The poor laws and the workhouse inspired their own genre of literature and art. Popular songs and poems ranged from the sentimental (e.g. In the Workhouse: Christmas day ), through the satirical (e.g. The New Poor Law in Force ), to the sheer gruesome (e.g. The Vorkhouse Boy ). James Greenwood's lurid account of A Night in a Workhouse spawned a whole line of undercover exposés of workhouse life by numerous "social explorers" including Jack London ( The People of the Abyss ) and George Orwell ( The Spike ) and the less well-known but equally significant Mary Higgs ( Glimpses of the Abyss ) and Olive Malvery ( The Soul Market ). In more recent times, commemoration of the Irish Famine has produced some remarkable works of sculpture.
Sold in the streets for a penny or halfpenny between the 16th and early 20th centuries, broadside ballads were popular songs of the day, often anonymous, and were performed in taverns, homes, or fairs. The examples below, relating to the poor-law and workhouse, are from the Bodleian Library's collection. A Bodleian shelfmark follows each title.
You can listen to a recording (MP3) of The Workhouse Boy (a variation on The Vorkhouse Boy above) performed by the Seven Dials Band (by kind permission of Beautiful Jo Records).
In the early 1900s, collectors of English folk songs such Cecil Sharp and Ralph Vaughan Williams, used workhouse inmates and casuals as one of the sources for their material. One of these (name unknown) was photographed at St Marylebone.
The American folk song collector James Madison Carpenter visited Britain in 1928 and recorded singers at the Ely the Insch parish poorhouse in Aberdeenshire. He also collected songs from the Ely workhouse in Cambridgeshire. The colllection has an online catalogue.
The workhouse was occasionally the subject of maudlin ballads such as Standing at the Workhouse Gate by B.K. Aungur in around 1908.
Modern songs by Peter Higginbotham.
The Archway of Tears — The Archway of Tears was the local name for the entrance to the Birmingham workhouse.
Charles, James Chelsea Workhouse: A Bible Reading (Our Poor) (1877) - in this unusually lyrical portrayal of a workhouse, a group of elderly female inmates sit at a table, bathed in sunshine. Some drink tea while one of their number reads from the bible.
Cope, Charles West Poor Law Guardians: Board Day Application for Bread (1841), was based on a visit to a meeting of the Staines Board of Guardians. It, shows a young widow pleading for relief for her four children.
Thomas Rowlandson St James Workhouse, 1809.
The Lurgan Workhouse Mural, 2006.
Chelsea Workhouse: A Bible Reading.
Board Day Application for Bread.
S.L.B. (1866) Description of a visit to an unnamed District School in the London area.
George Green Gabb, Master of Droitwich Workhouse Master (1849-53) compiled by Martin Gabb.
The colourful career of William Bragger, Master of Wrexham workhouse (1857-1863) researched by his great-great-great grandson Andrew Taylor.
Shaw, Charles (1903) When I was a Child includes memories of the Wolstatnton and Burslem workhouse at Chell which Shaw entered as a child in 1842.
A sardonic postcard from 1907 — "Wots the good of working when there's an 'ome for nothink"
A humorous interpretation of the play-title The Beauty of Bath, c.1906.
Punch Magazine's typically irreverent take on the Guardians interrogating a relief applicant in 1899.
Irish National Famine Monument at Croagh Patrick in Co. Mayo, 2000.
Casual Ward graffiti collectd by Poor Law Board Inspector Andrew Doyle in 1865.


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Signpost nabs $52M to help SMBs collect reviews and marketing collateral like bigger rivals
Ingrid Lunden	@ingridlunden / 3 years
Creating products aimed at smaller business users is a promising but challenging category in the area of enterprise software — they make up the vast majority of businesses today, but collectively are a fragmented customer base that is price sensitive and individually represents small ARPU. Today, a company that has found a way to tap the opportunity and capitalise on some of the challenges has raised a significant round of growth funding to tap into the opportunity.
Signpost, which has built a platform that today is used by thousands of local businesses to compete better against bigger companies — by letting them collect their own and subsequently use product reviews, construct and provide offers and manage contact lists of customers — has closed a round of $52 million, money that it will use to expand its technology and customer base.
The funding is described as "late-stage financing" (similar to the round it raised in 2016) and comes from HighBar Partners and BMO Bank of Montreal with previous investors Georgian Partners and Spark Capital also participating. Signpost has been around for about a decade, and its other backers have included OpenView Venture Parnters, Scout Ventures and GV (when it was still Google Ventures and Signpost was focusing on the viral marketing of local deals).
The Google connection is interesting: Signpost has gone through a few different iterations as a business and years ago was described as the "AdSense for local businesses." It's raised about $110 million to date, with its most recent funding (in 2016) at around $100 million, according to PitchBook data.
While Signpost is not disclosing valuation, I'm guessing it's definitely higher than this now, judging by the size of this latest round, and some of the other metrics the company is disclosing: the company cash flow break-even as of early 2019, with 43% year-over-year growth in its core product and less than one percent customer churn. Its also hiring in its three locations, New York (its HQ), Austin, and Denver.
The problem that the company is tackling is one that local, independent and brick-and-mortar stores may know all too well: they have it hard when it comes to marketing.
Traditionally, they have relied on word-of-mouth and (among physical businesses) incidental foot traffic to maintain and grow trade. But in the age of shopping malls, the internet and the economy of scale that comes with it in e-commerce, many of those small businesses have found themselves caught out in figuring out how best to connect with existing customers and reach new ones.
Added to this is the issue of user reviews. These have become an important — some would say essential — cornerstone for how people discover and decide to buy one product or service over another online, in conjunction with other services like Google Maps and Yelp.
Signpost notes stats that say at least 82% of smartphone shoppers conduct 'near me' searches and 90% are likely to click on the first set of results, but that those without reviews conversely getting nary a glance or attentive nudge from the ever-powerful search algorithm.
On the other hand, the rise of user reviews has led to a pernicious counter theme: the proliferation of fake reviews and spam. This is something that small businesses struggle to get a handle on, and I've seen more than once a business bemoaning the presence of these on Google Maps without knowing how best to try to fix the issue.
This is where a company like Signpost comes in, proving a platform that is easy to use to help those businesses get a grip on providing their own review and feedback tools, and taking charge of other kinds of basic marketing and CRM features, such as mailing lists and creating offers that introduce new customers and reward loyal, repeat ones.
These are all table stakes for bigger businesses, and the idea is that small businesses should have them, too.
"While large corporations have used powerful marketing technology to drive extraordinary growth for years, local businesses — which make up 99.7% of all U.S. businesses and account for nearly half of GDP — have been left behind," said Stuart Wall, founder and CEO of Signpost, in a statement. "Signpost puts the power of the world's largest marketing departments in the hands of local business owners and empowers them to instantly capture their customer data, get better online reviews, and win new repeat business automatically and effortlessly."
And in another feature that helps put small businesses on par with larger rivals, Signpost is also a big data powerhouse.
Signpost says that it has gathered behavioral data from over 70 million US consumers across all its local business customers, which it uses to feed its own algorithms to determine when and where to send personalised messages to a businesses' contacts to help drum up more sales. Signpost claims that its tools have led to a 34% rise in and revenue lifts of 14%, on average.
Brian Peters, Managing Director at HighBar Partners, is joining Signpost's Board of Directors with this round.
"HighBar's investment is a vote of confidence in the right approach to applying technology to level the playing field for local businesses," he said in a statement. "The turnkey setup and level of automation in Signpost's platform, as well as their ability to follow through on reviews and engage the modern customer well beyond their first interaction is what really sets them apart."
"We are excited to be working alongside Signpost – they have created an innovative solution to help small businesses compete and drive revenue growth," said Devon Dayton, Managing Director, Technology & Innovation Banking, BMO Bank of Montreal, in a statement. "The company has experienced a lot of momentum and we are looking forward to continuing this journey with them through their next phase of growth."

redpajama_set_name: RedPajamaCommonCrawl


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Law Firm Throws a Life Line to Active Coast Guard Military Personnel Who Aren't Getting Paid Due to Government Shutdown
Jan 23, 2019 | Business
George Sink, P.A. Injury Lawyers donates $5k to the Lowcountry CPOA (Coast Guard) and launches "Stay Afloat Challenge" urging others to help too.
CHARLESTON, SC, January 23, 2019 /24-7PressRelease/ — This morning, in a relief effort to help local Coast Guard personnel and their families stay financially afloat during the government shutdown, George Sink, P.A. Injury Lawyers announced a donation of $5,000 to the Lowcountry Chapter of the United States Coast Guard Chief Petty Officer Association (CPOA).
The firm also unveiled its online campaign, called the Stay Afloat Challenge, where people can learn how they can help Coast Guard personnel, and urges all who support the Coast Guard to use the hashtag #stayafloat to show their support on social media.
"I'm a veteran and I remember how tough it can be to get by," said George Sink, Sr., a Marine Corps veteran and the founding partner of George Sink, P.A. Injury Lawyers. "Right now, active military personnel in the Coast Guard aren't getting paid–but they still have to pay rent, buy food, and provide healthcare for their families. Some of these Guardians are having a hard time keeping their family afloat. We wanted to help."
860 Unpaid Coast Guard Personnel in Charleston
On Friday, January 18, ABC News4, published an interview with Coast Guard spokesperson PA3 Ryan Dickinson who explained in very real terms how the government shutdown was impacting the Coast Guard in Charleston.
Dickinson confirmed that all Charleston Coast Guard members had been affected but "continue to work diligently and faithfully," despite the fact that they have not received a paycheck since December 31, 2018. Although some administrative, maintenance, and recreational duties have been temporarily delayed/curtailed, the unpaid members of the Charleston Coast Guard are still responsible for essential services, including search and rescue, port and homeland safety and security, law enforcement, and environmental response, Dickinson explained in his interview.
The situation did not sit right with Sink.
"It's not right," Sink said. "Active military working without being paid. They're working to protect us, but who is protecting and providing for their families? It's a terrible situation, but it's an opportunity for us to serve them and that is what we intend to do."
Sink Puts His Money Where His Mouth Is
The decision to donate $5,000 to the Lowcountry CPOA was easy to make, as far as Sink is concerned. "Our military families need our support now," Sink said. "We're here to give it." The donation was presented to Lowcountry CPOA officials Wednesday morning.
Sink Challenges Others to Take the "Stay Afloat Challenge"
George Sink, P.A. Injury Lawyers has launched an online platform where people can learn how they can help. The page can be found at www.sinklaw.com/stayafloat. The firm is also urging the use of the hashtag #stayafloat in all social media posts related to the effort.
"The Coast Guard keeps our borders safe. They protect us. When we're in danger, they keep us afloat. Now it's time to help Coastie families stay afloat until the government shutdown is ended."
About George Sink, P.A. Injury Lawyers
George Sink, P.A. Injury Lawyers is a personal injury law firm with offices throughout South Carolina and in Georgia. The firm proudly plays a consistently supportive role in the community. For more information, visit www.sinklaw.com.
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Everything works beside the Outlook Address Book service.
I've tried tried modifying the the custom11.prf file, and I've also tried installing a freshly created .prf file.
Why won't the Outlook Address Book install ???
Could someone post a .PRF file that installs a Outlook Address Book.


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Hacking out in the countryside is bound to allow many wildlife encounters, hence the importance of preserving rights of ways - but that can be discussed another time… It is thrilling and special to capture glimpses of wildlife as it reminds us that we do share this planet, and of the critical importance of countryside conservation. But, I would like to write most specifically what we horse riders see, and in which situations, because it forms a basis of our work at CPRE.
Vulpes Vulpes, or, in English, the red fox. Culturally known as Mr Fox. This lovely cousin of the dog resides in the farmlands where I ride, taking advantage of the rabbit population, the fields and hedges. They like the green byways and shrubbery, human dwellings with available food sources and paddocks in the outlying areas of my village.
Snowdrops. Like all seasonal flowers, I had forgotten them. Occasionally I remembered them, but I had mainly forgotten them.


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1992, Sony Repertory, 140 min, UK, Dir: James Ivory
Class conflicts boil beneath the genteel surfaces of early 20th century England in this film of E.M. Forster's novel. Howards End is an estate that Ruth Wilcox (Vanessa Redgrave) intends to bequeath to her friend, Margaret (Emma Thompson), but Ruth's starchy spouse, Henry (Anthony Hopkins), is not about to let the house pass out of the family. Winner of three Oscars, for Best Actress (Thompson), Adapted Screenplay and Art Direction.
Ann Wingate
Ismail Merchant
Tony Pierce-Roberts
2013, The Weinstein Company, 132 min, USA, Dir: Lee Daniels
Forest Whitaker gives one of his finest performances as Cecil Gaines, a White House butler who serves eight consecutive U.S. presidents during a tumultuous period in American history. Director Lee Daniels neatly balances Gaines' personal story with the broader social implications of the times. With Oprah Winfrey, John Cusack, Robin Williams and Jane Fonda.
Buddy Patrick
Cassian Elwes
Danny Strong
1966, Warner Bros., 111 min, Italy, UK, Dir: Michelangelo Antonioni
In-demand fashion photographer David Hemmings shags half the models in London - before running into enigmatic beauty Vanessa Redgrave, who only wants a certain roll of film from him. A riddle within a riddle, Antonioni's most famous film has often been imitated (De Palma's BLOW OUT, Greenaway's THE DRAUGHTSMAN'S CONTRACT). Look for a young Jane Birkin in one of her first film appearances - and for The Yardbirds mimicking The Who's penchant for destruction in the film's frenzied nightclub scene!
David Hemmings


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remarks, that, " of the innumerable theories that have been started upon this subject, the three following are those which are chiefly entitled to our attention, namely: life is the result of a general harmony, or consent of action between the different organs of which the vital frame consists ; second, life is a principle inherent in the blood; third, life is a gas or aura, communicated to the system from without. Each of these theories has to boast of a very high degree of antiquity, and each, after having had its day, has successfully yielded to its rivals, and in its turn has appeared under a different modification in some subsequent age, and run through a new stage of popularity." Aristoxenius, one of the Grecian philosophers, and pupil of Xenophylus, and Erythraca of Lamptus, and Aristotle, conceived the principle of life resulting from "the system of harmony." His theory met with much favor both at Athens and at Rome. In the writings of Lactantius we find him thus speaking of it : " As in musical instruments an accord and assent of sound, which musicians term harmony, is produced by the due tone of the strings, so in bodies, the faculty of perception proceeds from a connexion and vigour of the numbers of
of the frame."— Vol. 5, p. 140. Sir Humpbrey Davy supposed the principle of life, or life itself, to consist in a "perpetual series of corpuscular changes, and the substrata or living body as the being in which these changes take place." Mr. John Hunter, the distinguished physiologist, regarded the blood itself as the principle of life, and treats the subject very ably in his work entitled, " Hunter on the Blood." Lavoisier sprang up with his brilliant powers of investigation, and pronounced oxygen to be the principle of life. Magendi, Dumas, Bichat, and Richerands, had all their peculiar theories, and their genius and ingenuity have given a probability at least to their various hypotheses. But no demonstrative proof has yet been given, capable of substantiating any theory ever yet advanced. When reduced to a tangible form, they really mean nothing ; we are as much in the dark as ever. Harmony may be the result of life, and the blood is necessary to its support, and oxygen for our existence; but they are not the principle of life itself. Harmony is there, the blood is there, the oxygen is there, and death ensuesand we have heretofore stood in the dark, with every theory crumbling into ashes. No theory is of real value unless it is susceptible of mathematical demonstration; and the author of the following, advances his only on the ground of his belief, that the proof of it exists in every thing that we behold in the material world. The
principle of life exists in the essential properties of matter, and it results from, or is caused by, the action of one thing upon another ; and why one thing, or substance, or element, acting upon another in a certain way, should produce life and sustain it, is no more a mystery, than why oxygen and hydrogen, in certain proportions, form water, or why anything is as it is. It is the result of properties of being sustaining in their movements their relations to each other, and no thing more. And the various theories that have been taught as to the principle of life, from Aristoxenius, who believed it proceeded from the system of harmony, down to the present day, must give way to the great fact, that the principle of life exists neither in the blood nor in the brain, nor in the nerves ; but that the substance of our being is depending upon the essential properties of the relationship of matter only, and which will be made self-evident from the following remarks, namely : We know that every elementary principle in the universe has an identity which distinguishes it from every thing else; we know that it has properties; we know that the principle of its being is action ; for the Deity reposes never ; neither do any of his works. We know that this disposition for every element to act, produces a combined action; we know that this combined action produces all that
exists. We know that certain elements and substances have an affinity for certain other elements and substances; that affinity, drawing these together, produce in their union new formations. And it is on this same great principle of action that life is produced and re-produced. We are composed of elements; these elements are in continual action from the instant we spring into being until we cease to be, when they start back to their original forms, and continue their action under some other manifestation. Not one instant do they repose ; and it is this great principle of action that is constantly producing and destroying. A certain action produces life-a certain action destroys it. Certain food which, of course, is composed of elements, supports life; certain minerals destroy it. Thus we come to see, that in the great progression of things, it is their essential properties of being to act upon one another, so as to produce and destroy—to form and re-form-to repeat again and again—to enter into combinations and separate—a continuous succession of changes. The production of to-day ceases to exist to-morrow; it has entered into something else. The vegetable has ceased to be a vegetable, and a part of its elements is flowing in our veins, It speedily will leave these and pass into new forms, and thus, throughout the whole universe of God, the order of its arrangement is motion, and from this inherent property of motion proceeds all that is. But this subject is susceptible of still further and more familiar proof. Take, for instance, a grain of corn placed in a vacuum; it would undergo no change whatever, because it cannot act upon itself. Place it, however, in the earth, and it at once undergoes a change. And what is this that we call a change ? Simply the action of one element upon another, and without such action no change would be produced whatever ; and herein is a familiar proof. This action is called life, and the change produces growth, and its cessation thus to act, death; and precisely so in relation to ourselves—our being is commenced, continued and ended, by the action of one thing upon another.
What is the great cause of the changes that are constantly taking place in all that exists? Why does not anything remain as it is? The flower that is so beautiful, so perfect in all its parts,—why does it not remain a flower ? The fruit so attractive to the eye—why does it decay? The young, the gay, the beautiful, why do they grow old? Why do the fairest things fade ?-why do the leaves fall in autumn ?-why do the winds blow? -why does the rain fall ?—why does any thing take place that does take place? It may be answered, they are the operations of nature. But this answer defines nothing. What is it? Where is the grand principle ? In what consists this ceaseless, eternal change? It is all brought about by the action of one element upon another, which action is the inherent principle of their being, (and here the reader must not take exception by endeavoring to separate this great action from its great author, but is requested to bear in mind what has previously been said on this subject.)* And if we reflect only for a few minuets, every object that we behold around us is indebted to this great law for its present form. There is nothing which has material being that is not composed of elements; and that which has caused these elements to act one upon another, so as to produce the given manifestation of any body, is their inherent principle of being. And although man has not heretofore understood the great cause of the existence of things, he has nevertheless acted upon it daily, and is a practical observer of the relationship of matter. For instance, the farmer sows his seed in the earth and enriches his soil, and waits for his crop, knowing he has acted in accordance with what he has always observed to produce crops heretofore; whereas, if he had laid his seed upon a rock, he would have had no hopes of a crop, because, the relationship of a seed to a
* Referring to a work now in the course of publication.
rock is such, that it would not cause it to produce. And he would have no hope, because he had not acted in accordance with the nature of the thing. That the principle of life exists in the essential properties of matter, can be as clearly demonstrated, as that the diameter of a circle is always equal to one-third its circumference, No matter under what form life is found, whether it be in a tree, flower, vegetable, animal, or in a man, it is indebted to the operations of this great law for its being; and the attention of even a common observer will have to be directed only to what is going on around him, to find the proof in every thing. It has heretofore been the case that we have been contented with observing the progress of things without seeking to penetrate into their causes. We know that if the seed were properly planted, it would ripen in time to grain or fruit; but of the causes of this process we know nothing, but in the operations of this law all is clearly and satisfactorily proven. Let us take three acorns, for instance; we lay one upon a rock, one is planted in the earth, and one is placed in a vacuum; trace now the result of each position. The one on the rock decomposesthe one in the vacuum remains the same—the one in the earth springs forth, and its sprout is seen struggling up through the ground. Now let us find the cause of the state of each. First, the one placed on the rock is decomposed; its relationship to matter has not been observed to make its germinating principle productive, notwithstanding it has not been permitted to retain its original form. No—it has been acted upon, and it has ceased to be. It was a spontaneous action, no extraneous or visible power was applied, and yet this change took place. Now, it is impossible for any change, however small, to take place in any particle of matter without a power; and wherein consists this power? Of necessity, it must be in the essential principle of matter, as it can be contained in nothing else; the proof of which is clearly found in the acorn which is placed in the vacuum ; this remains the same. Why? Because no element from without could come in contact with it, and therefore, no action could be produced. The one in the earth has passed through a different process from either, because the relationship to matter has been observed in the earth, and elements acted upon it so as to produce a sprout, the tree. The product is nothing more than the action of one element upon another. The acorn itself is composed of elements; the earth in which it is planted is composed of elements; the air up to which the sprout shoots is composed of elements; the heat of the sun, or caloric, is an element; the rain that waters it is coinposed of elements; and all these elements, acting upon each other, produce what is called life and growth, and the tree is the product of their combined action. And the same process of reasoning applies to whatever has life. The animal masticates the vegetable, which is composed of elements; it passes into the stomach. When the juices, which are composed of elements, act upon it, the nutritious part is conveyed to the blood, and the blood forms the solids of the system, perfectly simple and perfectly plain ; one element acting on another throughout, and in sustaining their relationship to each other, life is produced and continued. And it is impossible to find anything in the universe that lives, whose being is not begun and continued in this way. The action of one element upon another, is not only the cause of all that exists, but by it alone we are enabled to perform the simplest act. We are constantly talking of the power of steam, of electricity, of the mighty ocean, of the rushing wind. Calculations are made as to the horse-power, from the mighty Niagara to the little stream. Now, all power is obtained by opposing forces, and opposing forces are nothing but the essential principle of the being of matter, of elementary principles to act upon each other ; instance in detail the above. Steam is nothing more
than oxygen and hydrogen acted upon by caloric, and its expansive force is restrained by a pressure from without. We call it a steam-boiler, but the iron of it has been analyzed. It is composed of elernents, and the nature of its elements is such, in their present combination, as to enable them to withstand the power exerted against them by the elements, of what we call steam. Here, then, are two powers resisting each other ; the one the expansive power of steam, the other the cohesive power of iron. The power will continue to increase until the greater overcomes the weaker ; for this belongs also to the essential properties of matter, that the lesser force must always give way to the greater. But man does not carry things to this extent; for it is not his object, in constructing a steam-engine and boiler, and building his fire, merely to blow it up, as it is called; he has invented dipes, and tubes, and cylinders, by which means the action of these elements is brought to bear upon each other, and from this he obtains motion or power. The power of electricity is gained in the same way, by opposing forces. When it has no opposing forces it remains insensible; oppose it, and the sublime artillery of heaven is its voice, the sundered rock its victim, and the splintered oak its prey. It plays upon the world from the clouds on high, and the startled earth trembles beneath its power. The power of the ocean is derived from the winds, which disturb its equilibrium, and the law of gravitation which draws it back ; these forces contending, produce the mighty conflict which the mariner beholds with fear and trembling. The law of gravitation causes the water to descend with force; and a weaker force opposing it gives way, or is put in motion, as the water-wheel, for instance. It is not necessary to extend these remarks, as the information of every one upon these subjects is sufficient to enable them to make a further application. The rushing wind, or rather air, derives its force from the law of gravitation-heat and its nature, to maintain its equilibrium. Heat being much greater in some places than in others, causes the air to become rarefied or lighter, or more correctly speaking, enables it to overcome, to a greater degree, the force of the law of gravitation, and rise from the earth; the cool air from a distance commences moving forward to fill up the vacuum caused by the warm air ascending, and thus its motion or power is produced. Similar observations might be extended through volumes, but these are quite sufficient to show that all force or power is caused by opposing force or power; and the proof of this fact proves, also, that this universal opposing force so found, exists in the essential principles of matter to act upon each other, and this being proved, it is self-evident that all that exists is the product of the working of this great law; and unless we are willing to deny that we are matter, and not composed of elements, and not only to deny it but prove it, we must inevitably come under the operation of those laws which govern material beings. And to undertake to say we do not, when the testimony of every science proves that we do, would be merely an assertion of no weight; and to argue from the imagination rather than from the facts of the case, is as useless as it is unwise. The difficulty with us all is, we give ourselves up entirely too much to our imagination, and we have fixed the existence of a certain state of things that have no existence at all, and we take it for granted that we know all about things, of which in reality we know nothing; and notwithstanding it is one of the most pleasing traits bestowed upon us, yet it is one of the most dangerous, if not under proper control, for its tendency ever is to lead us into error ; every one having pictures in their minds, which, if they should take the trouble to investigate and reduce to a tangible form, would find they could actually have no reality; but from their being permitted to dwell there from year to year, they have taken a strong hold, and have become to be believed in as realities. I
will not stop to define what these pictures are, as they will be treated of in another place. I wish particularly here to request the reader to give himself up to the facts that have been set forth, and to free himself if possible from previously-conceived opinions and prejudices.* This work must be read as a whole, and what has been already stated in relation to the power and omnipotence of the Creator, borne constantly in mind; for the object of its author is to show our true relationship to him, and to explain our existence here, so that we may be enabled better to discharge the solemn duties incumbent upon us, and not, as some might at first think, to show that either we or his works exist in a kind of independence of him ; for any such attempt would be no less impossible than censurable; and we might just as well regard our limbs as some other person's limbs, or an enemy's, for instance, as to regard the works of the Almighty in any way separate from him; and, in proving that we exist from the essential principles of matter, sustaining in their movements their relationship to each other, it is only to be borne in mind, that not only the essential principles of matter, but matter itself, came fresh from the hands of the Infinite, and that all its combinations and formations are the result of his pleasure, which we always find to exist under certain laws, which were assigned when bes ing was given. I say it is only necessary to bear this in mind, in order to satisfy the most pious and conscientious Christian.
It has too long been the habit with us to endeavor to involve everything in mystery and surround it with impenetrability, as far as it relates to our material being here, and even our existence hereafter, rather than by a proper spirit of investigation to find out the real truth of the matter. No matter where error is found, it is dangerous, and the man who propagates it, is an enemy to his race. These remarks, then, serve to prove, without any detriment to revelation, first, that we exist precisely the same as everything we behold exists, namely, by the essential properties of matter, sustaining in their movements their relationship to each other ; that our life is commenced and continued by the operations of this great law; that there are two direct and opposing forces within us, which keep up the action, (on the same principle as has been previously explained, and in relation to force, page 69 ;) that these forces are, first, one exerting itself to preserve us in our present form of being, the other to change us into another form of be ing. The conflict is the action, which is life; and in compliance with the laws of nature, that a greater force always overcomes a lesser, we do what we call live or die. Whenever the elements acting upon each other, which cause life, are overcome by the opposing force of these elements, acting so as to free themselves from their present form of being, which we call death, then we die, when all action ceases in one sense, so far as new supplies go, and the counter-actions sweep our remains back to their elementary principles, the cause of which is, there has ceased to be an opposing force. Death, then, is brought about by the operation of the same great law as life, viz. : by the action of one element upon another in an opposite direction; and is nothing more than one force overcoming another, on the same principle that a projectile force is overcome by the laws of gravitation and resistance; or else a ball put once in motion would fly on forever. So with ourselves if there was no counteracting force, we would live on as we now are, forever. The medical profession will at once see that it is on the same principle of opposing force to force, that daily governs them in the treatment of all diseases. The equilibrium of the systein is disturbed, which superinduces some manifestation of disease; to counteract the force which has thrown it from its equilibrium, they make use of force under the form of a medicinal agent; the greater overcomes the weaker, and their patient is restored
* Referring to the work already spoken of.


text:
<?php

namespace EasyMinerCenter\Model\EasyMiner\Entities;

use LeanMapper\Entity;

/**
 * Class Rule
 * @package EasyMinerCenter\Model\EasyMiner\Entities
 * @author Stanislav Vojíř
 * @license http://www.apache.org/licenses/LICENSE-2.0 Apache License, Version 2.0
 *
 * @property int $ruleId
 * @property Task|null $task m:hasOne
 * @property-read RuleSetRuleRelation[] $ruleSetRuleRelations m:belongsToMany
 * @property string $text
 * @property string $pmmlRuleId = ''
 * @property Cedent|null $antecedent m:hasOne(antecedent)
 * @property Cedent|null $consequent m:hasOne(consequent)
 * @property int|null $a
 * @property int|null $b
 * @property int|null $c
 * @property int|null $d
 * @property float|null $confidence = null
 * @property float|null $support = null
 * @property float|null $lift = null
 * @property int|null $antecedentRuleAttributes = null
 * @property bool $inRuleClipboard
 * @property-read array $basicDataArr
 */
class Rule extends Entity{

  /**
   * @return array
   */
  public function getBasicDataArr() {
    $result=[
      'id'=>$this->ruleId,
      'text'=>$this->text,
      'a'=>$this->a,
      'b'=>$this->b,
      'c'=>$this->c,
      'd'=>$this->d,
      'selected'=>($this->inRuleClipboard?'1':'0')
    ];
    if ($this->confidence!=null){
      $result['confidence']=$this->confidence;
    }
    if ($this->support!=null){
      $result['support']=$this->support;
    }
    if ($this->lift!=null){
      $result['lift']=$this->lift;
    }
    if ($this->antecedentRuleAttributes!==null){
      $result['antecedentRuleAttributes']=$this->antecedentRuleAttributes;
    }
    if (!empty($this->row->task_id)){
      $result['task']=$this->row->task_id;
    }
    return $result;
  }

  /**
   * Method returning the relation of this rule to a concrete RuleSet
   * @param $ruleSet
   * @return RuleSetRuleRelation
   */
  public function getRuleSetRelation($ruleSet){
    if ($ruleSet instanceof RuleSet){
      $ruleSet=$ruleSet->ruleSetId;
    }
    if (!empty($this->ruleSetRuleRelations)){
      foreach($this->ruleSetRuleRelations as $ruleSetRuleRelation){
        if ($ruleSetRuleRelation->ruleSet->ruleSetId==$ruleSet){
          return $ruleSetRuleRelation;
        }
      }
    }
    return null;
  }

  public function getAntecedentRuleAttrbutes(){
    return $this->row->antecedent_rule_attributes;
  }

  public function setAntecedentRuleAttributes($value){
    if ($value===null){
      $this->row->antecedent_rule_attributes=null;
    }else{
      $this->row->antecedent_rule_attributes=$value;
    }
  }

  /**
   * @return array
   */
  public function getRuleHeadDataArr(){
    return [
      'text'=>$this->text,
      'task_id'=>$this->task->taskId,
      'pmml_rule_id'=>$this->pmmlRuleId,
      'a'=>$this->a,
      'b'=>$this->b,
      'c'=>$this->c,
      'd'=>$this->d
    ];
  }

}

redpajama_set_name: RedPajamaGithub


A City Can Be Green?
When someone says a "city," one thinks about dirty, and unhealthy. No one thinks that city living is the way to be more green. David Owen, the author of Green Metropolis proposes that living in the city is greener.
Everything is condensed so people either take public transportation, walk, or bike.
Only half of the population has cars, most are rarely used.
City dwellers use almost half the amount than people who live outside cities..
Living space is very small, which does not allow for the accumulation of things.
People have sprawled out over a large area, in order to get anything they have to drive there cars. They have more space in their cars. They have more space in their homes so they accumulate more.
Owen Claims that Solar encourages people to consume more energy and that the more green energy and cleaner energy for cars just continues to encourage the environmentally damaging behavior.
In order to change the car-dependent lifestyle he claims that driving has to be unpleasant so no one wants to do so. He also says that HOV lanes only allow for a better flow of traffic. In order for them to really work environmentally they need to eliminate regular lanes and charge single-occupant cars. I would have to argue that carpooling is not always a realistic expectation. My mother, for example, is a doctor who has very different shifts than anyone else, she can not realistically carpool with someone, I do not think it's fair to punish these people when there is no way they can carpool and no public transportation that is available and efficient.
While I do agree all these things make city dwellers have less of a direct environmental impact, I do not think that its impact is much less. Cities have a dense population leading to a dense concentration of pollution. Also what about all of the resources it takes to get goods and services to cities? Is that factored into his pollution equation? I think if this is factored in we might find that fossil fueled trucks, trains, and other forms of transportation are being used to supply these resources.
New York has the third highest cancer risk caused by airborne chemicals. Cities concentrate pollution that contributes to many health problems. Cities also have very high amounts of particulate pollution. With such a large population there is a large amount of garbage that needs to be disposed of. Although I realize that they would have a similar amount if they lived in the suburbs, cities do not have adequate plans for their garbage disposal. They often send it to other land fills in other states. Urban air pollution causes around 1.3 million deaths worldwide per year.
Children and the Elderly are especially susceptible to Health Effects of Ozone, fine particles, and other airborne toxicants.
The more polluted cities are with pollutants and smog, the more someone is at risk for hospitalization and early death.
I think that Owen makes good points but, there are many things he does not take into consideration that make the city solution appear less green. I think all communities need to have more self-sustaining practices, rather then relying on outside sources to bring in many of their goods. There will always be a portion of the population that wants to live in the city and a portion that wants to live in a rural setting. Irregardless of where one is living the use of public transportation, biking and walking should be encouraged. Communities should be as self-sustaining as possible utilizing local resources whenever possible. How will we get to a green world? Maybe a combination of ideas that Americans can adapt to and accept.
Should you read Green Metropolis?
I highly recommend that you read the book, it contains some excellent ideas and is very insightful. Although, I do not think it is the all answer to our environmental problems it does bring up some very good points. I highly recommend the reading to broaden your look at environmental issues and some possible solutions. I find Owen's writing to be very opinionated and his model for an environmental world to be very unrealistic.
What do you think? Is city living the way towards an environmentally friendly future?
Increased power is needed in the rapid developing world we have today.
Many World Organizations and the World Bank got behind the idea of hydropower as a 'sustainable' way of producing energy.
Large Scale Hydropower does not benefit the local population and environment.
There are over 450,000 dams that are over 5 stories high.
Large scale dams have been built all over the world.
Countries with large dams include; the United States, Canada, Argentina, South Africa, China, India, and Australia.
Displaced people are often not given the appropriate housing and living conditions (their cultural beliefs are not considered).
Displaced people are never given the compensation they were promised.
Those living downstream no longer have fisheries and water to support themselves.
Cultural artifacts and Archeological sites are lost.
Building dams destroys the land nearby through earthquakes created because of the pressure of the water collected by the dam.
Fish are unable to migrate upstream to spawn, decreasing the amount of fish available for food.
Huge amounts of habitat is filled with water.
Animals are disoriented by the sudden amount of water in their migration path, and they drown while trying to pursue their migration paths.
Downstream less water is flowing, which endangers the ecosystems.
Ecosystems are disrupted because many animals rely on the size, timing, and other seasonal changes to the water flow survive.
Wildlife living downstream can go extinct.
In w armer climates dams create a breading ground for diseases such as Malaria and Schistosomiasis.
Increase in Salinization can occur. This causes natural salt to rise to the surface. Elevated groundwater and increased salt content has killed many trees and wetlands have been eliminated.
Sedimentation also occurs in large dams.
Hypoxia, a reduced oxygen content is damaging to aquatic life.
The picture below shows what Salinization does to the land.
I would highly recommend reading Jacques Leslie's book Deep Water it is very educational and shows the consequences of dams in different places all over the world. I would warn you that the book is dry at sometimes, but the benefit from reading it is well worth it.
Large dams will affect the community living near the dam in the long-term. Take for example the Three Gorges Dam, as the above video stated, it has had a significant effect on the drought in China.
World Organizations, as well as the countries themselves, need to take into account the large impact building a big dam has on wildlife, culture, people, and weather. To be clear I am not against all dams, but there has to be a balance between energy demand and environmental consequences.
Did you know of the millions of people that have been displaced by dams? Do you think there is an equivalent compensation to them? There are many large scale dams being built in China, will anything be done to help the people who live there?
Do you know where your e-waste is going?
E-waste is electronic waste that makes up 5% of municipal solid waste worldwide. E-waste includes phones, computers, televisions, audio equipment, printers, DVD players, and any electronic device.Every year 25 million tons of e-waste is produced, and a large percent of it is dumped abroad.
In American society we have this notion that we need the newest gadget and we need it now! In each new generation there is an increasing need for instant gratification. Our society no longer purchases only their true needs, but over consumes without considering the environmental impact. American society is materialistic, and consumers only care about their own selfish needs.
There are harmful elements found in electronics like PVC, mercury, solvents, flame retardants, and lead. Lead can be very disastrous to many organs and tissues (heart, bones, intestines, kidneys), as well as cardiovascular, reproductive, and nervous systems. It can cause symptoms from headaches to seizures, coma, or death. Flame retardants can cause abnormal thyroid metabolism, and poor brain development. When burned, plastic and other materials in electronic products release toxic chemicals. Plastics release dioxins such as; PVC or hydrofluoric acid, which can damage the lungs of those exposed to it. Not only are people who are actively participating in this getting exposed, but also the surrounding communities ground water and air is contaminated. The contaminated water leads to contamination of crops, animals and anyone who drinks it.
The map above shows the known and suspected routes of e-waste dumping (sorry the picture is so small but it was the only one I could find that contained suspected routes). E-waste is shipped overseas to Asia, India, Africa and other developing countries. Because these people are so impoverished they work at 'recycling facilities.' At these facilities they do not wear any protection so their health and environment declines rapidly. These people earn about 1 meal a day while damaging their health. Many of these people are not aware of the risks involved with their work.
What can you do to ensure your e-waste does not end up overseas?
E-waste recycling programs that have been certified by e-stewards have to comply with certain standards. These include waste not being disposed of in solid waste landfills or incinerators, prohibits the exports and imports of electronics and specifically prohibits export of hazardous waste to developing countries. In order for this dangerous oversea shipping to stop, it is necessary for the public to be aware of this issue, and make sure that the e-waste recycling program will not ship overseas.
How are you disposing of your e-waste? Is your e-waste recycler certified?


Yaqshaan Medan, a Philadelphia native, has over 14 years of experience in the art of Dance. Since beginning dance at the tender age of 16, Yaqshaan has went on to obtain his Bachelor of Fine Arts in Dance from the prestigious University of the Arts in Philadelphia, PA. Shortly thereafter, he worked to build his career in Dance Performance nationally and internationally. He went on to work with notable companies such as Dance Theatre X, Philadelphia Opera Company, The Rock School, Nevada Ballet, Norwegian Cruise Line, and the acclaimed Las Vegas Classic Jubilee! His networking and impeccable work ethic allowed Yaqshaan to pursue his chance at choreography. The opportunity allowed Yaqshaan to become an award winning choreographer with Youth America Grand Prix, International Ballet Competition, and Regional Dance America. Currently Yaqshaan holds several positions with Charlotte Ballet, Winston Salem State University as an adjunct Professor and most recently, serves as a Residential Director of Jazz and Modern at Dance Davidson in Davidson, NC.


A detailed article about A Step By Step Guide to Sell eBooks Online.
It would be safe to assume that there is a writer hidden in every person who likes to express himself. Moreover, there is definitely a dream in every individual to lead a relaxed and easy life wherein one can do whatever one wishes and as much as one wishes. A life where there are no worries or pressures. While many people think that this kind of life is a myth, there are hundreds of people in the world who are leading it by just deciding to write and sell eBooks online.
If you decide to sell eBooks online and actually stick with the plan long enough, you will see that the whole process is extremely easy and highly lucrative. In fact, if you sell your eBooks online, you would not have to worry about any of the conventional business worries such as storage space, distribution costs or, for that matter, even production costs. In addition to this, if you can say to yourself that 'I will sell my eBook online and make it work', you would be able to establish an income stream that would last for the foreseeable future without any extra investment.
As is obvious, before you actually sell eBooks online, you would need to create one. Creating the eBook is, debatably, the most crucial aspect of the whole process of selling them online. The reason for this is that unless your eBooks are of good quality and cater to an already existing demand in the industry, you would not be able to sell enough of them.
Consequently, choosing the subject on which you will write and sell your eBooks can make or break your whole effort. There are several tips that can help you choose the right subject before you sell your eBooks online. The most important would be that you choose a subject that you are comfortable with. As you are going to be spending a lot of time writing these eBooks, you have to ensure that you are not only able to sustain your interest in them but also able to do justice to the chosen topic.
However, it is also important for you to research what kinds of eBooks are popular in the industry. For example, if there are a lot of people who are able to sell eBooks online of similar types then you can assume that such topics will sell for sure and focus on them.
Once you have a product that you can sell, your next task is to establish a virtual storefront from where you can sell your eBooks in comfort. This process involves choosing and purchasing the right domain name, followed by designing a website and finally putting it up.
You have to make sure that the website that you set up to sell your eBooks online has a design that is conducive to your area of concern. Moreover, the design of the website also needs to be good because research has shown that potential buyers can easily get put off by a poorly designed website.
After your virtual storefront is set up, you would have to establish a good backend system that will allow you to sell your eBooks to various customers easily. This means having a digital product shopping cart, a digital product delivery system and a payment system that is designed specifically for eBook sellers. The digital product shopping cart for eBooks would have specific features for people who want to sell eBooks online, the digital product delivery system would ensure that text products are piracy protected, and the payment system would guarantee smooth and flawless payments for eBooks sellers.
It is also worth pointing out that, if you want to sell your eBooks to customers hailing from different nations and regions of the world, then you should look for a digital product payment system that is compatible with multiple currencies, multiple languages and even multiple payment gateways. Payment gateways you should look to include Google Checkout, 2Checkout, Trial Pay and Amazon Payments amongst others. However, you should also know that the majority of business on the internet is done through PayPal which is why you need to sell eBooks through PayPal first and the rest after.
In fact, every customer that you want to sell your eBooks to would read the accompanying content to judge whether he wants to buy your eBooks or not. Hence, it is crucial that you pay as much attention to the accompanying descriptions as you did when you were writing your eBooks. It is also important for you to keep in mind the principles of search engine optimization when writing your website's content, because it would end up determining how many customers you attract and how much you are able to sell your eBooks.
In order to get people to come to your website and look at what you are offering, you have to first attract them. For this, you will need to put in place online marketing and search engine optimization techniques for your website. However, an even better way would be for you to start selling your eBooks on established digital product marketplaces such as PayLoadz. The reason why you should sell your eBooks in such marketplaces is that they give you access to a pre targeted customer base which, in turn, improves your chances to sell eBooks online.
In fact, sites like PayLoadz provide the full service to people looking to sell eBooks online. They not only have a digital product marketplace but also provide backend services such as customized shopping carts, elaborate digital product delivery systems and multiple payment systems compatibility. In addition to this, these types of websites are even more suitable for people looking to sell eBooks online because they have affiliate marketing systems that turn out to be very useful for such sellers.


text:
\section*{0\quad Introduction}

 A matrix is traditionally just a table of real or complex numbers. As
 eloquently put by Tobias Danzig,
 \begin{quote} $\ldots$ a theory in which a whole
 	array of elements is regarded as a number-individual. These filing cabinets are added and multiplied, and a whole calculus of
 matrices has been established which may be regarded as a continuation of the algebra of complex numbers. \cite[p.212]{TD1967}. \end{quote}
 What is magical about matrices is contained in the definition of the {\it multiplication} of matrices. We give a geometrical construction of $2^n \times 2^n$ matrix algebras $Mat_{2^n}(F)$ over the real and complex numbers, when $F=\mathbb{R}}  %\def\R{I\!\!R$, or $F=\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}$, respectively, in terms of the Kronecker Product of $2\times 2$-blocks of {\it anti-commuting} null vectors.

 Whereas this rather restricted class of matrix algebras grow in size quite rapidly,
  \[  Mat_{1}(F) , Mat_2(F), Mat_4(F) , \cdots, Mat_{2^n}(F) , \cdots \]
  they have unique geometric properties which call for special attention. Indeed,
  it is the geometric properties of the matrix algebras $Mat_2(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I} )$ and $Mat_4(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I} )$, that early in the $20^{th}$-Century was key to the development of the  fundamental theories of relativity and quantum mechanics. We explain in detail how the structure of the geometric algebras $\mathbb{G}} % \def\G{I\!\!\!G_{p,q}$ are embedded in the matrix algebras $Mat_{2^n}(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I} )$. The comprehensive geometric interpretation of matrices of these sizes has many ramifications, justifying the terminology {\it geometric matrices} \cite{S08}. One promising area of application is sparse random matrices, such as used in error correcting codes \cite{McKay}.

The {\it permutation} on $n$ letters is a good place to start a study of the
symmetric group $S_n$, and the corresponding natural {\it permutation representation} of the rows of a square $n\times n$ matrix \cite{kao2010}. Such matrices are easily represented in terms of the geometric matrices of our approach. Representation theory of finite groups has provided new insight into the structure of finite groups, with many applications in other areas of mathematics and science. More generally, the study of finite dimensional representations of Lie groups and Lie algebra, \cite{JL2001,FH1991}, is a powerful tool in the study of quantum mechanics and atomic structure \cite{Weyl1950}. Aside from its many important applications, it is a beautiful mathematical structure that has developed over the last 150 years, and is worthy of study on its own merits.

\section{Geometric matrices}

Real or complex matrices of size $2^n\times 2^n$ are built up, block by block, by considering pairs of null vectors satisfying the following two simple rules:

  \begin{itemize}
  	\item[N1)] $a^2 = 0 = b^2$ \quad {\it The non-zero vectors $a $ and $b $ are nilpotents}.
  	\item[N2)] $a b + b a =1$ \quad {\it The sum of $a b $ and $b a $ is $1$.}
   \end{itemize}
  Since we are assuming product associativity, the second property easily implies that $aba =a$.
  We take $k$ such pairs of vectors $a_1, b_1, \cdots, a_k, b_k$, and assume that
  the null vectors with distinct indexes are pare-wise anti-commutative. That is,
  \[ a_i a_j = - a_j a_i , \ b_i b_j = - b_j b_i, \ a_i b_j = -  b_j a_i \]
  for $i \ne j$.

  If $M= \{m_{ij} \}$ and $N = \{ n_{jk} \}$ are matrices over geometric numbers, we define
  the {\it left} and {\it right} {\it Kronecker product} of $M$ and $N$ by
 \[ M \overrightarrow\otimes N:=  \{ M n_{ij}  \}, \ {\rm and} \
  M \overleftarrow\otimes N:=\{ m_{ij} N  \}.  \]

    As an example, consider the pairs of null vectors $a_i$ and $b_i$ satisfying N1) and N2) above. Then
  \[ \pmatrix{1 \cr b_1}\overrightarrow \otimes \pmatrix{1 \cr b_2} :=
  \pmatrix{1 \cr b_1 \cr b_2 \cr b_{12}}  \ {\rm and} \  \pmatrix{1 & a_2} \overleftarrow  \otimes \pmatrix{1 & a_1} := \pmatrix{1 & a_1 & a_2 & a_{21}}.    \]
  Defining $B_i^T:=\pmatrix{1 & b_i}$, and $A_i^T:= \pmatrix{1 & a_i}$, the above
  relationships take the form
  \[  B_1 \overrightarrow{\otimes}B_2=\pmatrix{B_1 \cr B_1 b_2} \ {\rm and} \
      A_2^T  \overleftarrow{\otimes}A_1^T = \pmatrix{A_1^T & a_2 A_1^T}.        \]
  Shortly, we will use these expressions and their generalizations.

  In a previous paper \cite{hypre2017}, I showed that the geometric
  algebra defined by
  \[ \mathbb{G}} % \def\G{I\!\!\!G_{1,1}:=\Big\{  \mathbb{R}}  %\def\R{I\!\!R(a,b)| \ a^2 = b^2 = 0, \  ab+ba =1\Big\} , \]
  is specified by its {\it spectral basis}
  \[ \mathbb{G}} % \def\G{I\!\!\!G_{1,1} =B ab A^T=  \pmatrix{1 \cr b}ab \pmatrix{1 & a} =\pmatrix{ab  \cr b } \pmatrix{ab & a} = \pmatrix{ab & a \cr b & ba} .   \]
The matrix representation of a general element
\[ g = g_{11}ab  + g_{12} a+g_{21} b + g_{22} ba \in \mathbb{G}} % \def\G{I\!\!\!G_{1,1} \]
is given by
\[  g = B^Tu A  g B^Tu A =
     \pmatrix{1 & b}u\pmatrix{1 \cr a} g\pmatrix{1 & b}u \pmatrix{1 \cr a}            \]
   \begin{equation}  = \pmatrix{1 & b}u \pmatrix{ g & g b \cr a g & agb} u \pmatrix{1 \cr a}=
      B^Tu [g] A ,    \label{specbasisg} \end{equation}
where $u=ab$ and $[g]:= \pmatrix{g_{11} & g_{12} \cr g_{21} & g_{22}}$ is the real matrix of the geometric number $g\in \mathbb{G}} % \def\G{I\!\!\!G_{1,1}$ with respect to this spectral basis.
The matrix algebra $Mat_2(\mathbb{R}}  %\def\R{I\!\!R)$ is algebraically isomorphic, and thus fully equivalent to the geometric algebra $\mathbb{G}} % \def\G{I\!\!\!G_{1,1}$, and inherits a comprehensive geometric interpretation.

 By using the same spectral basis over the complex numbers, any geometric number $g\in \mathbb{G}} % \def\G{I\!\!\!G_{1,2}$ can be
represented by a $2\times 2$ matrix $[g]_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}$ over the complex numbers.
That is
\[  g =  \pmatrix{1 & b}ab [g]_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}  \pmatrix{1 \cr a} =B^Tu[g]_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}   A ,  \]
where the generating unit vectors   $e_1, f_1, f_2$ of $\mathbb{G}} % \def\G{I\!\!\!G_{1,2}$ are defined by
\[ e_1=a+b, f_1 = a-b, \ {\rm and} \ f_2:= e_1 f_1 i, \]
and $i:=e_1 f_{12} $.

Calculating the conjugation operations \cite[p.60]{SNF}, of {\it reversion} $g^\dagger$ and {\it grade inversion} $g^-$, gives
\begin{equation} g^\dagger = A^Tu^\dagger \overline{[g]}^T  B    = B^T u\pmatrix{\overline g_{22} & \overline g_{12} \cr \overline g_{21} & \overline g_{11}}_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}  A
  \label{dag-g}, \end{equation}
\begin{equation}    g^- =  \big( B^T\big)^- u\overline{[g]}  A^-      = B^Tu\pmatrix{\overline g_{11} & -\overline g_{12} \cr -\overline g_{21} & \overline g_{22}}_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}   A,
\label{invertg} \end{equation}
respectively,
and the reverse-inversion $g^*$ of $g \in \mathbb{G}} % \def\G{I\!\!\!G_{1,2} $ is
\begin{equation}  g^* = \big(A^T\big)^-u^\dagger {[g]}^T  B^-   = B^T u\pmatrix{ g_{22} &- g_{12} \cr - g_{21} &  g_{11}}_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}  A.    \label{star-g} \end{equation}
 Using the operation of reverse-inversion, the {\it determinant}
\begin{equation} \det [g] := gg^* = B^Tu [g][g^*] A = g_{11} g_{22}- g_{12} g_{21} . \label{detg} \end{equation}

Before generalizing to higher dimensional geometric algebras $\mathbb{G}} % \def\G{I\!\!\!G_{p,q}$, consider $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$. The spectral
basis of $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$ is
\[  \mathbb{G}} % \def\G{I\!\!\!G_{2,2} =B_1 \overrightarrow{\otimes}B_2 u_{12} A_2^T \overleftarrow{\otimes}A_1^T =\pmatrix{B_1 \cr B_1 b_2}u_{12} \pmatrix{A_1^T& a_2 A_1^T}\]
\[=
\pmatrix{B_1 u_{12}A_1^T & B_1 u_{1}a_2 A_1^T \cr B_1 b_2 u_{1}A_1^T &
	B_1 u_{1}u_2^\dagger  A_1^T }\]
\[=\pmatrix{1 \cr b_1}\overrightarrow\otimes \pmatrix{1 \cr b_2}
u_{12}  \pmatrix{1 & a_2} \overleftarrow \otimes \pmatrix{1 & a_1 }  \]
 \[ =\pmatrix{1 \cr b_1}\overrightarrow\otimes \pmatrix{u_{12} & a_2 u_1 \cr b_2 u_1 & u_1 u_2^\dagger} \overleftarrow \otimes  \pmatrix{1 & a_1}  \]
 \begin{equation} = \pmatrix{u_{12} & a_{1} u_2 & a_{2}u_1 & a_{21}
 		\cr b_1 u_2 & u_1^\dagger u_2 & b_1 a_2 &-a_2 u_1^\dagger   \cr
 		b_{2}u_1  & b_2 a_1 & u_1 u_2^\dagger & a_1 u_2^\dagger  \cr
 		b_{12} & -b_2 u_1^\dagger & b_1u_2^\dagger & u_{12}^\dagger} \label{specG22} \end{equation}
 	where $u_i:= a_ib_i$, $u_i^\dagger=b_i a_i$, $u_{12}:=u_1u_2$, and $u_{12}^\dagger = u_1^\dagger u_2^\dagger$.

For $g\in \mathbb{G}} % \def\G{I\!\!\!G_{2,2}=\mathbb{G}} % \def\G{I\!\!\!G_{1,1}\times \mathbb{G}} % \def\G{I\!\!\!G_{1,1}^\prime, $
where $\mathbb{G}} % \def\G{I\!\!\!G_{1,1}:=\mathbb{R}}  %\def\R{I\!\!R(a_1,b_1)$ and $\mathbb{G}} % \def\G{I\!\!\!G_{1,1}^\prime:=\mathbb{R}}  %\def\R{I\!\!R(a_2,b_2)$,
   \[ g= B_1^T \overrightarrow{\otimes}B_2^Tu_{12} \Big[  A_2 \overleftarrow{\otimes}A_1 g
   B_1^T \overrightarrow{\otimes}B_2^T  \Big] u_{12}A_2 \overleftarrow{\otimes}A_1     \]
   \[  = B_1^T \overrightarrow{\otimes}B_2^Tu_{12} [ g ] A_2 \overleftarrow{\otimes}A_1 = B_1^T u_1\Big[ B_2^Tu_2\pmatrix{[g]_{11} & [g]_{12} \cr [g]_{21} & [g]_{22}}A_2 \Big]A_1               \]
   \begin{equation}   =B_1^T u_1\pmatrix{g_1^\prime & g_2^\prime \cr g_3^\prime & g_4^\prime}A_1              \label{2blocks} \end{equation}
   where $[g]$ is the real $4\times 4$ matrix of $g$ with respect to the spectral basis (\ref{specG22}), $[g]_{kl}$ are its $2\times 2$ blocks, and $g_1^\prime,g_2^\prime, g_3^\prime, g_4^\prime$ are the elements in $\mathbb{G}} % \def\G{I\!\!\!G_{1,1}^\prime$ represented by each of these blocks, respectively.  Generalizing (\ref{2blocks}) for $\mathbb{G}} % \def\G{I\!\!\!G_{n+1,n+1}=\mathbb{G}} % \def\G{I\!\!\!G_{1,1}\times \mathbb{G}} % \def\G{I\!\!\!G_{n,n}^\prime $, gives for $g\in \mathbb{G}} % \def\G{I\!\!\!G_{n+1,n+1}$
      \begin{equation} g =B_1^T u_{1}\pmatrix{g_1^\prime & g_2^\prime \cr g_3^\prime & g_4^\prime}A_1,  \label{specGnn} \end{equation}
   for $g_1^\prime,g_2^\prime,g_3^\prime,g_4^\prime \in \mathbb{G}} % \def\G{I\!\!\!G_{n,n}^\prime$.

    For $g\in \mathbb{G}} % \def\G{I\!\!\!G_{2,2}=\mathbb{G}} % \def\G{I\!\!\!G_{1,1}\times \mathbb{G}} % \def\G{I\!\!\!G_{1,1}^\prime$, the reverse of $g$ is
   \[g^\dagger = A_1^T \overrightarrow{\otimes}A_2^Tu_{12}^\dagger [ g]^T
   B_2 \overleftarrow{\otimes}B_1 =\pmatrix{A_1^T & A_1^Ta_2}u_{12}^\dagger[g]^T \pmatrix{B_1 \cr b_2 B_1}.       \]
   We also have
   \[ [g] = \pmatrix{g_{11} & g_{12} & g_{13} & g_{14} \cr
   	g_{21} & g_{22} & g_{23} & g_{24} \cr
   	g_{31} & g_{32} & g_{33} & g_{34} \cr
   	g_{41} & g_{42} & g_{43} & g_{44} }, \ \
   	 [g^\dagger] = \pmatrix{g_{44} & g_{34} & -g_{24} & -g_{14} \cr
   	 	g_{43} & g_{33} & -g_{23} & -g_{13} \cr
   	 	-g_{42} & -g_{32} & g_{22} & g_{12} \cr
   	 	-g_{41} & -g_{31} & g_{21} & g_{11} }, \ \
   	   	\]
   	 and
   \[  [g^*] = \pmatrix{g_{44} & -g_{34} & g_{24} & -g_{14} \cr
   -	g_{43} & g_{33} & -g_{23} & g_{13} \cr
   	g_{42} & -g_{32} & g_{22} & -g_{12} \cr
   	-g_{41} & g_{31} & -g_{21} & g_{11} }, \ \
   \]

 The matrices $[a_1],[b_1],[a_2],[b_2]$ of the null vectors $a_1,b_1,a_2,b_2$, are

 \[  [a_1]= \pmatrix{0 & 1 & 0 & 0 \cr
 	                 0 & 0 & 0 & 0 \cr
 	                 0 & 0 & 0 & 1 \cr
 	                 0&0&0&0 }, \
 	   [a_2] =     \pmatrix{0 & 0 & 1 & 0 \cr
 	                 0 & 0 & 0 & -1 \cr
 	                 0 & 0 & 0 & 0 \cr
 	                 0&0&0&0 }   \]
  \[  [b_1]= \pmatrix{0 & 0 & 0 & 0 \cr
  	1 & 0 & 0 & 0 \cr
  	0 & 0 & 0 & 0 \cr
  	0&0&1&0 }, \
  [b_2] =     \pmatrix{0 & 0 & 0 & 0 \cr
  	0 & 0 & 0 & 0 \cr
  	1 & 0 & 0 & 0 \cr
  	0&-1&0&0 }  .  \]
  Note that in the special case when
  \[ g=g_{11} a_1b_1+g_{12} a_1+g_{21}b_1+g_{22}b_1 a_1\in \mathbb{G}} % \def\G{I\!\!\!G_{1,1}\subset \mathbb{G}} % \def\G{I\!\!\!G_{1,1}\times \mathbb{G}} % \def\G{I\!\!\!G^\prime_{1,1}  , \]
   then the $4\times 4$-matrix $[g]$ of $g$ is
  \[  [g]=\pmatrix{g_{11}&g_{12} & 0 &0 \cr g_{21} & g_{22} & 0& 0 \cr 0&0 & g_{11}&g_{12}  \cr 0&0& g_{21} & g_{22} } .  \]
  Alternatively, for $g^\prime =g_{11}^\prime a_2 b_2 + g_{12}^\prime a_2+ g_{21}^\prime b_2+g_{22}^\prime b_2a_2\in \mathbb{G}} % \def\G{I\!\!\!G_{1,1}^\prime$, then
    \[  [g]=\pmatrix{g_{11}^\prime & 0 &g_{12}^\prime & 0  \cr
    	 0 & g_{11}^\prime & 0& - g_{12}^\prime \cr
    	 g_{21}^\prime &0 & g_{22}^\prime & 0  \cr
         0 & -g_{21}^\prime & 0 & g_{22}^\prime } .  \]

    The {\it standard basis} of $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$ is defined by
    \[ \mathbb{G}} % \def\G{I\!\!\!G_{n,n} := \mathbb{R}}  %\def\R{I\!\!R(e_1, \cdots, e_n,f_1, \cdots f_n)
    =gen_\mathbb{R}}  %\def\R{I\!\!R \{e_i,f_i| \ e_i := a_i+b_i, f_i=a_i-b_i \},     \]
    for $1\le i \le n$.
    For the {\it complexified geometric algebra}
    \[ \mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}:= \mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}(e_1,\cdots,e_n,f_1,\cdots f_n)=gen_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I} \{e_i,f_i| \ e_i := a_i+b_i, f_i=a_i-b_i \} ,\]
    with $i:= e_{n\cdots 1}f_{1\cdots n+1}$. Defined in this way, the $2n+1$-vector $i$, playing the roll of the imaginary unit, is in the center of $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$, and $i^2=-1$.

  Consider now the geometric algebra $\mathbb{G}} % \def\G{I\!\!\!G_{p+1,q+1}=\mathbb{G}} % \def\G{I\!\!\!G_{p,q}^\prime\times \mathbb{G}} % \def\G{I\!\!\!G_{1,1}$. Using the spectral basis of $\mathbb{G}} % \def\G{I\!\!\!G_{1,1}=gen_\mathbb{R}}  %\def\R{I\!\!R\{a_1,b_1\}$, a general element $g\in \mathbb{G}} % \def\G{I\!\!\!G_{p+1,q+1}$ can be written
  \[  g= h_1 a_1b_1 + h_2 a_1 + h_3 b_1 + h_4 b_1a_1 = a_1b_1h_1 +  a_1 h_2^- +  b_1 h_3^- +  b_1a_1 h_4 , \]
  where $h_1,h_2,h_3,h_4 \in \mathbb{G}} % \def\G{I\!\!\!G_{p,q}^\prime$. Then, for $u_1=a_1b_1$,
  \begin{equation}  g = \pmatrix{1 & b_1}u_1 \pmatrix{1 \cr a_1 }g \pmatrix{1 & b_1}u_1  \pmatrix{1 \cr a_1}
  =  B^T u_1\pmatrix{h_1 & h_2 \cr h_3^- & h_4^- } A. \label{pqdecomp} \end{equation}
   Thus, the elements of $\mathbb{G}} % \def\G{I\!\!\!G_{p+1,q+1}$ can be expressed as $2\times 2$ matrices over
  $\mathbb{G}} % \def\G{I\!\!\!G_{p,q}^\prime$.

The spectral basis referred to in \cite[p.206]{SNF} is equivalent to
\[ \pmatrix{1 & a_1 & a_2 & a_{21}}^T b_1 a_1 b_2 a_2 \pmatrix{1 & b_1 & b_2 & b_{12}}   \]
The {\it reverse $\dagger$} of this basis is
\[   \pmatrix{1 & b_1 & b_2 & b_{21}}^T a_1 b_1 a_2b_2 \pmatrix{1 & a_1 & a_2 & a_{12}} ,   \]
which is {\bf not} equivalent to the spectral basis (\ref{specG22}), meaning there is no inner automorphism which will take this spectral basis into the spectral basis (\ref{specG22}), and at the same time leave invariant of the null vectors
$a_i$ and $b_i$.

 \section{The permutation group algebra of $ S_n $}

    The symmetric group $S_n$ consists of all permutations on $n$ letters. We use the
  usual cycle notation. For example,
  \[ S_3 =\{ 1, (12),(13),(23),(123),(132)\},   \]
  with group multiplication of cycles. An example cycle multiplication, from right to left, is \[ (23)=(12)(13)(12) .\]

The {\it permutation representation} of $S_n$ is generated by real $n\times n$ matrices of the form
 \[  (1k)={\bf 1}_{(1k)}, \]
 where the matrix ${\bf 1}_{(1k)}$ is obtained by interchanging the first and the
 $k^{th}$ rows of the $n\times n$ of the identity matrix $\bf 1$. The generators of the permutation matrix group algebra ${ S}_3$ are
 \[{ S}_3 =gen \big\{(12),(13) \big\} = gen\Big\{ \pmatrix{0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1}, \pmatrix{0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0} \Big\}. \]

 In studying the permutation representation, and the closely related {\it standard representation} of $S_n$, it is expedient to introduce the {\it all ones matrix}
 \[ {\cal A}_n:= \pmatrix{1 & \cdots &1 \cr \cdot & \cdots & \cdot
 	 \cr \cdot & \cdots & \cdot \cr
 	            1 & \cdots & 1},   \]
 and the {\it Casimir matrix} \cite{Kun2015},
 \[  {\cal C}_n := {\cal A}_n - 1_n,  \]
 where $1_n$ is the $n\times n$-identity matrix. For example
 \[ {\cal C}_3 = \pmatrix{0 & 1&1 \cr 1 & 0 & 1 \cr 1 & 1 & 0}.    \]

 Not surprisingly, both the $n$-all ones matrix and the $n$-Casimir matrix commutes with all the regular symmetric matrices $ S_n$. It is also easy to verify the simple relationships
 \[ {\cal A}_n^2 = n {\cal A}_n \ \ {\rm and} \ \ {\cal C}_n^2=  (n-2){\cal C}_n +(n-1)1_n ,  \]
 implying that
  \begin{equation}  min({\cal A}_{n}) := x(x-1) , \ {\rm and} \ min({\cal C}_{n}) :=(x+1)\big(x-(n-1)\big),  \label{minpolys}  \end{equation}
  are, respectively, the minimal polynomials of ${\cal A}_n$ and ${\cal C}_n$.

  \section{The geometric group algebra of $S_n$}

The geometric algebras $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$ and $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$ provide a comprehensive geometric interpretation for the elements of the group algebra of the symmetric group $ S_n$, and give a new
way of studying its properties. One problem of this approach is that these geometric group algebras only exist for dimensions $2^n$, corresponding to the matrix
representations using $2^n \times 2^n$ matrices over the real or complex numbers. However, our approach provides both new tools and a geometric interpretation of the results.

We begin by examining the geometric group algebras of the smaller symmetric groups, and then generalize these results to $S_n$. The group algebra of
$S_2$ is generated by the single element $(12)=\pmatrix{0&1 \cr 1 & 0}$. The
corresponding generating element in the geometric group algebra is $(12)=a_1 +b_1 \in \mathbb{G}} % \def\G{I\!\!\!G_{1,1} $. Thus, the permutation representation of $S_2$ is
\[ S_2 = span \{ \pmatrix{1& 0 \cr 0 & 1} ,\pmatrix{0&1 \cr 1 & 0} \}=span_\mathbb{R}}  %\def\R{I\!\!R \{1, a_1+b_1\}=gen_{\mathbb{R}}  %\def\R{I\!\!R } \{ a_1+b_1 \}         . \]

To get the permutation geometric group algebra of $S_3$, we must go to the larger geometric algebra $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$. Using the spectral basis (\ref{specG22}), we find the geometric permutation representation
\begin{equation} S_3 =gen_\mathbb{R}}  %\def\R{I\!\!R \{(12),(13)\} = gen_\mathbb{R}}  %\def\R{I\!\!R \{1+(a_1+b_1 -1)u_2, 1+ (a_2+b_2-1)u_1 \}. \label{permS3a} \end{equation}
For $S_4$, we have
\begin{equation} S_4=S_3 \cup \{(14)\}=S_3 \cup \{ 1+ (a_{21}+ b_{12} - u_{12} - u_{12}^\dagger) \}.         \label{permS3b} \end{equation}
In a different approach, \cite[p.201-222]{SNF}, a special {\it twisted product} was developed, but has not yet been fully explored.

There is an {\it irreducible representation}, called the {\it standard representation}, which is easily obtained from the permutation representation of $S_n$. We illustrate the general method first for $S_5$ in $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$.
The minimal polynomial (\ref{minpolys}) for ${\cal C}_4$ is
\[ min({\cal C}_4)= (x+1)(x-3).  \]
The {\it spectral basis} \cite[p.125]{SNF}, \cite{S2,S0}, for this minimal polynomial is
\[ s_1=\frac{{\cal C}_4-3}{-4}=\frac{{\cal A}_4-4}{-4}, \ \ {\rm and} \ \ s_2=\frac{{\cal C}_4+1}{4}=\frac{{\cal A}_4}{4},   \]
where $s_1$ and $s_2$ are mutually annihilating idempotents with the property that $s_1+s_2=1$.

Using (\ref{specG22}), from the matrices ${\cal A}_4$ and ${\cal C}_4$, we calculate
the corresponding geometric numbers $A_4$ and $C_4$ in $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$, getting
\[ A_4 = 1+a_1+b_1 +(a_2+b_2)\Big( (a_1-b_1)+2 a_1 \wedge b_1 \Big)    \]
\[ = A_2\Big( 1+2(a_2+b_2)\Big)a_1\wedge b_1 =1+ C_4.  \]
where $A_2:= (1+a_1+b_1) \in \mathbb{G}} % \def\G{I\!\!\!G_{1,1}\subset \mathbb{G}} % \def\G{I\!\!\!G_{2,2}$.

More generally, for the minimal polynomials (\ref{minpolys}), the spectral basis
of $C_{2^n}\in \mathbb{G}} % \def\G{I\!\!\!G_{n,n} $ is given by
 \begin{equation} s_1=\frac{C_{2^n}-2^n+1}{-2^n}=\frac{A_{2^n}-2^n}{-2^n}, \ \ {\rm and} \ \ s_2=\frac{C_{2^n}+1}{2^n}=\frac{A_{2^n}}{2^n}.   \label{specCn} \end{equation}
  We have the useful
 recursive relation
 \[  A_{2^{n+1}} = A_{2^n}\Big(1+2^n(a_{n+1}+b_{n+1})(a_1\wedge b_1) \cdots (a_{n}\wedge b_{n})\Big) =1+C_{2^{n+1}} \in \mathbb{G}} % \def\G{I\!\!\!G_{n+1,n+1},\]
 where $A_{2^n} \in \mathbb{G}} % \def\G{I\!\!\!G_{n,n}$ for $n\ge 1$. For $n=1$, $A_1 =1$ and $C_1=0$.

 Using the spectral basis (\ref{specCn}),
 \[  C_{2^n} = (-1) s_1 + (2^n-1)s_2. \]
Since the Casimir geometric number $C_{2^n}$ commutes with the permutation representations of $S_{2^n}$,
it follows that the inner automorphism of the geometric number
\begin{equation}   g_c=s_1(1-u_{1\cdots n}^\dagger) +s_2u_{1\cdots n}^\dagger \label{gcsurgery} \end{equation}
 will {\it diagonalize} $C_n$, meaning that
 \[ [g_c^{-1}C_n g_c ] =\pmatrix{-1 & 0 &\cdots & 0 &0 \cr
 	        0 & -1 &\cdots & 0 &0\cr
 	         &\cdots  & \cdots &  \cr
 	          & \cdots &\cdots &  \cr
 	        0 & 0 &\cdots &- 1 & 0 \cr
 	        0 & 0 &\cdots &  0 & 2^n-1             }.    \]
 If we now apply the inner automorphism of $g_c$ to the permutation representation
 of $S_{2^n}$ in $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$, we get the standard irreducible representation of
 $S_{2^{n}} $ in $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$.

 The explicit calculations for the standard irreducible representation of $S_4$ in $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$ are as follows:
  \[ C_4 = (-1)s_1 + 3 s_2 \]
  for $s_1 = \frac{C_4 -3}{-4}$, and $s_2= \frac{C_4+1}{4}$. Next, we preform surgery,
  removing the last column of $s_1$, and replacing it with the last column of $s_2$, giving
  \[ g_c = s_1(1-u_{12}^\dagger)+ s_2 u_{12}^\dagger ,  \]
which is a matrix whose column vectors are the eigenvectors of $C_4$. Checking,
  \begin{equation} [g_c^{-1} C_4 g_c] = \pmatrix{-1 & 0 & 0 & 0 \cr
  	                               0 & -1 & 0 & 0 \cr
  	                               0 & 0 & -1 & 0 \cr
  	                               0 & 0 & 0 & 3 }, \label{gcsurgery2}  \end{equation}
 showing the $g_c$ diagonalizes $C_4$ as expected.

 Applied to the permutation representations $(12),(13)$ and $(14)$ of $S_4$ in $\mathbb{G}} % \def\G{I\!\!\!G_{2,2}$, given in (\ref{permS3a}) and (\ref{permS3b}), $g_c$ {\it block
 diagonalizes} the matrix generators of these permutations, giving the {\it standard irreducible representation} of $S_4$,
 \[  (12)=\pmatrix{0 & 1 & 0 & 0 \cr
 	1 & 0 & 0 & 0 \cr
 	0 & 0 & 1 & 0 \cr
 	0 & 0 & 0 & 1 } =[1+(a_1+b_1-1)u_2]   ], \]
  \[  (13)=\pmatrix{0 & 0 & 1 & 0 \cr
                          	0 & 1 & 0 & 0 \cr
                        	1 & 0 & 0 & 0 \cr
                          	0 & 0 & 0 & 1 } =[1+(a_2+b_2-1)u_1]\]
 and
 \[  (14)=\pmatrix{-1 & 0 & 0 & 0 \cr
             	    -1 & 1 & 0 & 0 \cr
 	                -1 & 0 & 1 & 0 \cr
 	                0 & 0 & 0 & 1 }=[1-(2+b_1+b_2)u_{12}].     \]
Note, in this representation the permutation matrices of $(12)$ and $(13)$ are unchanged, but the representation of $(14)$ is now a $3\times 3$ sub-block.

From what we have learned, it is easy to write down a standard representation of $S_5$, completing the permutation representations for the $2$-cycles $(12),(13,(14)$,
with
 	          \[   (15) = \pmatrix{-1 & 0 & 0 & 0 \cr
 	          	-1 & 1 & 0 & 0 \cr
 	          	-1 & 0 & 1 & 0 \cr
 	          	-1 & 0 & 0 & 1   }=[1-(2+b_1+b_2+b_{12})u_{12}] .    \]

\section{Group characters}

In studying a particular group algebra, defined by its permutation representation as a subgroup of the symmetric group, it is very useful to find a matrix which commutes
with all the conjugacy classes of the symmetric group, because these elements are one way of finding the respective irreducible representations \cite{newman67}. Here we study the conjugacy classes of the group algebra of $S_4$.

There are five conjugacy classes of $S_4$:
The group identity $1$, the six $2$-cycles, the $8$ 3-cycles, the 6 $4$-cycles, and 3 double $2$-cycles, making up the 24 group elements of $S_4$. The general matrix that commutes with the generating 2-cycles $(12),(13)$, and $(14)$, of $S_4$ will necessarily commute with all of the elements of $S_4$. The matrix
$[g_{ij}]$ that commutes with $S_4$ is
\begin{equation} g_{all}:=\pmatrix{s & t & t& t \cr
	          t & s & t& t \cr
	          t & t & s& t \cr
	          t & t & t& s },     \label{22D3CY} \end{equation}
dependent on the two independent parameters $s,t$. The chacteristic and minimal polynomials of $g_{all}$ are $\Big(\lambda-(s-t)\Big)^3\Big( \lambda-(3t+s)\Big) $ and
\[ min(g_{all})=\Big(\lambda-(s-t)\Big)\Big( \lambda-(3t+s)\Big) , \]
respectively.

 The matrix $g_{alt}$ which commutes with the conjugacy class of the alternating group, or even subgroup of $S_4$ is
 \begin{equation} g_{alt}=\pmatrix{s_1 & t_1 & s_2& t_2 \cr
 	t_1 & s_1 & t_2& s_2 \cr
 	s_2 & t_2 & s_1& t_1 \cr
 	t_2 & s_2 & t_1& s_1 },   \label{4CY} \end{equation}
dependent on the four parameters $s_1, t_1, s_2,t_2$. The minimal polynomial of
$g_{alt}$ is
\[  min(g_{alt})=\Big(\lambda -(t_2-s_2-t_1+s_1)\Big)\Big(\lambda- (-t_2+s_2-t_1+s_1)\Big)\] \[\Big(\lambda-(-t_2-s_2+t_1+s_1)\Big) \Big(\lambda- (t_2+s_2+t_1+s_1)\Big).    \]
 Clearly, the condition (\ref{4CY}) of commuting with all even-cycles is less restrictive  than the condition (\ref{22D3CY}) of commuting with all of $S_4$.

 Representation theory is built up on an {\it inner product} defined on the {\it traces} of matrices which define the {\it characters} of a representation. Given an element
 $g \in \mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$, together with its formally {\it complex matrix} $[g]$,
 \[  tr[g] := 2^n \langle g \rangle_{0+(2n+1)}, \quad {\rm and} \quad
      tr\overline{[g]} := 2^n \langle g^- \rangle_{0+(2n+1)},           \]
 where $i:= e_{1\cdots n}f_{1 \cdots n+1}^\dagger$. A {\it representation} of a group $G$ in
 the geometric algebra $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$ is a mapping
 \[    f: G \to \mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}\]
  of the group $G$ into the geometric algebra $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$ which preserves
  the group operation, and for which $f_e = 1$, where $e \in G$ is the group identity. It follows that $f_{gh} = f_g f_h$ for all $g,h \in G$.
  Let $f$ be a representation of the group $G$ in $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$. The
  {\it group character}
  \[   \psi_f: G \to \mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}  \]
  is defined by $ \psi_f(g):=tr(f_g)$ for each $g\in G$. A comprehensive introduction to representation theory is found in \cite{FH1991}.

 \section{New tools}

 Consider an arbitrary geometric number $g\in \mathbb{G}} % \def\G{I\!\!\!G_{2,2}$, with the matrix
 $[g_{ij}]$ in the basis (\ref{specG22}). We can perform ``surgery" on this matrix. For example,
    \[  [g-g u_{2}^\dagger-u_2^\dagger g] = \pmatrix{g_{11} & g_{12} & 0 & 0 \cr
    	                                            g_{21} & g_{22} &0 & 0 \cr
    	                                            0 & 0 &-g_{33} & 0 \cr
    	                                            0 &0&0&g_{44}}     \]
 and
  \[ [g-g u_{12}^\dagger-u_{12}^\dagger g] = \pmatrix{g_{11} & g_{12} & g_{13} & 0 \cr
   	g_{21} & g_{22} &g_{23} & 0 \cr
   	g_{31} & g_{32} &g_{33} & 0 \cr
   	0 &0&0&-g_{44}}.     \]
      The same such surgery was performed on $s_1$ and $s_2$ in (\ref{gcsurgery}) and
   (\ref{gcsurgery2}) to obtain the respective automorphisms defined by $g_c$.
   We can also directly extract rows or columns of the matrix $[g]$. For example,
   \[ [g b_{1}u_2]= \pmatrix{g_{12} &0&0&0 \cr
   	                      g_{22} &0&0&0 \cr
   	                      g_{32} &0&0&0 \cr
   	                      g_{42}&0&0&0} \quad {\rm and} \quad
   	          [gb_{12}]= \pmatrix{g_{14} &0&0&0 \cr
   	          	g_{24} &0&0&0 \cr
   	          	g_{34} &0&0&0 \cr
   	          	g_{44} &0&0&0}           \]

   Direct representations of matrix algebras by Clifford algebras $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$  of neutral signature restricts the classes of matrices considered to sizes of $2^n\times 2^n$. However, the operation of surgery ameliorates this restriction by allowing us to extract matrices of smaller sizes. Because of their geometric significance, matrices over the complex numbers of size $2\times 2$, the {\it Pauli Matrices}, and of size $2^2\times 2^2$, the {\it Dirac Matrices}, have proven themselves to be of immense importance in physics. It is this extra geometric structure that geometric algebra makes explicit in its many different guises \cite{geoS2017}.

 The {\it standard spectral basis} of the geometric algebras $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$ and $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}:=\mathbb{G}} % \def\G{I\!\!\!G_{n,n}(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I})$ are easily constructed, starting with the primitive idempotent $u_{1\cdots n}:=u_1\cdots u_n$
 where each $u_i = a_i b_i$. The first column is then written down, followed by successive columns defined by the Kronecker products, with the dual blocks in reverse order,
 \[ \mathbb{G}} % \def\G{I\!\!\!G_{n,n}:= B_1 \overrightarrow{\otimes} \cdots \overrightarrow{\otimes} B_n u_{1\cdots n} A_n^T \overleftarrow{\otimes} \cdots \overleftarrow{\otimes} A_1^T. \]
 We have already demonstrated the method for $n=2$ in (\ref{specG22}); we now
 give the standard spectral basis for $\mathbb{G}} % \def\G{I\!\!\!G_{3,3}$.

 The standard spectral basis defining $\mathbb{G}} % \def\G{I\!\!\!G_{3,3}$ is
 \[ \mathbb{G}} % \def\G{I\!\!\!G_{3,3}:= B_1 \overrightarrow{\otimes} B_2 \overrightarrow{\otimes} B_3 u_{123}
  A_3 \overleftarrow{\otimes} A_2 \overleftarrow{\otimes} A_1.    \]
  In expanded form, $\mathbb{G}} % \def\G{I\!\!\!G_{3,3}=$
 \begin{equation} \pmatrix{u_{123} & a_1 u_{23} & a_2 u_{13} & a_{21} u_{3} & a_3 u_{12}
 	            & a_{31} u_2 & a_{32} u_1 & a_{321}  \cr
 	            b_1 u_{23} & u_1^\dagger u_{23} & b_1 a_2 u_{3} & -a_{2}u_1^\dagger u_{3} & b_1 a_3 u_{2}
 	            & -a_{3} u_1^\dagger u_2 & b_1 a_{32} & a_{32}u_1^\dagger \cr
 	          b_{2} u_{13} & b_2 a_1 u_{3} & u_2^\dagger u_{13} & a_{1}u_2^\dagger u_{3} & b_2 a_3 u_{1}^\dagger
 	            & b_2 a_{31} & b_2 a_{32} u_1 & a_{13} u_{2}^\dagger \cr
 	           b_{12} u_{3} & -b_2 u_1^\dagger u_{3} & b_1 u_2^\dagger u_{3} & u_{12}^\dagger u_{3} & a_3 b_{12}
 	            &b_2 a_{3} u_1^\dagger & a_{3}b_1  u_2^\dagger & a_{3}u_{12}^\dagger \cr
 	            b_3 u_{12} & b_3 a_1 u_{2} & b_3 a_2 u_{1} & b_3 a_{21} & u_{12} u_{3}^\dagger
 	            & a_{1} u_2 u_3^\dagger & a_2 u_{1} u_3^\dagger & a_{21} u_3^\dagger  \cr
 	           b_{13} u_{2} & -b_3 u_1^\dagger u_{2} &b_{13} a_2 & b_3 a_{2} u_{1}^\dagger & b_1 u_{2} u_3^\dagger
 	            & u_{13}^\dagger u_2 & b_1 a_{2} u_3^\dagger & -a_2 u_{13}^\dagger \cr
 	           b_{23} u_{1} & b_{23} a_1 & -b_3 u_1 u_{2}^\dagger & -b_3 a_{1} u_{2}^\dagger & b_2 u_1 u_{3}^\dagger
 	            & b_2 a_{1} u_3^\dagger & u_1 u_{23}^\dagger & a_1 u_{23}^\dagger \cr
 	            b_{123} & b_{23} u_{1}^\dagger & -b_{13} u_{2}^\dagger & b_{3} u_{12}^\dagger & b_{12} u_{3}^\dagger
 	            & -b_2 u_{13}^\dagger & b_1 u_{23}^\dagger & u_{123}^\dagger  }   \label{stdspecG33} \end{equation}

  We can now easily construct the geometric permutation representation of
  $S_9$ in $\mathbb{G}} % \def\G{I\!\!\!G_{3,3}$. Several examples of the geometric permutations are
  \[ (12) = 1+ (a_1+b_1-1)u_{23}, \ \ (13)=1+(a_2+b_2-1)u_{13} ,  \]
   \[ (16)= 1+ (a_{31}+b_{13} -u_{13}-u_{13}^\dagger)u_2   \]
   and
 \[ (19)= 1-u_{123} - (1 +b_1)(1+b_2)(1+b_3)u_{123}. \]
  It is interesting to write down the matrix for the 9-cycle $(123456789)$. We have
 \[   (123456789) = \pmatrix{0& 0&0&0&0&0&0&-1 \cr
                            1& 0&0&0&0&0&0&-1 \cr
                            0& 1&0&0&0&0&0&-1 \cr
                            0& 0&1&0&0&0&0&-1 \cr
                            0& 0&0&1&0&0&0&-1 \cr
                            0& 0&0&0&1&0&0&-1 \cr
                            0& 0&0&0&0&1&0&-1 \cr
                            0& 0&0&0&0&0&1&-1 }    \]
      \[=[b_1+b_2a_1+b_3 a_{21} - (a_{321} +a_{32}+a_3 +a_{21}-a_2+a_1+1)u_{123}^\dagger],          \]
  which is a $9^{th}$ root of unity.

  Let us use our new tools to decompose the regular representation of $S_3$ in $\mathbb{G}} % \def\G{I\!\!\!G_{3,3}$ into the sum of its irreducible parts \cite[p.127]{JL2001}. We begin by writing
  \[ X=x_0 + x_1 (18)+x_2 (19) +x_3 (89) + x_4 (189) + x_5 (198), \]
  represented by an $8\times 8$ matrix $[X]$. We have used the generating $2$-cycles $(18)$ and $(19)$ to take advantage of the irreducible standard representation of $S_3$ in $Mat_2(\mathbb{R}}  %\def\R{I\!\!R)$. Letting $x_{0\cdot 5}=x_0 + \cdots + x_5$, the matrix $[X]$ of $X$ is
  \[ \pmatrix{x_0-x_2+x_3-x_5& 0& 0 &0&0&0&0&x_1-x_3-x_4+x_5 \cr
                -x_2-x_5&x_{0\cdot 5}& 0 &0&0&0&0&0 \cr
               -x_2-x_5& 0 & x_{0\cdot 5} &0&0&0&0&0 \cr
                -x_2-x_5& 0&0&x_{0\cdot 5}&0&0&0& 0 \cr
                -x_2-x_5& 0&0&0&x_{0\cdot 5}&0&0& 0 \cr
                -x_2-x_5& 0&0&0&0&x_{0\cdot 5}&0& 0 \cr
                -x_2-x_5& 0&0&0&0&0&x_{0\cdot 5}& 0 \cr
                	x_1-x_2+x_4-x_5& 0&0&0&0&0&0& x_0+x_2-x_3-x_4 }.              \]

   Except for the strange arrangement of $2\times 2$ block, the block diagonal matrix $[X]$ has desired form. The matrix $[X]$ can be fully diagonalized, however this would not give the desired irreducible decomposition of $[X]$. Instead, noting that $x_{0\cdot 5}$ is an eigenvalue of the remaining $2$-block, we block diagonalize $X$ with the eigenvectors of the matrix $[X]_{x_0\to 1, x_1 \to 2, \cdots, x_5 \to 6}$,
    getting the decomposition of $[X]$ into the sum of its irreducible representations
   \[  \pmatrix{x_{0\cdot 5}& 0&0&0&0&0&0&0 \cr
   	 0& x_{0\cdot 5}&0 &0&0&0&0&0 \cr
   	 0& 0 &x_{0\cdot 5}&0&0&0&0&0 \cr
   	0& 0&0&x_{0\cdot 5}&0&0&0& 0 \cr
   	0& 0&0&0&x_{0\cdot 5}&0&0& 0 \cr
   	0& 0&0&0&0&x_{0\cdot 5}&0& 0 \cr
   	0& 0&0&0&0&0&	x_0+x_1-x_3-x_4& x_2-x_3-x_4+x_5 \cr
   	0& 0&0&0&0&0&-x_1+x_2+x_4-x_5& 	x_0-x_1+x_3-x_5 }.          \]
   Since we are representing the group algebra of $S_6$ in $\mathbb{G}} % \def\G{I\!\!\!G_{3,3}$, we can
   reduce the size of our representation by acting on the $6$-dimensional column vector
   \[ x=\pmatrix{0 & 0 & x_0 & x_1 & x_2 & x_3 & x_4 & x_5}^T, \]
    which essentially eliminates the first two rows and columns of the matrix.

    \section*{Appendix: Lower dimensional geometric algebras}

    Recall that the standard basis of the geometric algebras $\mathbb{G}} % \def\G{I\!\!\!G_{n,n}$ and
    $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$ are defined by
    \[ \mathbb{G}} % \def\G{I\!\!\!G_{n,n}:= \mathbb{R}}  %\def\R{I\!\!R(e_1, \cdots , e_n,f_1,\cdots ,f_n ) \widetilde = Mat_{2^n}(\mathbb{R}}  %\def\R{I\!\!R) \] and
     \[ \mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}:= \mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}(e_1, \cdots , e_n,f_1,\cdots ,f_n ) \widetilde = Mat_{2^n} (\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}),   \]
    where $e_i := a_i+b_i$ and $f_i := a_i-b_i$ for $i=1, \cdots, n$.
    The geometric algebra $\mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}$ can be obtained as a real geometric algebra,
    \begin{equation} \mathbb{G}} % \def\G{I\!\!\!G_{n,n+1}:= \mathbb{R}}  %\def\R{I\!\!R(e_1, \cdots, e_n,f_1, \cdots ,f_n,f_{n+1}),  \label{geospq} \end{equation}
    where $f_{n+1}:= e_{1\cdots n}f_{1\cdots n}^\dagger i $.

 If $p+q \le 2n$, the $2^{2n}$ dimensional geometric algebras $\mathbb{G}} % \def\G{I\!\!\!G_{p,q}$ can be considered as the real geometric algebras,
  \[  \mathbb{G}} % \def\G{I\!\!\!G_{p,q} := \mathbb{R}}  %\def\R{I\!\!R(e_1, \cdots , e_p, f_1, \cdots, f_q),     \]
   by noting that $(ie_i)^2=-1$ and $(i f_i)^2=1$. If $p>n$ or $q>n$, we simply change the required number of basis vectors $f_i$ or $e_i$, to $e_i$'s or $f_i$'s, respectively, by multiplying the $f_i$ or $e_i$ by $i$.
     Additional $2^{2n+1}$-dimensional real geometric algebras can be obtained from
     (\ref{geospq}), by replacing an {\it even} number of the generating vectors by $i$ times that generator. For example,
     \[ \mathbb{G}} % \def\G{I\!\!\!G_{n-2,n+3}:= \mathbb{R}}  %\def\R{I\!\!R(e_1, \cdots, e_{n-2}, i e_1,i e_2,f_1, \cdots, f_{n+1} )
     \widetilde = Mat_\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}(2^n)  .    \]

     Below is a list of lower dimensional $2^{2n+1}$ geometric algebras that can be represented in terms of the matrix algebras
     $Mat_2(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I})$, $Mat_{2^2}(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I})$, and $Mat_{2^3}(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I})$,  or vice-versa.

     \bigskip

    \centerline{$\bf  \mathbb{G}} % \def\G{I\!\!\!G_{1,2}:= \mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}(e_1,f_1) \widetilde =Mat_2(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}) $}

    \[ \mathbb{G}} % \def\G{I\!\!\!G_{1,2} = \mathbb{R}}  %\def\R{I\!\!R(e_1,f_1,f_2), \quad \mathbb{G}} % \def\G{I\!\!\!G_{3,0}= \mathbb{R}}  %\def\R{I\!\!R(e_1,i f_1, i f_2) ,  \]

       \centerline{$\bf  \mathbb{G}} % \def\G{I\!\!\!G_{2,3}:= \mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}(e_1,e_2,f_1,f_2) \widetilde =Mat_4(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}) $}

       \[ \mathbb{G}} % \def\G{I\!\!\!G_{2,3} = \mathbb{R}}  %\def\R{I\!\!R(e_1,e_2,f_1,f_2,f_3), \quad \mathbb{G}} % \def\G{I\!\!\!G_{4,1}= \mathbb{R}}  %\def\R{I\!\!R(e_1,e_2,if_1, if_2,f_3) ,  \]
       \[ \mathbb{G}} % \def\G{I\!\!\!G_{0,5} = \mathbb{R}}  %\def\R{I\!\!R(ie_1,ie_2,f_1,f_2,f_3) .  \]

        \centerline{$\bf  \mathbb{G}} % \def\G{I\!\!\!G_{3,4}:= \mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}(e_1,e_2,e_3,f_1,f_2,f_3,f_4) \widetilde =Mat_8(\mathbb{C}} %\def\C{I\!\!\!C} %\def\C{{C\kern-.647em I}) $}

        \[ \mathbb{G}} % \def\G{I\!\!\!G_{3,4} = \mathbb{R}}  %\def\R{I\!\!R(e_1,e_2,e_3,f_1,f_2,f_3,f_4), \quad \mathbb{G}} % \def\G{I\!\!\!G_{5,2}= \mathbb{R}}  %\def\R{I\!\!R(e_1,e_2,e_3,if_1, if_2,f_3,f_4) ,  \]
        \[ \mathbb{G}} % \def\G{I\!\!\!G_{7,0} = \mathbb{R}}  %\def\R{I\!\!R(e_1,e_2,e_3,i f_1,i f_2,i f_3,i f_4), \quad  \mathbb{G}} % \def\G{I\!\!\!G_{1,6} = \mathbb{R}}  %\def\R{I\!\!R(e_1,i e_2,i e_3, f_1,f_2,f_3,f_4) .   \]
      A complete classification of geometric algebras is given in the
      paper, {\it Geometrization of the Real Number System} \cite{geoS2017}.

redpajama_set_name: RedPajamaArXiv