [Knu03] Allen Knutson  “A Schubert calculus recurrence from the noncomplex W-action on G/B”, 2003  arXiv:math/0306304 [math.CO]
  * [Kum02] Shrawan Kumar  “Kac-Moody groups, their flag varieties and representation theory” 204, Progress in Mathematics  Birkhäuser Boston, Inc., Boston, MA, 2002, pp. xvi+606  DOI: 10.1007/978-1-4612-0105-2
  * [MNS22] Leonardo C. Mihalcea, Hiroshi Naruse and Changjian Su  “Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K-theory”  In _Int. Math. Res. Not. IMRN_ , 2022, pp. 12096–12147  DOI: 10.1093/imrn/rnab049
  * [MNS24] Leonardo C. Mihalcea, Hiroshi Naruse and Changjian Su  “Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes”, 2024  arXiv:2312.17200 [math.AG]
  * [Tho87] R. W. Thomason  “Algebraic $K$-theory of group scheme actions”  In _Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983)_ 113, Ann. of Math. Stud.  Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563
  * [Uma14] V. Uma  “Equivariant $K$-theory of flag varieties revisited and related results”, 2014  arXiv:1101.5911 [math.AG]
  * [Wil07] Matthieu Willems  “A Chevalley formula in equivariant $K$-theory”  In _J. Algebra_ 308.2, 2007, pp. 764–779  DOI: 10.1016/j.jalgebra.2006.07.031