text:
Why is Cameron more stupid than Erdogan?
Internationalist Observer | 11.11.2013 20:46 | Analysis | Anti-Nuclear | Ocean Defence
It is not a matter of verification but of explanation: The quote-unquote nuclear renaissance – more precisely to be described as the scorched Earth economy – of the London regime follows a pattern of penetration of an uninformed public that is known from regions where atomic power is mystified as the last step of colonial independence and pushed into the faces of the people regardless of specific value for geopolitical triangulation. The post-colonial regimes are repeating the self-destructive mistakes of the colonialist and its heirs in order to overtake their failing economies and avoid more of the same, and the internal contradiction of that ambition transforms into mathematically suicidal projections of atomic power generation. Now since the last big reactor meltdown has slashed all economic foundations for atomic power in Europe, a financial blowback from these spoiled colonies serves as the new centerpiece of the so-called nuclear ideology there. That means Japanese reactors at the Black Sea for the Turkish politician Erdogan to assert himself of what he seems to feel as a lack of power, and now Chinese-financed French reactors in Britain. This pattern is being called suicidal because it is designed to evade meltdown compensation claims so that a catastrophe there would be likely to take down the local currency with it, since neighbours might find it even more difficult to take external enablers into account. In such a distorted situation the people do no longer have reactors, but the reactors have the people. And the most stupid politician would be the one betting everything on an absurd expectation that when the reactor meltdown comes everyone else would look even more stupid.
In fact the choice of location does not matter, because already a quick glance at the geography of the islands reveals that there is no spot where such a facility could be inserted without causing a speculation bubble involving neighbours, the local mouth of the ocean and beyond, and arbitrary territories outside the currency. In addition to all the political friction caused by atomic power stations within the euro currency but taking space across political borders, in that instance there also is economic friction because this enabling construct is reaching across economic borders, in an attempt of geopolitical triangulation in the global trade war which makes the former colonialist appear as a colony of a terminal compensation evasion scheme. And since compensation requests would also have to be made within the euro currency, even if that of Britain was to be considered of no value at last, against the corporation producing the reactors, it can also be expected to undermine the current political containment of these disputes there.
The closer look at the British situation reveals that ever since the islands were populated, large settlements have been avoided at coast lines bearing unframeable oceanic risks that now seem to be the driving force of the compensation evasion scheme. The speculation is not only not avoiding the dangerous risk, it is deliberately seeking it in order to distribute as much of it as it can to others. These others involve both its enablers and its opponents, and it is at this point where there might appear constellations putting an end to the atomic power farce. But China is a confused territory or set of territories of the clumsiness of a colonial colossus, and so is its no less spoiled South-Western neighbour. The proliferation of reactors literally built on a commercial bubble in South-Eastern Asia has preceded that development in Europe with Japanese reactors in Vietnam, and the Berlin regime is putting reactors into Brasil for which it is not prepared to shoulder the cost of accidental damage they might cause, only these of contractors dropping out are covered.
But stupidity never is adequately expressed only in terms of the risks being taken, as even though only theoretically there always might be a bigger risk making everything completely different. A solar storm could cause more damage in a moment than all atomic bombs of the morbid empire and its potential successors together. And it is true that the imperialist policy apparatus which has made itself entirely dependent upon the economic aberration it is still promoting, in its fundamental cluelessness is being influenced much more by the matter of where a fly leaves a dot or not than by the political will of the people. It does not require a technical device of any kind to blind the evil spying system whose name now is in everybody´s mouth, although a more useful purpose for it could hardly be found. That is already being achieved by any arbitrary sensation heating up its baseless speculations to a level where they can no longer stand their own pressure, which can be as little as scholastically imaginable as long as it is appearing in a conflict area targeted with spying. Certainly a system at that stage of measurable instability cannot be declared capable of the administration of atomic facilities and their remainders. Under pressure of overwhelming developments which might somehow occur it would risk to lose focus in such vanities when unexpected circumstances require technically functional shutdown.
The assassination policy reinforces that point, since a morbid system which is so trigger-happy that it is even trying to implicate others as symbolic enablers must not be expected to produce rational decisions over wide-reaching risks. But the sun is so patient not to point directly at these issues and even offer practical choices to avoid them for the time being, because as the central star of this place it is aware that however postponed they will still come up the other day. None of the bigger risks can mandate taking the risk of reactors melting down. That alone could still be just a monstrous error but the ingredient which turns it into a monstrous stupidity is how that error is being pushed in ways any person could only successfully convince themselves of by deliberately ignoring the risks. Of course the only thing more stupid than deliberate ignorance is the leaving of evidence for deliberate ignorance, such as specifically designed incentives for it. This is terminally stupid because once intent is proven it amounts to political-economical suicide, like any dead-end strategy based on internal contradiction.
Cameron argued for atomic power to "kickstart" the economy, and that atomic was not a fossil energy although it is based on uranium mining. In the treacherous metaphor of the organic combustion engine that is being kickstarted in a functional procedure, he reveals that inorganic fossil energy is no less fossil, it is even using the same technical metaphors although they are lacking any functional context there since a steam engine cannot be set running that way. But since uranium is just another type of fossil fuel it cannot provide the reliability its proponents are trying to assure, because its calculations are based on the compensation evasion bubble. At this point the economy of the fission of the atom mirrors its technology – all the risk along the process is being shifted to the most risky scenario in order to make the entire process appear less dysfunctional than it really is. For Cameron this means that in his expectation he will already have spent the money and the imagination that he had actually achieved something when the worst case, which then would not just remain one of the reactors, happens to occur.
Given that he inherited a set of power stations which literally are in mafia conditions, with the old waste piling up front of them for a disposal service that never arrives, the deliberate ignorance of the future is a quality whose lack would disqualify for the function of an useful idiot of energy corporations, but that cannot exonerate anything. In fact the only surprise is, that such a reaction to the failure of the speculation that the atomic waste problem would be solved within the operation span of these power stations does first occur in Britain and not in Unitedstates. In the latter the problem of worn-out reactors (first of all inoperable due to getting lost in stashes of spent fuel rods) is even more pressing, and probably the only reason why the grab for new reactors to join the existing cycle of corruption before old ones have been dismantled did not occur there first but in Britain is that in its ongoing decline the morbid empire in the sector of reactor marketing has already been overtaken by its likewise greedy rivals. For Britain this means that without the Chinese investment there would be no atomic power station rather than an American-financed one. It also indicates that China is not doing this to take away a market share from a rival but that faced with the math of reactor meltdowns it is only strong enough to grab there where that rival has already become too weak to do so.
This externalised rivalry suggests that there is yet another central contradiction in the power structure it is resulting from, the one between the fact that the largest opponents of empire are not inspired by antiimperialist intent on one side, and on the other the fact that the capacities for empire are running out quicker than the current one is trying to make itself believe. Yet when their apparent strength depends on the imaginary assumption of unlimited resources that aren´t, it also is their strongest weakness. In the case of Britain this means that Camerons suicidal stupidity makes it and specifically the reactor project a benchmark for the measuring of a qualitative gap between these two powers. As long as the subprime atomic mortgage is upon the country, the Bejing regime has not morally overtaken the Washington regime. And when a meltdown occurs before it comes to that and the nonsensical project is canceled, the economic implications thereof alone are going to make the stupid politician with the inappropriate name appear as having been unaware of the fact that the speculative bubble of the last such instance of scorched Earth economy in Europe was suddenly washed away by a tsunami on the other side of the planet and the politicians promoting it are not even being remembered for what they had been corrupted for.
Internationalist Observer
And YOUR solution is? — MDN
TL; DR — short attention span
I disagree with your analysis. — Anonymous


Since 2001, it's been our mission to develop the highest-quality, innovative housing solutions for communities. The Commonwealth Development team is unique in our breadth of knowledge, experience and ability to deliver housing that exceeds expectations which also adds value and enhances the lives of the residents we serve, as well as communities in which we work. We pursue these goals as a community partner by working closely with neighborhood leaders, urban planners, state and local organizations and governments.
Commonwealth is unique in our breadth of knowledge, experience and ability to deliver housing that exceeds expectations, adds value and enhances the lives of the residents we serve and the communities in which we work. Our new construction developments are constantly evolving to provide our residents with the amenities and design features that they are looking for in the place they call home.
We call it inspiring a community.
Others call it renovation. Every community has obsolete structures that continue to sit vacant and underutilized. The concept of adaptive reuse is to use these historically or architecturally significant buildings in a new, innovative and relevant way. Commonwealth has successful developed and built apartment communities with this sustainable approach.
Preserving the architectural legacy of your community.
Commonwealth believes in the preservation of buildings of historic significance. We seek to preserve and restore the architectural legacy of the communities in which we work. Commonwealth's expereience with these highly sensitive projects ensures that the utmost care is taken when developing and renovating a historic building.
We continue to actively look for opportunities through the development and acquisition of low, moderate and mixed-income properties across the country. Commonwealth is well-positioned to buy out existing owners who lack the resources to recapitalize and expertly rehabilitate the units. Due to our development, construction and management teams' ability to work in concert to re-develop and manage existing properties and portfolios, we are able to execute quickly on the acquisition of these interests.


Our personal information is constantly being collected by third parties without our realization. Every time we use one of our devices, we expose personal details and information to collection by any number of entities that use the data for various purposes. Privacy is clearly on the decline as the use of one device or another becomes a standard and unavoidable part of life. An individual cannot be part of modern society absent a cell phone and/or computer. These facts all have serious implications for criminal defendants.
In 1979, the United States Supreme Court decided Smith v. Maryland. There the Court discussed what has become known as the "Third Party Doctrine", which provides that individuals do not have a reasonable expectation of privacy in information voluntarily given to a third party (e.g., telephone carrier or bank). This information is then available to Government agencies, including law enforcement agencies.
The Court is now scheduled to hear a case that asks what the police and prosecutors can legitimately do with personal data that is collected from third parties. Carpenter v. United States could greatly alter Fourth Amendment principles and procedures as they must be applied in cases involving data resulting from the use of cellphones, computers, and similar devices.The defendant was involved in a series of robberies over a two-year period. Law enforcement obtained cell site location information arising from his use of a cell phone. The police did not obtain a search warrant, but a judge did execute an Order under the Stored Communications Act, which sets the standard for a search at reasonable suspicion, as opposed to probable cause. The cell site information placed him near the crime scenes during the timer periods relevant to the robberies.
A major issue here arises from the fact that Smith was decided approximately 30 years ago, and our modern-day cellphones and computers – things that we all use extensively today – were not conceived of at that time. Yet, a broad application of Smith could erase Fourth Amendment protections for information generated by our use of what have become commonplace devices. by opening up personal data that has been collected by third parties.
In Riley v. California, decided in 2014, the Court ruled that the police need a warrant to search a smartphone recovered from someone who has been arrested. This was based on, among other things, the pervasiveness of cell phones in modern life and their power to store large amounts of data. Riley's holding was also based on the First Amendment and Freedom of speech issues related to sell phone usage. Thus, the Court has recognized the central role that cellphones have come to play in modern life.
But what happens when the data is not in the defendant's cell phone, personal computer or iPad? These devices are invariably connected in some way to a third party. Indeed, it is probably impossible to use them without some transfer of personal data to a third party. Further, the user does not even know this is happening since the data transfer is automatic, and probably does not even know what data is being transferred. But any number of third parties have a considerable store of information about each of us because of our devices. Carpenter presents the Court with the opportunity to (re)structure basic Fourth Amendment protections in light of our use of electronic devices which, in turn, lies at the heart of almost every aspect of modern life. This could easily one of the most far-reaching Fourth Amendment decisions that the Court has decided in recent memory.
James S. Friedman is a criminal defense attorney based in New Brunswick, New Jersey, who handles criminal cases in the Superior Court of New Jersey, the New York State criminal courts located in Manhattan and Brooklyn, all federal courts in New Jersey and New York City, and all New Jersey Municipal Courts.


Body: L-SPARK, a new company based in Kanata, Ontario, Canada that provides incubator and accelerator programs to SaaS companies has partnered with BiteSite to produce a series of videos ranging from promotion to thought leadership and more. BiteSite has been covering L-Spark's events and interviewing their mentors and staff to provide a full picture of what the company is all about. Check out the results at L-Spark's YouTube channel here.


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Project Aspiro:
A career planning and employment resource for people who are blind or partially sighted
Sponsored by World Blind Union
EmployersCurrently selected
Project Aspiro
Working with people with vision loss
Employer Concerns - Productivity
Learning about workers who are blind or partially sighted.
This section is designed for employers who are considering a job candidate who is blind or partially sighted or who currently have an employee who is losing vision. People with vision loss appreciate employers who are informed and interested in working with them – good for you for learning more about this valuable and underrepresented segment of the workforce.
People who are blind or partially sighted are as diverse as the rest of the population – some will be unable to see and others will have considerable useable vision. What's important to understand is that regardless of whether a prospective or current employee has functional vision, they will have received training and will use tools to enable them to do the job required of them.
You can expect as much work from a worker who is blind or partially sighted as you can from your other workers. You can expect an employee with vision loss to be as diligent, attentive to detail, punctual, honest, loyal, and everything else you expect from any other employee.
In this section, you will find answers to general concerns about working with people who are blind or partially sighted. You can also see some tools and accommodations in use by viewing videos of successfully employed individuals in the Success Stories section. These stories include interviews with the worker as well as their employer, who discusses their experience working with an employee with vision loss.
If you have just hired, or are considering hiring, an employee with vision loss you may want to review Working with people with vision loss.
If you have concerns about how someone who is blind or partially sighted can perform certain tasks independently and safely, this section includes articles which address:
Getting around or traveling without good vision
Pertinent links to additional information are also available in the Employers' Resources section.
Find useful links to organizations, programs, directories, and more.
Audio Feature
Hiring someone with vision loss? In this interview, two employment specialists address common questions from employers.
About Project Aspiro
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CNIB: Program Partner
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redpajama_set_name: RedPajamaCommonCrawl


The primary purpose of the Skipper's Chair is to allow any disabled individual full control over the steering of the sailboat. Designed for approximately ninety percent of the population, the major goal of the current project is to maximize the number of readily available parts within the system. In turn, this would reduce the labor needed to manufacture each part to custom specs., while reducing cost. The system also needs to be greatly reduced in weight. Ultimately, the chair must be usable in training, recreation, and competitive activities.
The two main changes that we are looking to improve upon from the last iteration, P13032, are the steering mechanism and the overall stability of the system.
The sailboat model that this will be tailored to will be the Sonar Series.


text:
Alfred Buri, född 1868 och död 1915, var en schweizisk konstnär.

Buri studerade i München och Paris och tog starka intryck av Ferdinand Hodler. Han målade främst Schweiziska landskap och landsbygdsbor i kraftig dekorativ teckning och friska färger.

Källor
Svensk uppslagsbok, Lund 1930

Schweiziska konstnärer
Födda 1868
Avlidna 1915
Män

redpajama_set_name: RedPajamaWikipedia


Availability: Monday morning and evening, Thursday afternoon, Friday morning, Saturday morning.
I am a consultant orthopaedic surgeon at Spire Hartswood in Brentwood, Essex, offering private spinal surgery and general orthopaedic surgery in our clean and comfortable private hospital.
I specialise in medical management and surgery for neck and back pain and spinal disorders, including lumbar stabilisation, disc replacement and fusion.


text:
News Blog >
14/09/2013 - NHS warns killer Legionnaires' disease could be in your garden as bug shuts down school and hospitalises four people
The particular strain of the bug, which shut down Cargilfield Preparatory school and left two elderly victims fighting for their lives yesterday, can be found in compost.
Four Scots gardeners have been struck down by deadly Legionnaires' disease - caught from compost.
Two elderly victims were fighting for their lives in intensive care, while two others have been treated in hospital.
A school was also closed down yesterday after a member of staff was diagnosed with the bug.
Health bosses issued a warning to all gardeners over the rare strain of the killer bug. They are being urged to wear facemasks when handling compost.
All four gardeners affected by the Legionella longbeachae strain come from the Lothian area and are aged between 62 and 84.
NHS Lothian consultant in public health Dr Richard Othieno said "This type of Legionella is quite rare in that, unlike other strains, it has never been identified in man-made water systems, such as cooling towers.
"We are working with experts to trace the source of the infection and samples of the compost have been sent for testing"
"We know that each of the four cases are keen gardeners and had purchased different products containing compost to acquiring the infection".
http://www.dailyrecord.co.uk/news/scottish-news/nhs-warns-killer-legionnaires-disease-2272637


Celeb Pooches is an emerging brand that is tailored to boost the culture of having pets and the bond between pets and owners in Nigeria globally.
Since Celeb Pooches is about celebrating the benefits of having pets, each pet image was strategically chosen to connect personally with the pet owner – current or potential – and engender feelings of importance and nostalgia, i.e. your pet is a star; so you're an even bigger star.
On the posters, banners, etc., we played around a lot with dog puns that would generate eye rolls and chuckles – tools to stimulate memory, attention and the ultimate, brand patronage.
Font: Main – Rockwell; Sub – Montserrat.
Brand Colour Palette: Brown, Yellow, Monochrome.
Brand Idea Representation: A softly curved rectangle with an extending claw tip silhouette.


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Reporters on the Job
• A CITY FROZEN IN TIME: For today's story about Burma (Myanmar), reporter Simon Montlake made his first visit to the capital of Rangoon in five years. "What struck me immediately upon arrival is how Rangoon has stood still during that time, while the rest of Southeast Asia - Bangkok, Singapore, and even Jakarta - has moved into the 21st century," he says. "It was one of the richest British colonies when it gained independence [in 1948]. But now it's a dilapidated place.
"Of course, the preserved-in-aspic feeling of old Asia can be quite charming," Simon notes. "Men and women here still wear the traditional longyi - a sarong wrap. There are still rickshaws in the streets."
• AS SEEN ON TV: The Monitor's Robert Marquand says that evangelical Christians are relatively commonplace in of Seoul. "You can see evangelical choirs in the street and neon crosses," he notes. But his interest in the Yoido Full Gospel Church - the largest Christian church in Asia - was piqued by a pro-US demonstration. "I attended an anti-US rally in Seoul a few months ago, which was followed by a rally in support of American policies. There were 30,000 to 40,000 people at the second rally. I learned that about 25,000 were members of one church, Yoido Church."
Bob says he attended a Sunday service, but didn't get into the main sanctuary. "If you don't come early enough, you won't get a seat. I had to watch the service in the overflow area on a large- screen TV."
• SETTLEMENTS AND THE MEDIA: Israeli settlers are sometimes hostile to the media, apparently out of frustration with how they are represented, says the Monitor's Cameron Barr. So when a reporter visits a settlement - especially the more radical outposts - it means "managing a little anxiety," he says.
But while reporting today's story on the West Bank outposts being dismantled by the Israeli government, Cameron detected a change in attitude.
"This time, compared with settlement forays a couple of years ago, I noticed that people were more kindly disposed toward The Christian Science Monitor than they once were. I can only assume that the rising level of support among some Christians in the US for Israel is rubbing off favorably on the Monitor.
David Clark Scott
World editor
Global News Blog Asian, African megachurches overtaking US congregations
On barren hilltop, Israeli settler vigilantism blurs into Jewish theocracy
Israel evacuates 300 residents from illegal settlement outpost


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Tag / Black Gangsters
November 18, 2021 November 21, 2021 by brandycavalli
Rapper Young Dolph Shot & Killed in Hometown of Memphis
Hollywood, Hollywood Crime, Media, True Crime
Black Gangsters, Gangland, Real Rap, Young Dolph Killed
I can't believe I'm writing this.
Memphis rapper Adolph 'Young Dolph' Thornton Jr. was truly one of the last real ones left. He was not only the CEO of his own very successful record label, Paper Route Empire, but he was also a family man and often times a mentor to troubled youth in his community.
Young Dolph, known as simply "Dolph" was shot & killed on November 17, 2021 while entering Makeda's Cookies in his hometown of Memphis, TN. He was 36 years old.
Back in 2017, Dolph was shot in Hollywood and survived with non life threatening injuries. So far there is not a lot of official details as to who killed Dolph & more importantly — why. The internet is already talking and connecting his murder to long-standing beefs Dolph may have had with other Memphis rappers. The streets are such a dirty place, it's really no telling what the motive behind this is. Sometimes pure hate and jealousy will do it.
I'm a firm believer that when you 'make it big' you leave your hometown — especially if you're from small towns and/or poverty ridden areas aka "the hood". We have seen time and time again where people make it out and they go back "home" to show love and they end up dead. Nipsey Hussle to give a recent example.
As someone who left my hometown over 20 years ago, I completely understand wanting to stay connected to your roots and give back, etc. — but do it discreetly and sparingly. I don't recommend being too visible or trackable these days. Be extremely mindful of who you allow in your space as you begin to elevate. Dare I say it, but some friends and family must be left behind.
Sure crime happens any and everywhere, but it's something extremely dangerous about the people you started with watching you become successful. It creates envy and envy is sometimes deadly.
What are your thoughts? Do you agree that people should leave their old stomping grounds for good once they make it out? Why or why not?
REST IN PEACE YOUNG ADOLPH.
November 4, 2021 by brandycavalli
New Prison Photo Surfaces of BMF Kingpin Big Meech
Hollywood, Hollywood Crime, Media, Photos, True Crime
50 Cent, Behind Bars, Big Meech, BIG Meech 2021, Black Gangsters, Black Mafia Family, BMF, BMF ON STARZ, Cartels, Demetrius Flenory, Dope Tweetz, Gangland, Gangs, kingpins, Real Rap
(L) Demetrius "Big Meech" Flenory 2021
BMF: Bleu Davinci Delivers Warning to 50 Cent Over "BMF" Series
Crime News, Hollywood, Hollywood Crime, True Crime
50 Cent, Big Meech, Black Gangsters, Black Mafia Family, Bleu Davinci, Cartels, Gangland, Gangs, JBO BMF, Lil Meech, Real Rap
"BMF" finally made its way to Starz Network and it is a smash. The streets appear to be loving the made for television series and the internet is on fire with former members trading both threats and shots online.
Apparently Bleu Davinci, one of the original members of the Black Family Organization is at odds with JBo another former leader, who is also involved with the production of the series. JBo posted a meme to Instagram, allegedly of Davinci as Eddie Kane from The Five Heartbeats trying to get back in the group.
Rolling Out — "The only thing I'm saying is that if they handle my name wrong, I'm coming to see you. The writers, the producers, the directors, the executive producers, y'all play with my name — I'm telling y'all right now. Y'all play with my name, it's up. Period. There won't be no more shooting. No more filming that s—. I'm gonna try to find a way to f— up everything about it if you play with [my] name. Soon as I get the first report, 'Aye, bro, they playing' with your name,' I'm coming. Just already know. Have your security beefed up … 'cause I'm gonna give it to they a—, too. I ain't finna play with my name. I'm gonna go all the way out," he stated in the IG post.
50 Cent apparently was not worried about the threats either as he shared JBo's meme as well as a letter from Big Meech to his son Demetrius Flenory Jr, who plays his father in the series.
In the letter, the incarcerated Flenory also salutes 50 Cent for helping turn his family legitimate so they can make their livelihoods from other ways than selling drugs.
Have you been watching the series? Check it out on STARZ NETWORK.


text:
\section*{Introduction}
Fix a reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ with Cartan and Borel subalgebra $\mathfrak{h}\subset \mathfrak{b}$. The abelian category $\mathcal{O}(\mathfrak{g},\mathfrak{b})$ of $\mathfrak{g}$-modules associated to this data was introduced by Bernstein, Gelfand and Gelfand in \cite{BGG}. The simple modules are labelled by $\mathfrak{h}^\ast$. The indecomposable integrable blocks are described by orbits of the Weyl group $W=W(\mathfrak{g}:\mathfrak{h})$ in $\mathfrak{h}^\ast$, and non-integrable blocks by orbits of the relevant integral Weyl subgroups. In \cite{So}, Soergel proved that a block in $\mathcal{O}(\mathfrak{g},\mathfrak{b})$ is, up to equivalence, determined by the data of the relevant integral Weyl subgroup $U< W$ (viewed as a Coxeter system) and the parabolic subgroup $U'<U$ which `stabilises' the block. In particular, this shows that every non-integral block is equivalent to an integral block of a different Lie algebra. Furthermore, we can unambiguously write $\mathcal{O}(U,U')$ to denote the block.

It is known that there are more equivalences between blocks in $\mathcal{O}$ than described by Soergel's theorem. A trivial example is given by the maximally singular block $\mathcal{O}(W,W)$, which is equivalent to the category of vector spaces, for each Weyl group $W$. More refined examples are listed in Theorem~2 below. Our main result is the following theorem.
\begin{thm}
Consider finite Weyl groups $W,U$ with parabolic subgroups $W'<W$ and $U'<U$. The categories $\mathcal{O}(W,W')$ and $\mathcal{O}(U,U')$
 are equivalent if and only if the partially ordered sets $(W/W',\le_B)$ and $(U/U',\le_B)$ are isomorphic, with $\le_B$
the Bruhat order.\end{thm}
Of course, if we have an isomorphism of Coxeter groups $W\stackrel{\sim}{\to}U$ which maps $W'$ to $U'$, the two Bruhat orders are isomorphic and we recover Soergel's theorem. All non-trivial isomorphisms of Bruhat orders correspond to the list in Theorem~2 below.

Unfortunately, we only have a conceptual proof of one direction of Theorem~1, contrary to the results in \cite{So}. Concretely, we prove that any finite dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure. This applies to blocks of category $\mathcal{O}$ and hence any equivalence must be one of highest weight categories. By the BGG theorem, the Bruhat order is an invariant of the highest weight structure of $\mathcal{O}$. Note that this actually implies a stronger statement than the one in Theorem~1, namely any equivalence between two blocks in $\mathcal{O}$ induces an isomorphism of Bruhat orders.

To prove the other direction of Theorem~1 we investigate to which extend we can recover the pair $(W,W')$, for an arbitrary Coxeter group $W$ with parabolic subgroup $W'$, from the poset $(W/W',\le_B)$. A lot of information can be reconstructed from general methods. For instance, we show that for label free simple Coxeter graphs, any Coxeter pair can be reconstructed from the Bruhat order on its parabolic quotient. Applying this and other methods to finite Weyl groups shows that isomorphisms between such posets are extremely rare. In all cases where there exists such an isomorphism, the blocks of category $\mathcal{O}$ are known to be equivalent.

In order to classify blocks in $\mathcal{O}$, we can restrict to simple Lie algebras, see Theorem~\ref{Thm4} below. The following thus yields a complete classification of blocks in category~$\mathcal{O}$.

\begin{thm}
Consider two irreducible finite Weyl groups $W,U$ with parabolic subgroups $W'<W$ and $U'<U$. All non-trivial equivalences $\mathcal{O}(W,W')\simeq\mathcal{O}(U,U')$, excluding the cases $W=W'$ and $U=U'$, are:
\begin{enumerate}
\item $\mathcal{O}(A_{2n+1},A_{2n})\;\simeq\; \mathcal{O}(B_{n+1},B_n),\;\;$ with $n\ge 2$;
\item $\mathcal{O}(B_n, A_{n-1})\;\simeq\; \mathcal{O}(D_{n+1},A_n),\;\;$ with $n\ge 3$;
\item $\mathcal{O}(A_3,A_2)\;\simeq\; \mathcal{O}(B_2,A_1)$;
\item $\mathcal{O}(A_5,A_4)\;\simeq\; \mathcal{O}(G_2,A_1)\;\simeq \mathcal{O}(B_3,B_2)$.
\end{enumerate}
\end{thm}

The paper is organised as follows. In Section~\ref{SecPrel} we recall the necessary background. In Section~\ref{SecQH} we prove the uniqueness of quasi-hereditary structures on algebras with simple preserving duality and mention an application to cellular algebras. In Section \ref{SecRecover} we study the reconstruction of Coxeter pairs from Bruhat orders, culminating in a classification of finite Weyl group pairs with isomorphic Bruhat order. In Section~\ref{SecO} we apply all of the above to prove Theorems 1 and 2 above. We also mention some potential applications in the study of category $\mathcal{O}$ for Kac-Moody algebras and Lie superalgebras.

\section{Preliminaries}\label{SecPrel}
We set $\mathbb{N}=\{0,1,2,\ldots\}$ and $\mathbb{N}^\infty=\mathbb{N}\cup \{\infty\}$. We also use $\mathbb{Z}_{>0}=\{1,2,3,\ldots\}$ and $\mathbb{Z}_{>0}^\infty=\mathbb{Z}_{>0}\cup \{\infty\}$.
Whenever we will refer to the cardinality $|E|$ of some set $E$, the latter will be countable. Hence we can unambiguously consider $|E|$ as an element of $\mathbb{N}^\infty$.

\subsection{Partial orders}

\subsubsection{} Consider a poset $(X,\le)$.
We say that $a\in X$ is a greatest element if $b\le a$ for every $b\in X$. We say that $a\in X$ is a maximal element if $a\le b$ for $b\in X$ implies $b=a$. We also have the dual notions of least and minimal elements.
For $a,b\in X$, we say that $a$ covers $b$ if $b< a$ and $b\le c\le a$ implies $c\in\{a,b\}$. Then we write $b\lhd a$.
We will always use the same notation for the partial order $\le$ restricted to a subset $Y\subset X$.

 \subsubsection{Pointed posets}
A graded poset is a poset $(X,\le)$ is equipped with a rank function $\rho:X\to\mathbb{N}$ such that
$$b\le a\mbox{ implies }\rho(b)\le \rho(a)\quad\mbox{and}\quad b\lhd a\mbox{ implies }\rho(b)=\rho(a)-1.$$
We make the additional normalisation assumption that $\rho^{-1}(0)$ is not empty and write $X_i=\rho^{-1}(i)$, for all $i\in\mathbb{N}$.
The length of a graded poset is defined as
$$\mathcal{L}(X,\le)\;=\; \sup\{i\in\mathbb{N}\,|\, X_n\not=0\}\;\in\,\mathbb{N}^\infty.$$
For example, a finite linear poset $(X,\le)$ can be graded in the obvious way and we have $\mathcal{L}(X,\le)=|X|-1$.

If a graded poset $(X,\le,\rho)$ has a least element then clearly $X_0$ is the singleton containing the least element. Furthermore, the rank function $\rho$ is then uniquely determined by our convention. We call a poset {\bf pointed} if it has a least element and admits a rank function.

\subsection{Quasi-hereditary algebras}
Let $A$ be a finite dimensional associative algebra over a field $k$.

\subsubsection{}Let $\Lambda$ denote the set of isomorphism classes of simple left modules of $A$ and denote by $L(\lambda)$ a representative of $\lambda\in\Lambda$. The category of finite dimensional left $A$-modules is denoted by $A$-mod. We denote the projective cover and injective hull of $L(\lambda)$ in $A$-mod by $P(\lambda)$ and $I(\lambda)$. We use the numbers
$$d_\lambda\,:=\,\mathrm{dim}_k\mathrm{End}_A(L(\lambda))\in\mathbb{Z}_{>0},\quad\mbox{for all $\lambda\in\Lambda$.}$$

 \subsubsection{}When $A$ is considered with a partial order $\le$ on $\Lambda$, we write $(A,\le)$. For such a partial order, following \cite{DR}, for each $\lambda\in\Lambda$ the {\bf standard module} $\Delta(\lambda)$ is defined as the maximal quotient of $P(\lambda)$ such that all simple constituents are of the form $L(\mu)$ with $\mu\le\lambda$. Dually, the costandard module $\nabla(\lambda)$ is the maximal submodule of $I(\lambda)$ with the same condition on simple constituents.

 We say that $(A,\le)$ is quasi-hereditary if its module category is a highest weight category for $\le$. Concretely, this means the following.
\begin{definition}[\cite{CPS}]\label{defqh}
The algebra $(A,\le)$ is quasi-hereditary if for all $\lambda\in\Lambda$:
\begin{enumerate}
\item We have $[\Delta(\lambda):L(\lambda)]=1$.
\item The module $P(\lambda)$ has a filtration with each quotient isomorphic to a standard module $\Delta(\mu)$ with $\lambda\le\mu$.
\end{enumerate}
\end{definition}
If $(A,\le)$ is quasi-hereditary, then \cite[Lemma~2.5]{DR} implies we have the equality
$$(P(\lambda):\Delta(\mu))d_\mu\;=\;[\nabla(\mu):L(\lambda)]d_\lambda,\quad\mbox{for all $\lambda,\mu\in\Lambda$}.$$

\subsubsection{}
If $(A,\le)$ is quasi-hereditary, then so is $(A,\le')$ for every extension $\le'$ of $\le$. Moreover, the (co)standard modules for the two partial orders are identical. This motivates the following definition.
\begin{definition}\label{defess}
Let $(A,\le)$ be quasi-hereditary. We define the essential partial order $\le^e$ of $\le$ on $\Lambda$ as the partial order transitively generated by the following two relations. For $\lambda,\mu\in\Lambda$, we have $\mu\le^e\lambda$ when $[\Delta(\lambda):L(\mu)]\not=0$ or $(P(\mu):\Delta(\lambda))\not=0$.
\end{definition}
Clearly $\le$ is an extension of $\le^e$. We say that two quasi-hereditary structures $(A,\le_1)$ and $(A,\le_2)$ are equivalent if they generate the same essential partial order. This is the same as demanding their standard modules be
 identical.

 \begin{example}
 If $A$ is semisimple, $(A,\le)$ is quasi-hereditary for every partial order $\le$ on $\Lambda$. We then always find $\mu\le^e\lambda$ if and only if $\mu=\lambda$. In particular, all quasi-hereditary structures are equivalent.
 \end{example}

\subsubsection{} Now assume that $A$-mod has an involutive contravariant autoequivalence $\mathbf{d}$ inducing the identity on $\Lambda$. We will simply say that `$A$ has a
simple preserving
duality $\mathbf{d}$'. It follows that $\mathbf{d} P(\lambda)\simeq I(\lambda)$ and, for every partial order, $\mathbf{d}\Delta(\lambda)\simeq \nabla(\lambda)$. In particular, if $(A,\le)$ is quasi-hereditary we find
\begin{equation}
\label{BGG}
(P(\lambda):\Delta(\mu))d_\mu\;=\;[\Delta(\mu):L(\lambda)]d_\lambda,\quad\mbox{for all $\lambda,\mu\in\Lambda$}.
\end{equation}
Consequently, the essential order of $\le$ of Definition~\ref{defess} is in this case generated by the relation $\mu\le^e\lambda$ when $[\Delta(\lambda):L(\mu)]\not=0$.
Equation~\eqref{BGG} also implies that
\begin{equation}
\label{LL}
[P(\lambda):L(\lambda)]\;=\;\sum_{\mu}\frac{d_\lambda}{d_\mu}[\Delta(\mu):L(\lambda)]^2\;=\;\sum_{\mu}\frac{d_\mu}{d_\lambda}(P(\lambda):\Delta(\mu))^2.
\end{equation}

\subsection{Coxeter groups}

\subsubsection{}A Coxeter matrix consists of a countable set $S$ and a symmetric function $m:S\times S\to\mathbb{Z}_{>0}^\infty$ with $m(s,s')=1$ if and only if $s=s'$. To a Coxeter matrix we associate the Coxeter group $W$ with generating set $S$ and relations $(ss')^{m(s,s')}=e$, for all $s,s'\in S$. Whenever we mention a Coxeter group $W$, it is considered as a group together with its defining set of generators $S$, {\it i.e.} as a Coxeter system $(W,S)$. We will freely use the standard results on Coxeter groups from \cite[Section~1]{BB}.

We have the length function $\ell:W\to\mathbb{N}$ of \cite[Section~1.4]{BB} which satisfies in particular $\ell(e)=0$ and $\ell(s)=1$ for $s\in S$.

Following \cite[Section~1.1]{BB}, we can graphically represent a Coxeter matrix as a Coxeter graph. In Appendix~\ref{App} we list those graphs for all finite Weyl groups. Of two vertices $s,t$ in the Coxeter graph ({\it i.e.} two elements in $S$) we say that they are neighbours if there is an edge between them, which is when $m(s,t)>2$, which is in turn equivalent with $st\not=ts$ in $W$.

\subsubsection{Coxeter pairs and bw-Coxeter graphs}  Following \cite[Section~2.4]{BB}, for a subset $J\subset S$ we let $W_J$ be the `parabolic subgroup' of $W$ generated by the set $J$. By a {\bf Coxeter pair} we will mean a pair $(W,W_J)$ of a Coxeter group $W$ (viewed as a Coxeter system) and a parabolic subgroup $W_J<W$. Two Coxeter pairs $(W,W_J)$ and $(U,U_K)$ are isomorphic if there exists an isomorphism $\phi:W\stackrel{\sim}{\to}U$ of Coxeter systems with $\phi(W_J)=U_K$. For example, all the Coxeter pairs of the form $(A_n,A_{n-1})$ are isomorphic, whereas those of the form $(A_n,A_{n-2})$ yield two isomorphism classes.
For a Coxeter pair $(W,W_J)$, we have the set $W^J\subset W$ of shortest (with respect to $\ell$) representatives in $W$ of the parabolic quotient $W/W_J$.
For $K\subset J\subset S$, we simplify the notation $(W_J)^K$ to $W_{J}^K$.

The information of a Coxeter pair $(W,W_J)$ can be graphically represented by the Coxeter graph of $(W,S)$, where the vertices in $J$ are white and the remaining ones black. We call such a graph a {\bf bw-Coxeter graph}. Examples are given in \ref{BU} and the proof of Theorem~\ref{Thm3} below.

\subsubsection{Bruhat orders} The Bruhat order $\le_B$ on $W$ is given in \cite[Definition~2.1.1]{BB}. We will usually employ the equivalent definition in \cite[Corollary~2.2.3]{BB}. By \cite[Theorem~2.5.5]{BB}, the poset $(W^J,\le_B)$ is pointed, for any $J\subset S$. Namely, it is graded with rank function $\ell$ and has least element $e$. If $|W|<\infty$, the length of such a poset is given by
$$\mathcal{L}(W^J,\le_B)\;=\;\ell(w_0w_0^J)\;=\;\ell(w_0)-\ell(w_0^J),$$
with $w_0$ and $w_0^J$ the longest elements of $W$ and $W_J$.

\subsubsection{Coxeter factors}\label{factors} A Coxeter graph naturally decomposes into connected components. A connected graph is known as irreducible and we will likewise refer to Coxeter groups with connected graph as `irreducible Coxeter groups'. For each Coxeter system $(W,S)$ where the graph has finitely many connected components, say $S=\sqcup_iS_i$, we have a canonical  group isomorphism
$W\simeq\prod_iW_i$,
with $W_i:=W_{S_i}$. For $J\subset S$ we write $J_i=J\cap S_i$. When $J_i=S_i$, the factor $W_i$ plays no role in $W^J$. It is therefore convenient to leave out those terms in the factorisation and write
$$W\;\stackrel{J}{=}\;\prod_{i=1}^n W_i,\quad\mbox{if we have}\quad W\,=\, \prod_{i=1}^mW_i\;\mbox{ with $W_i\subset W_J$ if and only if $n<i\le m$.}$$

\subsection{Category $\mathcal{O}$}\label{CatO}
In this section, we work over the field $\mathbb{C}$ of complex numbers. We recall some facts about the BGG category $\mathcal{O}$ from \cite{BGG}. Since we will only need very specific facts later on, we do not give any details.
\subsubsection{}\label{DefO}
For a reductive Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebra $\mathfrak{h}\subset\mathfrak{b}\subset\mathfrak{g}$, category $\mathcal{O}=\mathcal{O}(\mathfrak{g},\mathfrak{b},\mathfrak{h})$ is the full subcategory of the category of finitely generated $\mathfrak{g}$-modules containing the modules which are locally $U(\mathfrak{b})$-finite and semisimple as $\mathfrak{h}$-modules. The simple modules in $\mathcal{O}$ are labelled by $\mathfrak{h}^\ast$. The block decomposition of $\mathcal{O}$ is given in \cite[Section~1.13]{Hu}. For an integral weight $\lambda\in\mathfrak{h}^\ast$, see \cite[Section 0.6]{Hu}, the (isoclasses of) simple modules in the block containing $\lambda$ are labelled by the orbit $W\cdot\lambda$ for $W=W(\mathfrak{g},\mathfrak{h})$ the Weyl group of $\mathfrak{g}$. This Weyl group is a Coxeter group with generators given by reflections with respect to simple roots of $\mathfrak{b}$. Now take $\lambda$ to be anti-dominant, see \cite[Section~3.5]{Hu}, and $W_J<W$ the stabiliser of $\lambda$. The block containing $\lambda$ is denoted by $\mathcal{O}_\lambda$. We thus find a  bijection between $W^J$ and the set of simple modules in $\mathcal{O}_\lambda$, given by $w\mapsto w\cdot\lambda$. Non-integral blocks can be described similarly.

It is shown in \cite[Theorem~11]{So} that, up to equivalence of categories, $\mathcal{O}$ only depends on (the isomorphism class of) the Coxeter pair $(W,W_J)$. We will therefore write $\mathcal{O}(W,W_J)$ instead of $\mathcal{O}_\lambda(\mathfrak{g},\mathfrak{b},\mathfrak{h})$.

\subsubsection{}\label{ResSoergel}By \cite[Proposition~3.13]{Hu}, a block in $\mathcal{O}$ is equivalent to $A$-mod for a finite dimensional algebra, and by \cite[Theorem~3.2]{Hu} it has a simple preserving duality.
 By \cite[Theorem~3.10]{Hu}, the algebra $A$ is quasi-hereditary for the partial order $\le$ as defined in \cite[Section~0.6]{Hu}. Moreover, it is shown in \cite[Section~5.1]{Hu} that the essential order $\le^e$ is the Bruhat order $\le_B$ on $W^J$. Actually, in this special case, the partial order $\le^e$ only contains the generating relation(s) in Definition~\ref{defess}.

\section{Quasi-hereditary algebras with simple preserving duality}\label{SecQH}

Consider a finite dimensional algebra $A$ over a field $k$.

\subsection{Uniqueness of highest weight structure}\label{SecUnique}
The main result of this section states that when $A$ has a simple preserving duality, it admits at most one quasi-hereditary structure, up to equivalence.
\begin{theorem}\label{ThmQH}
Assume $A$ has a simple preserving duality. If for two partial orders $\le_1$ and $\le_2$, both $(A,\le_1)$ and $(A,\le_2)$ are quasi-hereditary, then $\le_1^e=\le_2^e$. Equivalently, the standard modules for both partial orders coincide.
\end{theorem}
\begin{proof}
We denote the standard modules with respect to the partial orders $\le_1$ and $\le_2$ by $\Delta_1(\lambda)$ and $\Delta_2(\lambda)$. We will start by proving that $[\Delta_1(\lambda)]=[\Delta_2(\lambda)]$, in the Grothendieck group $K_0(A)$, by induction on $\lambda$ along $\le_1$.

First assume that $\lambda$ is maximal in $\le_1$. By Definition~\ref{defqh}(2), we find $P(\lambda)=\Delta_1(\lambda)$. By Definition~\ref{defqh}(1), this means in particular that $[P(\lambda):L(\lambda)]=1$. Hence, \eqref{LL} applied to $(A,\le_2)$ implies $P(\lambda)=\Delta_2(\lambda)$. In particular, we find $\Delta_1(\lambda)=\Delta_2(\lambda)$.

Now we fix $\lambda\in\Lambda$ such that for all $\lambda<_1\nu$ we already know that $[\Delta_1(\nu)]=[\Delta_2(\nu)]$. In particular, we define
$$n_\nu:=[\Delta_1(\nu):L(\lambda)]=[\Delta_2(\nu):L(\lambda)]\qquad\mbox{for $\lambda<_1\nu$.}$$
By \eqref{LL} we have
$$1+\sum_{\lambda<_1\nu}\frac{d_\lambda n_\nu^2}{d_\nu}\;=\; [P(\lambda):L(\lambda)]\;=\;1+\sum_{\lambda<_1\nu}\frac{d_\lambda n_\nu^2}{d_\nu}+\sum_{\lambda\not\le_1\mu} \frac{d_\lambda}{d_\mu}[\Delta_2(\mu):L(\lambda)]^2.$$
We thus find $[\Delta_2(\mu):L(\lambda)]=0$ unless $\lambda\le_1\mu$.
 By \eqref{BGG} we find that in $K_0(A)$ we have
$$[\Delta_1(\lambda)]\;=\;[P(\lambda)]-\sum_{\lambda<_1\nu}\frac{d_\lambda n_\nu}{d_\nu}[\Delta_1(\nu)]\;=\;[\Delta_2(\lambda)].$$
This concludes the proof that $[\Delta_1(\lambda)]=[\Delta_2(\lambda)]$ for all $\lambda\in\Lambda$. By definition of the essential order, this shows $\le_1^e=\le_2^e$.
\end{proof}

\subsection{Examples and applications}

\subsubsection{Cellular algebras} Assume that $A$ has a cell datum as in \cite[Definition~1.1]{GL}, {\it i.e.} $A$ is cellular. Part of the cell datum is an anti-involution $\star$ of the $k$-algebra $A$, which thus defines a duality on $A$-mod. By \cite[Definition~1.1(C2)]{GL} and \cite[Theorem~3.4]{GL}, this duality is simple preserving.

In \cite[Theorem~1.1]{KX} it is proved that the cell datum on $A$ induces a quasi-hereditary structure on $A$ if and only if the global dimension of $A$ is finite. This shows in particular that if the cell datum does not lead to a quasi-hereditary structure, $A$ does not admit any quasi-hereditary partial order. Together with Theorem~\ref{ThmQH}, this thus implies that the only quasi-hereditary structure a cellular algebra can admit comes from its cell datum.

This observation is for instance applicable to (walled) Brauer algebras, Temperley-Lieb algebras and partition algebras, see~\cite{GL, KX}.

\subsubsection{Category $\mathcal{O}$}\label{CorO}
Recall the conclusions from Section~\ref{CatO}.
Assume that we have a $\mathbb{C}$-linear equivalence
$$\mathcal{O}(W,W_J)\;\stackrel{\sim}{\to}\; \mathcal{O}(U,U_K)$$
for finite Weyl groups $W, U$ and parabolic subgroups $W_J<W$ and $U_{K}<U$. By Theorem~\ref{ThmQH}, this must be an equivalence of highest weight categories. In particular, considering the equivalence on simple objects must induce an order isomorphism
$$(W^J,\le_B)\;\stackrel{\sim}{\to}\; (U^K,\le_B).$$

\section{Recovering Coxeter pairs from Bruhat orders}\label{SecRecover}
Motivated by the observations in \ref{CorO} we investigate to which extend a Coxeter pair $(W,W_J)$ can be recovered from the poset $(W^J,\le_B)$.

\subsection{Coxeter systems} It is easy to see that a Coxeter system $(W,S)$ can be recovered from its Bruhat order. In this subsection we prove a slightly more general statement, namely that the poset $(W,\le_B)$ is isomorphic to some Bruhat order $(U^K,\le_B)$ if and only if $U$ and $W\times U_K$ are isomorphic as Coxeter groups.

\subsubsection{} Consider an arbitrary pointed poset $(X,\le)$. For a subset $I\subset X_1=\rho^{-1}(1)$, we define
$$X_2(I)\;=\;\{x\in X_2\,|\, y\lhd x \mbox{ if and only if }y\in I\}.$$
For $I=\{a_1,\ldots,a_n\}$ we simplify notation as $X_2(\{a_1,\ldots,a_n\})$ to $X_2(a_1,\ldots,a_n)$. We also set
$$X^0(I)\;=\;X_0\sqcup I \sqcup\{x\in X_{>1}\,|\, y\le x \mbox{ with }y\in X_2\mbox{ implies }y\in X_2(I)\}$$
and
$$X^\infty(I)\;=\;\{x\in X\,|\, y\le x \mbox{ with }y\in X_1\mbox{ implies }y\in I\}.$$
The subsets $X^0(I)\subset X^\infty(I)$ of $X$ inherit the structure of a pointed poset. Note that for $a\in X_1$, we have $X^\infty(a)=X^0(a)$. In general, we have $X^\infty(I)\cap X_1=I$ and $X^0(I)\cap X_2=X_2(I)$.

\begin{definition}\label{DefTrip}
To a pointed poset $(X,\le)$ we associate the triple $(X_1, \mu,\nu)$ comprising the set $X_1$ with two functions
$$\mu:X_1\times X_1\to\mathbb{Z}_{>0}^\infty,\;(a,b)\mapsto \mathcal{L}(X^0(a,b))\quad\mbox{and}\quad \nu:X_1\to\mathbb{N}^\infty,\; a\mapsto |X_2(a)|.
$$

\end{definition}

\begin{theorem}\label{Thm1}
For a Coxeter system $(W,S)$ with matrix $m:S\times S\to\mathbb{Z}_{>0}^\infty$ and a subset $J\subset S$, consider the pointed poset $(X,\le,\rho):=(W^J,\le_B,\ell)$. The triple $(X_1,\mu,\nu)$ of Definition~\ref{DefTrip} is determined by the following properties:
\begin{enumerate}
\item The subset $X_1\subset W^J$ is precisely $S\backslash J$.
\item For distinct $s_1,s_2\in S\backslash J$, we have $m(s_1,s_2)=\mu(s_1,s_2)$.
\item For $s\in S\backslash J$, the number of neighbours of $s$ in $J$ is given by $\nu(s)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Parts (1) and (3) are obvious. Now we prove part (2). For $\{s_1,s_2\}\subset S\backslash J=X_1$ it is clear that $X_2({s_1,s_2})$ contains only $s_1s_2$ and $s_2s_2$, where the two elements might be identical. We will prove in the next paragraph that $X^0(s_1,s_2)= W_{\{s_1,s_2\}}.$ The length of the Bruhat order on $W_{\{s_1,s_2\}}$ is clearly $m(s_1,s_2)$.

That $W_{\{s_1,s_2\}}\subset X^0(s_1,s_2)$ is obvious. Now take $x\in X^0(s_1,s_2)$ and assume that there exists $t\in S\backslash\{s_1,s_2\}$ which appears in a reduced expression of $x$. Without loss of generality, we assume that $t$ is the right-most such reflection in the relevant reduced expression.
If $t\in J$ then $t$ cannot appear on the right in $x\in W^J$, in particular it cannot commute with both $s_1$ and $s_2$. But this means that $ts_i\in X_2$ and $ts_i\le x$ for at least on $i$, a contradiction by the definition of $X^0(s_1,s_2)$. Hence $t\not\in J$. But this means that $s_it$ and $ts_i$ are in $X_2$, for both $i$. At least one of them is under $x$ in the Bruhat order and again we find a contradiction. This proves part (2).
\end{proof}

\begin{corollary}
Assume that two Coxeter groups $W,U$ with a parabolic subgroup $U_K<U$ yield an order isomorphism $(W,\le_B)\simeq(U^K,\le_B)$. Then we have
an inclusion of Coxeter groups $W\hookrightarrow U$, which together with the canonical inclusion $U_K\hookrightarrow U$ induces an isomorphism
$W\times U_K\stackrel{\sim}{\to} U. $
\end{corollary}
\begin{proof}Denote the Coxeter systems by $(W,S)$ and $(U,T)$.
By Theorem~\ref{Thm1}(1), an isomorphism $(W,\le_B)\stackrel{\sim}{\to}(U^K,\le_B)$ induces an injection $S\hookrightarrow T$ where the image $S'$ has complement $K$. By Theorem~\ref{Thm1}(3), $S'$ consists of vertices in the Coxeter graph which are not neighbouring to any vertex in $K$. In particular, $U\simeq U_{S'}\times U_K$ as Coxeter groups. Finally, Theorem~\ref{Thm1}(2) implies that the Coxeter matrices of $(W,S)$ and $(U_{S'},S')$ are identical, hence $W\simeq U_{S'}$ as Coxeter groups.
\end{proof}

\subsection{Irreducible factors}

In this subsection we reduce the problem of recovering Coxeter pairs from their Bruhat orders to the case of irreducible Coxeter groups.
\begin{theorem}\label{Thm2}
Consider two Coxeter systems $(W,S)$ and $(U,T)$ with $J\subset S$ and $K\subset T$. Assume the Coxeter graphs $S$ and $T$ have finitely many connected components. If the partial orders $(W^J,\le_B)$ and $(U^K,\le_B)$ are isomorphic, there exists $n\in\mathbb{N}$ such that we can order the factorisations into irreducible Coxeter groups as
$$W\;\stackrel{J}{=}\;\prod_{i=1}^{n}W_i\qquad\mbox{and}\qquad U\;\stackrel{K}{=}\;\prod_{i=1}^{n}U_i$$
such that we have order isomorphisms $(W_i^{J_i},\le_B)\simeq (U_i^{K_i},\le_B)$ for all $1\le i\le n$.
\end{theorem}
By definition in \ref{factors}, the irreducible factors of $W$ and $U$ which are ignored in the theorem all satisfy $|W_j^{J_j}|=1=|U_l^{K_l}|$ and hence trivially also lead to isomorphic posets.
Before proving the theorem, we need some preparatory results and notions.

\subsubsection{}\label{leadsto} Consider again an arbitrary pointed poset $(X,\le)$. We define a binary relation $\leadsto$ on $X_1$, where $a\leadsto b$ means that $\mu(a,b)=2$ and there exists $y\in X^0(a)$ and $\{y_1,y_2\}\subset X$ with $y_1\rhd y\lhd y_2$ and $y_1\ge b\le y_2$.
We also introduce two equivalence relations $\leftrightsquigarrow$ and $\sim$ on $X_1$. The relation $\leftrightsquigarrow$ is the minimal equivalence relation with $a \leftrightsquigarrow b$ whenever $a\leadsto b$ or $b\leadsto a$.
The relation $a\sim b$ is the minimal equivalence relation with $a\sim b$ whenever $\mu(a,b)>2$ or $b\leftrightsquigarrow a$.

\begin{prop}\label{PropIrr}
For a Coxeter system $(W,S)$ and a subset $J\subset S$, consider the pointed poset $(X,\le,\rho):=(W^J,\le_B,\ell)$.
\begin{enumerate}
\item For $a,b\in X_1=S\backslash J$ with $\mu(a,b)=2$, we have $a\leadsto b$ if and only if there exists a path in the Coxeter graph of $(W,S)$ which connects $a$ and $b$ via vertices exclusively in $J$. In particular, the binary relation $\leadsto$ is symmetric.
\item For $a,b\in X_1$, we have $a\sim b$ if and only if they belong to the same connected component of the Coxeter graph of $(W,S)$.
\item For an equivalence class $I\in X_1/\hspace{-1mm}\sim$, let $\hat{I}$ denote the connected component in the Coxeter graph of $S$ which contains $I$. Also set $\mathring{I}=\hat{I}\backslash I=J\cap \hat{I}$.
The poset $(X^\infty(I),\le)$ is isomorphic to $(W_{\hat{I}}^{\mathring{I}},\le_B)$.
\end{enumerate}
\end{prop}
\begin{proof}
For the proof of part (1), we consider arbitrary $a,b\in X_1$ with $\mu(a,b)=2$.
Assume first that there exists $n\in\mathbb{Z}_{>0}$ and distinct $r_1,r_2,\ldots, r_n\in J$ such that each two consecutive elements in the sequence $a,r_1,\ldots,r_n, b$ are neighbours. Then we can define
$$y=r_nr_{n-1}\cdots r_1a\;\in\, X^0(a), \quad y_1=br_nr_{n-1}\cdots r_1a\mbox{ and }y_2=r_nr_{n-1}\cdots r_1ab.$$
Clearly $y_1,y_2\in W^J$ satisfy the conditions of \ref{leadsto} to conclude $a\leadsto b$. This the ``if'' direction of part (1).

To prove the other direction of part (1), we observe that every reduced expression of an element $y\in X^0(a)$ is a word of elements in the connected component of the Coxeter graph of $J\sqcup \{a\}$ which contains $a$. By Theorem~\ref{Thm1}(2), $a$ and $b$ commute. If $b$ also commutes with all elements of $J$ used in the reduced expressions of $y$, then there can be only one $z\in X$ which satisfies $y\lhd z$ and $b\le z$, namely $z=yb=by$. Hence $a\leadsto b$ implies that we can form a path as desired, which concludes the proof of part (1).

Part (2) is an immediate application of part (1) and Theorem~\ref{Thm1}(2).

Now we prove part (3). Clearly we have $W_{\hat{I}}^{\mathring{I}}\subset X^\infty(I)$. Assume there exists $x\in X^\infty(I)$ with a reduced expression containing a simple reflection $t\in S$ not in the connected component $\hat{I}$. Without loss of generality, we assume that $t$ is the right-most element in a reduced expression and thus $t\not \in J$. Hence we have $t\in X_1\backslash I $ and $t\le x$, a contradiction.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Thm2}]
An isomorphism $(X,\le)\stackrel{\sim}{\to}(Y,\le)$ between two pointed posets must restrict to an isomorphism $(X_1,\sim)\stackrel{\sim}{\to}( Y_1,\sim)$. In particular, for an equivalence class $I\in X_1/\sim$, there must be $I'\in Y/\sim$ such that the posets $X^\infty(I)$ and $Y^\infty(I')$ are isomorphic.
The conclusion thus follows from Proposition~\ref{PropIrr}(2) and (3).
\end{proof}

\subsection{Simple graphs}By Theorem~\ref{Thm2} it is justified to consider connected Coxeter graphs. Since our motivation comes from finite Weyl groups we will furthermore focus on graphs without cycles. When  a Coxeter graph satisfies these properties, {\it i.e.} the underlying graph (obtained by ignoring labels) is simply connected as a topological space, we say it is simple.

\subsubsection{} Let $(X,\le)$ be a pointed poset. We define the subset $V\hspace{-1mm}X\subset X$ as
$$V\hspace{-1mm}X\;=\;\{x\in X_{>0}\,|\, \mbox{for every $i\in\mathbb{N}$, there is at most one $y\in X_i$ with $y\le x$}\}.$$
To a pointed poset $(X,\le)$ we associate a bw-Coxeter graph $\mathcal{G}(X,\le)$ as follows:
\begin{itemize}
\item the black vertices are identified with $X_1\subset V\hspace{-1mm}X$;
\item the white vertices are identified with $V\hspace{-1mm}X\backslash X_1$;
\item between two distinct black vertices $\{a,b\}\subset X_1$ we write an edge if $\mu(a,b)\ge 3$, and the edge is labelled with $\mu(a,b)$ if  $\mu(a,b)> 3$;
\item for each covering $x\lhd y$ with $x,y\in V\hspace{-1mm}X$ we add an edge between the corresponding vertices;
\item for $a\in X_1$ and for $a\not\le x\in V\hspace{-1mm}X$ minimal in $(X,\le)$ with the property that there exist $\{x_1,x_2\}\subset X$ with $x_1\rhd x\lhd x_2$ and $x_1\ge a\le x_2$, we add an edge between $a$ and $x$.
\end{itemize}
By construction of $\mathcal{G}(X,\le)$, we can only have labels on edges between two black vertices.

\begin{lemma}\label{unique}
Consider a Coxeter system $(W,S)$ with simple graph. For $J\subset S$, we set $(X,\le):=(W^J,\le_B)$. The subset $V\hspace{-1mm}X\subset W^J$ consists of those $w\in W^J$ for which
\begin{enumerate}
\item every $v\in W^J$ with $v\le_Bw$ has unique reduced expression;
\item precisely one element of $S\backslash J$ appears in reduced expressions of $w$.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove the claim by induction along $\ell: W^J\to\mathbb{N}$. For $X_1=S\backslash J=\ell^{-1}(1)$ there is nothing to prove.
Now consider $w\in W^J$ with $\ell(w)>1$ and assume the claim has been proved for all $u\in W^J=X$ with $\ell(u)<\ell(w)$.

First assume that $w$ satisfies (1)-(2) in the lemma. Consider $y\in W$ with $y\lhd w$. In particular, $y$ is obtained by removing one simple reflection from the reduced expression of $w$. If this was the right-most one, then by assumption (2) we find $y\not\in W^J$.
Assume that the removed reflection was not the left-most one either. Since $w$ has unique reduced expression and the Coxeter graph is simple, there are two adjacent letters in the reduced expression of $y$ inherited from $w$ which are the same or commute. In the first case we have $y\not\lhd w$, in the second case $y$ does not have unique reduced expression and hence $y\not\in W^J$ by (1). In conclusion, there is only one element in $W^J$ covered by $w$. Moreover, by construction that element satisfies (1)-(2) and is thus by induction hypothesis in $V\hspace{-1mm}X$. It now follows immediately that $w\inV\hspace{-1mm}X$.

Assume conversely that $w\inV\hspace{-1mm}X$. Clearly (2) is satisfied. Assume that (1) is not satisfied. Consider $x\in X$ with $x\le w$ which does not have unique reduced expression, and has minimal $\ell(x)$ under those assumptions. By minimality, $x$ has two reduced expressions with a different simple reflection on the left. Omitting those first letters in the respective expressions yields two distinct $x_1,x_2\le x\le w$ with $x_1,x_2\in W^J$ and $\ell(x_1)=\ell(x_2)$, which is a contradiction.
\end{proof}

\subsubsection{}\label{BU} Consider a simple bw-Coxeter graph which only has labels on edges between two black vertices. To that input we will associate a new connected bw-Coxeter graph. The black vertices and the edges between them, including possible labels, remain the same. On the other hand, each connected component of the subgraph of white vertices in the original graph will appear $n$ times in the new graph, identically attached to the black vertices, with $n$ the number of black vertices which neighbour the connected component.

For example, for the Coxeter pair $(W,W_J)=(E_6,A_3\times A_1)$ with bw-Coxeter graph as below on the left, the above procedure would yield the bw-Coxeter graph on the right
\begin{center}
\begin{tikzpicture}
\draw[fill=black]

(0,0)
      circle [radius=.1]  --
(1,0)
      --
(2,0)
       --
    (3,0)
    circle [radius=.1]  --
      (4,0)

    (2,0) --++(90:1)

    (6.4,0) node {$\Rightarrow$}

    (9,0)
    circle [radius=.1]

    (11,.5)
    --
    (12,0) circle [radius=.1]
;

\draw

(1,0)
      circle [radius=.1]
(2,0)
      circle [radius=.1]
      (2,0) --++(90:1)         circle [radius=.1]
      (4,0)
    circle [radius=.1]

    (9,0) --
    (10,0.5) circle [radius=.1]
    --
   (11,.5) circle [radius=.1]--
   (11,1.5) circle [radius=.1]

    (9,0) --
    (10,-0.5) circle [radius=.1]
    --
   (11,-.5) circle [radius=.1]--
   (11,-1.5) circle [radius=.1]

   (11,-.5) --
   (12,0) --
   (13,0) circle [radius=.1]

   (13.5,0) node {.}
    ;

\end{tikzpicture}
\end{center}
 It is clear that this procedure maps non-isomorphic bw-Coxeter graphs to non-isomorphic bw-Coxeter graphs.

\begin{theorem}\label{ThmNew}
Consider a Coxeter system $(W,S)$ with simple graph and with $J\subset S$ such that no edge which meets $J$ is labelled. The bw-Coxeter graph $\mathcal{G}(W^J,\le_B)$ is obtained from the one of $(W,W_J)$ by procedure~\ref{BU}.
\end{theorem}

\begin{proof}Set $(X,\le)=(W^J,\le_B)$.
Let $V$ denote the set of pairs $(s,t)$ with $s\in S\backslash J$ and $t\in J$ such that the unique minimal path from $s$ to $t$ in the Coxeter graph of $(W,S)$ contains, besides $s$, only elements of $J$. It follows easily from Lemma~\ref{unique} that we have a bijection
$$\phi:V\;\stackrel{1:1}{\to}\; V\hspace{-1mm}X\backslash X_1,\quad (s,t)\mapsto t r_d r_{d-1}\cdots r_1s.$$
where $r_i$ label the vertices in the Coxeter graph along the minimal path from $s$ to $t$ in the obvious way. In case $m(s,t)>2$, we thus have $d=0$.

By Theorem~\ref{Thm1}, the Coxeter subgraphs of $\mathcal{G}(X,\le)$ and the bw-Coxeter graph of $(W,W_J)$ consisting of black vertices (and labelled edges between them) are canonically isomorphic.
By the description of $\phi$, we have an edge between $\phi(s,t)$ and $\phi(s,t')$ in $\mathcal{G}(X,\le)$ if and only if there is an edge between $t$ and $t'$. By definition of $\mathcal{G}(X,\le)$ there is no edge between $\phi(s,t)$ and $\phi(s',t')$ whenever $s\not=s'$. The proof of Proposition~\ref{PropIrr}(1) shows that
there is an edge between $\phi(s,t)$ and $s'\in X_1$ in $\mathcal{G}(X,\le)$ if and only if $t\in J$ neighbours $s'$.

It now follows that $\mathcal{G}(X,\le)$ is obtained from the bw-Coxeter graph of $(W,W_J)$ by the procedure of \ref{BU}.
Concretely, consider a connected component $\mathring{I}$ in the white part of the graph of $(W,W_J)$ and label the neighbours of $\mathring{I}$ in $S\backslash J$ by $\{s_i\,|\, 1\le i\le n\}$. Then the subgraph $\mathring{I}$ of $S$ yields $n$ copies in $\mathcal{G}(X,\le)$ and they correspond to the sets of vertices $\{\phi(s_i,t)\,|\, t\in\mathring{I}\}$, for $1\le i\le n$.\end{proof}

\subsection{Finite Weyl groups}

\begin{theorem}\label{Thm3}
The only non-isomorphic Coxeter pairs $(W,W_J)$ and $(U,U_K)$, with $W,U$ irreducible finite Weyl groups which lead to isomorphic posets $(W^J,\le_B)\simeq (U^K,\le_B)$ are
\begin{enumerate}
\item $(A_{2n+1},A_{2n})\;\longleftrightarrow\; (B_{n+1},B_n),\;\;$ for $n\ge 2$;
\item $(B_n, A_{n-1})\;\longleftrightarrow\; (D_{n+1},A_n),\;\;$ for $n\ge 3$;
\item $(A_3,A_2)\;\longleftrightarrow\; (B_2,A_1)$;
\item $(A_5,A_4)\;\longleftrightarrow\; (G_2,A_1)\;\longleftrightarrow\; (B_3,B_2) $;
\item $(W,W)\;\longleftrightarrow\; (U,U)$, for any $U\not\simeq W$.
\end{enumerate}
\end{theorem}

We note that case (3) can be seen as a limit of both series in (1) and (2). Before proving the theorem, we establish the following lemma.

\begin{lemma}\label{LemNew}
Besides the cases listed in Theorem~\ref{Thm3}, the only non-isomorphic Coxeter pairs $(W,W_J)$ and $(U,U_K)$ with $W,U$ irreducible finite Weyl groups which lead to isomorphic bw-Coxeter graphs $\mathcal{G}(W^J,\le_B)\simeq\mathcal{G}(U^K,\le_B)$ are
\begin{itemize}
\item $(F_4,A_2)\;\longleftrightarrow\; (D_5,A_3)$;
\item $(F_4,B_3)\;\longleftrightarrow\; (E_6,D_5)$;
\item $(B_{m+n},P\times  A_{n-1})\;\longleftrightarrow\; (D_{m+n+1},P\times  A_{n} )$, for every parabolic subgroup $P<A_{m-1}$, using the canonical inclusions $A_{m-1}\times A_{n-1}<B_{m+n}$ and $A_{m-1}\times A_{n}<D_{m+n+1}$, for $n\ge 2$ and $m\ge 1$;
\item $(B_{m+n}, Q\times B_{n-1})\;\longleftrightarrow\; (A_{m+2n-1}, Q\times A_{2n-2})$, for every parabolic subgroup $Q<A_{m}$, using the canonical inclusions $A_{m}\times B_{n-1}<B_{m+n}$ and $A_m\times A_{2n-2}<A_{m+2n-1}$ for $n\ge 2$ and $m\ge 1$ (where $B_{n-1}$ for $n=2$ is to be interpreted as $A_1$).
\end{itemize}
\end{lemma}
\begin{proof}
By Theorem~\ref{ThmNew}, a necessary condition for an isomorphism $\mathcal{G}(W^J,\le_B)\simeq\mathcal{G}(U^K,\le_B)$ is that at least one of the original bw-Coxeter graphs of $(W,W_J)$ and $(U,U_K)$ contains a labelled edge linked to a white vertex. First we calculate $\mathcal{G}(W^J,\le_B)$ for all pairs $(W,W_J)$ with such an edge and for $W\in \{F_4,G_2\}$. On the left-hand side we write the original bw-Coxeter graph of $(W,W_J)$, on the right-hand side $\mathcal{G}(W^J,\le_B)$.
\begin{center}
\begin{tikzpicture}
\draw[fill=black]
(-1.5,0) node {$(F_4,B_3)$}

(0,0)
    circle [radius=.1]  --
(1,0)
      --node [midway,above] {$4$}
(2,0)
      --
(3,0)

   (5,0) node {$\Rightarrow$}

(7,0)
    circle [radius=.1]  --
(8,0)

    ;

\draw
(0,0)
      circle [radius=.1]  --
(1,0)
      circle [radius=.1] --node [midway,above] {$4$}
(2,0)
      circle [radius=.1] --
(3,0)
    circle [radius=.1]

    (7,0)
      circle [radius=.1]  --
(8,0)
      circle [radius=.1] node [above] {} --
(9,0)
      circle [radius=.1] node [above] {}    --
    (10,0)
    circle [radius=.1] node [above] {} --
      (11,0)
    circle [radius=.1]
    (9,0) --++(90:1)         circle [radius=.1]
;

\end{tikzpicture}

\vspace{2mm}

\begin{tikzpicture}
\draw[fill=black]
(-1.5,0) node {$(F_4,B_2)$}

(0,0)
    circle [radius=.1]  --
(1,0)
      --node [midway,above] {$4$}
(2,0)
      --
(3,0)
circle [radius=.1]

   (5,0) node {$\Rightarrow$}

(8,0.5)
    circle [radius=.1]  --
(9,0.5)

         (10,0.5) --
         (11,-0.5)
         circle [radius=.1]
    ;

\draw
(0,0)
      circle [radius=.1]  --
(1,0)
      circle [radius=.1] --node [midway,above] {$4$}
(2,0)
      circle [radius=.1] --
(3,0)

    (9,0.5)
    circle [radius=.1] {} --
    (10,0.5)
    circle [radius=.1] --
    (11,0.5)
    circle [radius=.1]--
    (12,0.5)
    circle [radius=.1]

    (8,0.5) --
    (9,-0.5) circle [radius=.1] {} --
    (10,-0.5) circle [radius=.1] {} --
    (11,-0.5)
    (7,-.5)circle [radius=.1] {} --
    (8,-.5) circle [radius=.1] {} --
    (9,-.5)
      ;

\end{tikzpicture}

\vspace{2mm}

\begin{tikzpicture}
\draw[fill=black]
(-1.5,0) node {$(F_4,A_2)$}

(0,0)
    circle [radius=.1]  --
(1,0)
      circle [radius=.1] --node [midway,above] {$4$}
(2,0)
      --
(3,0)

   (5,0) node {$\Rightarrow$}

(7,0)
    circle [radius=.1]  --
(8,0)
        circle [radius=.1] --
        (9,0)

    ;

\draw
(2,0)
      circle [radius=.1] --
(3,0)
    circle [radius=.1]
        (9,0)
    circle [radius=.1] node [above] {} --
      (9,0)
    circle [radius=.1] node [above] {}

(9,0)
      circle [radius=.1] node [above] {}

(9,0) --++ (30:1.15)
      circle [radius=.1]

(9,0) --++ (-30:1.15)
      circle [radius=.1]
;

\end{tikzpicture}

\begin{tikzpicture}
\draw[fill=black]

    (5.5,0) node {$(G_2,A_1)$}
    (7,0)
      circle [radius=.1] --node [midway,above] {$6$}
(8,0)

(12,0) node {$\Rightarrow$}

(14,0) circle [radius=.1] --
(15,0)
      ;

\draw

(8,0)
      circle [radius=.1]
      (15,0)
       circle [radius=.1] --
       (16,0)
       circle [radius=.1] --
       (17,0)
       circle [radius=.1] --
       (18,0)
       circle [radius=.1]

       ;
\end{tikzpicture}
\end{center}
The resulting graph for $(F_4,B_2)$ lacks the symmetry to come from a label free case via procedure~\ref{BU}.  It will follow from the case-by-case study for type $B$ below that $\mathcal{G}(F_4/B_2,\le_B)$ is also different from those cases.
The other right-hand sides are the (original) bw-Coxeter graphs, invariant under procedure~\ref{BU}, of respectively $(E_6,D_5)$, $(D_5,A_3)$ and $(A_5,A_4)$.

The remaining Coxeter pairs with a labelled edge linking to the parabolic subgroup are

(a) $(B_n,A_{n-1})$ for $n\ge2$,

(b) $(B_{m+n},P\times  A_{n-1})$ for some parabolic subgroup $P<A_{m-1}$ for the canonical inclusion $A_{m-1}\times A_{n-1}<B_{m+n}$ for $n\ge 2$ and $m\ge 1$,

(c) $(B_n,B_{n-1})$ for $n\ge 3$,

(d) $(B_{m+n}, Q\times B_{n-1})$ for some parabolic subgroup $Q<A_{m}$ for the canonical inclusion $A_{m}\times B_{n-1}<B_{m+n}$ for $n\ge 2$ and $m\ge 1$.

It is again straightforward to calculate the corresponding graphs $\mathcal{G}(W^J,\le_B)$, for instance using Lemma~\ref{unique}. For (a) the graph $\mathcal{G}(W^J,\le_B)$ is the (original) bw-Coxeter graph of $(D_{n+1},A_{n})$ if $n\ge 3$, and of $(A_{3},A_{2})$ if $n=2$. For (c) it is the (original) bw-Coxeter graph of $(A_{2n-1},A_{2n-2})$. As an immediate extension of that, the cases (b) and (d) yield isomorphisms of bw-Coxeter graphs as displayed in the lemma.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Thm3}]
That the listed pairs lead to isomorphic posets is well-known and easily checked. It remains to show that the cases in Lemma~\ref{LemNew} do not lead to isomorphic posets. Using the values in appendix~\ref{App}, we find
$$\mathcal{L}(F_4/A_2)-\mathcal{L}(D_5/A_3)=7,\quad\mathcal{L}(F_4/B_3)-\mathcal{L}(E_6/D_5)=-1,$$
$$\mathcal{L}(B_{m+n}/(P\times  A_{n-1}))-\mathcal{L}(D_{m+n+1}/(P\times A_n))=-m,\quad\mbox{and}$$
$$\mathcal{L}(B_{m+n}/(Q\times B_{n-1}))-\mathcal{L}(A_{m+2n-1}/(Q\times A_{2n-2}))=\frac{1}{2}m(m+1).$$
Consequently, the lengths of the posets never agree,
which concludes the proof. \end{proof}

\section{Category $\mathcal{O}$}\label{SecO}
In this section we work over the field $\mathbb{C}$ of complex numbers.
\subsection{Reduction to simple Lie algebras}

\subsubsection{}\label{SetUpDelO}Fix a reductive Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h}\subset\mathfrak{b}$. Assume furthermore that we have a non-trivial Lie algebra decomposition $\mathfrak{g}=\mathfrak{g}_1\oplus \mathfrak{g}_2$. This defines Borel and Cartan subalgebras $\mathfrak{h}_i\subset\mathfrak{b}_i\subset\mathfrak{g}_i$, for $1\le i\le 2$ and a factorisation of Coxeter groups $W=W_1\times W_2$. Fix also an antidominant weight $\lambda\in\mathfrak{h}^\ast$, which restricts to weights $\lambda^i=\lambda|_{\mathfrak{h}_i}$. For $1\le i\le 2$, we abbreviate $\mathcal{O}_{\lambda_i}(\mathfrak{g}_i,\mathfrak{b}_i,\mathfrak{h}_i)$ to $\mathcal{O}^i$ and $\mathcal{O}_{\lambda}(\mathfrak{g},\mathfrak{b},\mathfrak{h})$ to $\mathcal{O}$.

\subsubsection{} Following \cite{BGG}, we can define a projective object $P^i_\mu$ in $\mathcal{O}^i$ for each $\mu$ in $W_i\cdot\lambda_i$, such that we have a canonical isomorphism
\begin{equation}\label{caniso}\mathrm{Hom}_{\mathfrak{g}_i}(P^i_{\mu},M)\;\stackrel{\sim}{\to}\; \mathrm{Hom}_{\mathfrak{h}_i}(\mathbb{C}_\mu,M),\qquad\mbox{for every $M\in\mathcal{O}^i$}.\end{equation}
Note that the \eqref{caniso} remains valid for $\mathfrak{g}$-modules $M$ in the ind-completion $\mathrm{Ind} \mathcal{O}^i$, which is defined as in \ref{DefO} but without the condition of finite generation.
Isomorphism \ref{caniso} leads to $\mathbb{C}$-linear equivalences
$$\mathrm{Hom}_{\mathfrak{g}_i}(\oplus_\mu P^i_\mu,-):\,\mathcal{O}^i\,\stackrel{\sim}{\to}\,A_i\mbox{-mod},\qquad\mbox{with}\quad A_i:=\mathrm{End}_{\mathfrak{g}_i}(\oplus_\mu P^i_\mu)^{\mathrm{op}}.$$

Each $\kappa\in W\cdot\lambda$ can be written uniquely a sum $\kappa^1+\kappa^2$ with $\kappa^i\in W_i\cdot\lambda^i$.
It follows from a direct calculation, using the extension of \eqref{caniso} to $\mathrm{Ind}\mathcal{O}^i$, that the tensor product $P_\kappa:=P^1_{\kappa^1}\otimes_{\mathbb{C}}P^2_{\kappa^2}$ is a $\mathfrak{g}$-module which induces an isomorphism
$$\mathrm{Hom}_{\mathfrak{g}}(P_{\kappa},N)\;\stackrel{\sim}{\to}\; \mathrm{Hom}_{\mathfrak{h}}(\mathbb{C}_\kappa,N),\qquad\mbox{for every $N\in\mathcal{O}$}.$$
In conclusion, we find a $\mathbb{C}$-linear equivalence
$$\mathcal{O}\,\simeq\,(A_1\otimes A_2)\mbox{-mod}.$$
In order to formulate this result without mentioning of the auxiliary algebras $A_i$, we can employ Deligne's tensor product of abelian $\mathbb{C}$-linear categories $-\boxtimes-$, see \cite[Section~5]{De}.
\begin{prop}
For a reductive Lie algebra $\mathfrak{g}=\mathfrak{g}_1\oplus\mathfrak{g}_2$ as in \ref{SetUpDelO}, we have a $\mathbb{C}$-linear equivalence
$$\mathcal{O}_\lambda(\mathfrak{g},\mathfrak{b},\mathfrak{h})\;\simeq\; \mathcal{O}_{\lambda_1}(\mathfrak{g}_1,\mathfrak{b}_1,\mathfrak{h}_1)\boxtimes \mathcal{O}_{\lambda_2}(\mathfrak{g}_2,\mathfrak{b}_2,\mathfrak{h}_2).$$
\end{prop}

We can reformulate this result in terms of Weyl groups, based on the discussion in \ref{ResSoergel}.

\begin{corollary}\label{CorOSimp}
Let $W$ be a finite Weyl group, with factorisation $W=\prod_{i=1}^dW_i$ into irreducible Coxeter groups and generating set $S=\sqcup_{i=1}^d S_i$. For any $J\subset S$ with $J_i=J\cap S_i$, we have a $\mathbb{C}$-linear equivalence
$$\mathcal{O}(W,W_J)\;\simeq\;\mathcal{O}(W_1,{W_1}_{J_1})\boxtimes \mathcal{O}(W_2,{W_2}_{J_2})\boxtimes\,\cdots\,\boxtimes \mathcal{O}(W_d,{W_d}_{J_d}).$$
\end{corollary}

\subsection{Main result}

\begin{theorem}\label{Thm4}
Let $(W,S)$ and $(U,T)$ be finite Weyl groups and take $J\subset S$ and $K\subset T$. Consider the factorisations into irreducible Weyl groups
\begin{equation}W\;\stackrel{J}{=}\;\prod_{i=1}^{n_1}W_i\quad\mbox{and}\quad U\;\stackrel{K}{=}\;\prod_{i=1}^{n_2}U_i.\label{Fact2Weyl}\end{equation} The following properties are equivalent
\begin{enumerate}
\item There exists a $\mathbb{C}$-linear equivalence $\mathcal{O}(W,W_J)\simeq \mathcal{O}(U,U_K)$.
\item There exists an order isomorphism $(W^J,\le_B)\simeq (U^K,\le_B)$.
\item We have $n_1=n_2$ and can (re)order the factors in \eqref{Fact2Weyl}
such that we have order isomorphisms $(W_i^{J_i},\le_B)\simeq (U_i^{K_i},\le_B)$ for all $i$.
\item We have $n_1=n_2$ and can (re)order the factors in \eqref{Fact2Weyl}
such that we have $\mathbb{C}$-linear equivalences $\mathcal{O}(W_i,{W_i}_{J_i})\simeq \mathcal{O}(U_i,{U_i}_{K_i})$ for all $i$.
\end{enumerate}
\end{theorem}
\begin{proof}
That (1) implies (2) is observed in \ref{CorO}. That (2) implies (3) is a special case of Theorem~\ref{Thm2}. That (4) implies (1) follows from Corollary~\ref{CorOSimp}.

In order to show that (3) implies (4) it suffices to prove that for two irreducible finite Weyl groups $W$ and $U$ an isomorphism $(W^J,\le_B)\simeq (U^K,\le_B)$ implies an equivalence $\mathcal{O}(W,W_J)\simeq \mathcal{O}(U,U_K)$. In other words, we need to prove an equivalence for each of the order isomorphisms (1)-(4) listed in Theorem~\ref{Thm3}. The equivalences for \ref{Thm3}(3) and (4) follow from the explicit description in \cite[Section~5]{St}.
The equivalences for \ref{Thm3}(1) and (2) are well known. For instance, the Koszul duals (see \cite{BGS}) of the two algebras describing the respecting blocks are observed to be Morita equivalent in \cite[\S 1.7]{ES}.
\end{proof}

\subsection{Outlook: Generalisations of category $\mathcal{O}$}

\subsubsection{Kac-Moody algebras }\label{KM} Theorem~\ref{ThmQH} extends immediately to the setting of {\em upper finite highest weight categories} of \cite[Section~3.3]{BS}. Concretely, if we have a simple preserving duality on such a category, then \eqref{BGG} is still valid and we have maximal elements in the poset. This is all that is required for the proof. This means that our results could be relevant also in the study of category $\mathcal{O}$ for Kac-Moody algebras, see \cite[Section~5.2]{BS}.

\subsubsection{Lie superalgebras} Theorem~\ref{ThmQH} does {\bf not} extend to the setting of {\em essentially finite highest weight categories} of \cite[Section~3.2]{BS}. The proof does not extend, due to lack of maximal elements in the poset. A concrete counter example to Theorem~\ref{ThmQH} in that setting is given by category $\mathcal{O}$ for a basic classical Lie superalgebra, which is an essentially finite highest weight category with a simple preserving duality, see~\cite[Section~5.5]{BS}. Different (non-conjugate) Borel subalgebras with same underlying even Borel subalgebra can lead to non-equivalent highest weight structures.

Some results on the classification of the blocks in category $\mathcal{O}$ for $\mathfrak{gl}(m|n)$ are obtained in \cite{BG, CS}. We hope that the full classification for Lie algebras can help in the classification problem for $\mathfrak{gl}(m|n)$.

\subsubsection{Root reductive Lie algebras}
Similarly to \ref{KM}, category $\mathcal{O}$ for $\mathfrak{gl}(\infty)$ has blocks which are upper finite highest weight categories in the sense of \cite[Section~3.3]{BS}. Our methods hence apply to that case. As a concrete example, the methods in Section~\ref{SecUnique} show that the blocks compared in \cite[Lemma~4.7.2]{CP} are not equivalent.

redpajama_set_name: RedPajamaArXiv


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Home > Insights > International > PRIVACY SECURITY MANAGEMENT IN TIMES OF EMERGENCY
PRIVACY SECURITY MANAGEMENT IN TIMES OF EMERGENCY
Today, with the uncertainty on the extension in time of the current situation caused by the coronavirus and the future possibility of new similar situations, the trend initiated earlier by a large number of companies with the implementation of telework in specific areas and with established guidelines, the present situation has led this to the extreme both in relation to time and to the business processes involved.
The unexpected way in which it has arrived, in both speed and scope, has involved the generalization of telework to levels unthinkable a few weeks ago, introducing in turn a whole series of risks very relevant to the entities.
In the ideal environment, this situation should have been referred to in the Business Continuity Plans of the entities, allowing us to anticipate the situation, thinking, reflecting and trying to minimize the risks associated with it. In reality, the majority of entities, and covering the wide range, from entities that had a Business Continuity Plan but that with the probability associated with a specific threat of pandemic had not implemented the related recovery plan, to those entities that had a partial or outdated plan, and all those many entities that did not have one at all, the entities have implemented telework, not always managing or at least being aware of the risks associated with it.
Forced by the unexpected current situation, this non-planned and non-structured implementation, and bringing with it the variables of time, number of employees and business processes, lead to a very high volume and, in many cases, sensitivity of the data (with the obvious example of health data this is perfectly illustrated) that are moving across networks, subject to much higher risks and different from those that existed until a few weeks ago, in the implementation of the same business processes.
If privacy was already on the agendas of all entities before the Covid-19, now the current context is giving it a plus of protagonism and will make it both now and in the return to normality, one of the main lines on which entities should pilot certain policies and decisions.
Making a reflection, the basis of the new regulation on the protection of personal data is based on the concept of risk management and, specifically, in the management that each entity, based on the typology of data and the processing that it makes of these, values the existing risks and on this basis applies the security measures that it deems most appropriate to preserve the privacy of the data. The current situation based on telework involves significant and, in many cases, new risks, which are the result of new platforms for sharing documents, group work tools, remote access, different business flows, etc. that involve a review of the life cycle of data, and an update of the different risk analyses (Privacy Impact Analysis) that are based on this. We must make sure to manage the security of privacy at present and in the future.
This environment is the perfect setting for new threats and the Spanish Data Protection Agency itself warns of increased information security risks because of threats such as:
Non-secure networks. A large number of remote accesses occur via the Internet, and organizations normally do not have control over the security of external networks used.
Phishing attacks. Phishing campaigns that use as a pretext the interest that captures the progress of the COVID-19 have increased. These seek to steal personal information, spread fake news and/or deliver unwanted advertising.
Ransomware. Prevents users from accessing their system or personal files and requires the payment of a ransom to be able to re-access them.
On the basis of the new risks that may affect the confidentiality, integrity and availability of data and with the need to provide security to the information, we point out some guidelines for managing the security of privacy, on the basis of the key elements in the new situation. The security measures must ensure the remote access solutions, including the security of the server, location and security of the client's software, covering authentication, authorization and access control for remote access solutions; and at the same time, ensure the client's telework devices and protect the data in them:
Maintain up-to-date and patched servers, operated on the basis of the security settings defined by the organization and managed only from trusted hosts by authorized administrators.
Assess the factors of device performance, traffic examination, non-protected traffic and the NAT (Network Address Translation) to determine the locations of remote access servers.
Ensure that all internal remedies made available through remote access are adequately protected by firewalls and other access control mechanisms.
Application of cryptography to sensitive information from remote access communications, wireless networks and other non-reliable networks.
Have remote access security solutions available via VPN (Virtual Private Network) to access the organization's network from an external location.
Assess solutions of virtual desktop infrastructure (VDI), consisting of employee desktops controlled by the company and that make up safe work environments. Any security policy applied to the organization's physical devices, such as updates, backups, software control, DLP tools, etc., can be transferred to virtual desktops of employees.
Enable remote authentication mechanisms such as robust passwords, double authentication factor or mutual authentication.
Implement a demanding access control policy based on profiles, functions and the implementation of access based on the principle of least privilege.
To the extent possible, provide employees corporate devices, since in these are applied the security policies that the company considers appropriate.
We must ensure the continuity of business processes through the implementation, updating and periodic test of the business continuity plan (BCN) and the disaster recovery plan (DRP). This element, of different implementation in organizations today, is allowing the continuation of business activity, and is enabling companies to continue to operate more or less efficiently in the current situation.
In the same way, we must have an incident response plan of the information system so that in the event of a risk materializing, employees have the necessary procedures and channels to communicate it, and the systems and security area, the security information events management systems (SIEM), data visualization tools, and/or capabilities of automation and artificial intelligence (AI), which allow to monitor, detect and respond effectively and efficiently to the incidents, and at the same time, comply with the terms of art. 33 of the GDPR.
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Enric Doménech
Partner. Head of Risk Advisory Services
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Carriera
Specializzato nella farfalla, ha conseguito la medaglia di bronzo sulla distanza dei 200m ai mondiali di Kazan' 2015.

Ai Campionati europei di nuoto di Glasgow 2018 ha vinto, con i connazionali Konrad Czerniak, Jakub Kraska e Kacper Majchrzak e Jan Hołub, la medaglia di bronzo nella staffetta 4x100 metri stile libero.

Palmarès
Mondiali
Kazan' 2015: bronzo nei 200m farfalla.

Europei
Glasgow 2018: bronzo nella 4x100m sl.

International Swimming League

Note

Altri progetti

Collegamenti esterni


To aid in building high-performance computational environments, INTERLACE offers a framework for linking reusable computational engines in a heterogeneous distributed system. The INTERLACE model provides clients with access to computational servers which interface with "wrapped" computational engines. The wrappers implement mechanisms to translate client requests to engine actions and to move data across the server interface. These mechanisms are programmable, allowing engines of different type to be integrated. The framework takes advantage of the HPC++ runtime system to access servers through distributed object operations. The INTERLACE framework has been demonstrated by building a distributed computational environment with MatLab engines.


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The cruise industry continues to grow as cruise lines expand their itineraries and look for and establish new ports of call. Those in the industry, port managers and marketers alike, must stay abreast of the current trends.
Charles "Bud" Darr, Senior Vice President, Technical & Regulatory Affairs, Cruise Lines International Assoc., Inc.