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Talk:List of writing systems

Classification of writing systems
Maybe someone could explain the characteristics of the different writing systems (what do Logographic, Syllabaries etc. actually stand for). Kokiri 13:36, 12 Mar 2004 (UTC)

 * The first sentence of the article says See writing system for a description of the different kinds of writing systems. -- pne 10:07, 17 May 2004 (UTC)

Canadian syllabics
Are Canadian syllabics (e.g. Cree, Ojibwe, Inuktitut) really syllabaries? As far as I know, syllables beginning with the same consonant use the same shape and rotate and/or reflect it and/or add dots to signify the vowel sound, which means that they should probably be classified as abugidas. -- pne 09:59, 24 May 2004 (UTC)

 * I know you believe that, Philip, but I think it is contentious. The rotation is not applied regularly between all the series, and then there is the series of 'finals' which in fact are sometimes used as initials (in consonant clusters). The script is called a syllabary, and has always been. I think "reassigning" it to abugida is just overly clever.

 * No, it's just being accurate. Canadian syllabics is a "true" abugida in the sense that Ethiopic is, where the consonants are significantly and sometimes irregularly modified by their vowels. As for the finals and clusters, those are alphabetic, and clearly segmental, so it's even further from a syllabary.

 * As for the terminology, in the 19th century, the Arabs, Hebrews, Hindus, and Siamese all wrote in "syllabics" (everything we'd call an abugida or abjad today). Part of the reason was that, as everyone knew at the time, the Greeks made all the great advances of civilization, including the alphabet, and the less civilized Asiatics (like the Canadian Indians) made due with inferior "syllabic" writing. The term became entrenched with the Canadian Aboriginal script, almost as if it were in a kind of cultural backwater for the last half century. But "syllabics" (as a noun) is not equivalent to "a syllabary" in the modern use of the term. It only means that not every consonant and vowel is written out in a linear fashion. kwami 11:49, 2005 July 14 (UTC)

Hangul - featural?

 * I don't think that the "featural" category belongs for Hangul either.Evertype 17:56, 2004 May 25 (UTC)

 * Hangul is featural in its conception, but alphabetic in how it's learned. However, Ethiopic is an abugida in its conception, but a syllabary in how it's learned. I think it's worth drawing attention to Hangul as something special. Perhaps "Featural alphabet"? kwami 11:49, 2005 July 14 (UTC)

Mayan - logographic vs syllabic
Someone had moved Mayan from logographic to syllabic. I put it back. It's clearly logographic. All logographic systems are actually "logosyllabic": Sumerian, Chinese, Mayan, they all used glyphs for their sound values, and the most common of the glyphs were syllabic (or at least used as monosyllables). 90% of Chinese characters are "logosyllabic" (that is, composed of a determiner combined with syllabic phonetic element). Egyptian was somewhat different, due to the basic abjad-like nature of the script; it's perhaps best described as logo-consonantal. That should be explained in the articles on logographies, but Mayan was not exceptional, except of course in as far as all these scripts are ideosyncratic. Mayan feels a lot like Japanese, or Egyptian, in its application. kwami 12:09, 2005 July 14 (UTC)

 * Kwami, 'twas I who had moved Mayan from logographic to syllabary; in the absence of a mid-way logosyllabic designation, I felt that its syllabic features were important enough to be featured. Also, the very concept that Mayan was primarily logographic or even ideographic in nature actually held back the script's decipherment; it was only when Knorosov and others demonstrated the essential syllabic componentry that major breakthroughs were made. Many Mayanist sources habitually (tho' casually, perhaps) refer to it as a Maya syllabary, although one could argue that they are merely confining themselves to the subset of syllabic glyphs. Nonetheless the number (but not necessarily the frequency) of logographic glyphs does outweigh that of syllabic ones, and I agree with you that the logogram article should be expanded to emphasise more proper designation as logosyllabic. I have now expanded on the logographic definition in this listing here to hopefully clarify these points. --cjllw |  TALK  08:52, 2005 July 19 (UTC)

Old Persian cuneiform
One of the scripts that is arguable is Old Persian cuneiform. It was syllabic in its conception, and you'd get that impression from seeing a chart of it, but it's alphabetic in its application. Since all vowels are written explicitly, I've entered it as an alphabet. kwami 12:09, 2005 July 14 (UTC)

Braille, Kanji, Chinese
"Braille" is not a single script. Japanese Braille represents kana, and so is syllabic. Korean Braille, by the way, is not a featural script, though it does distinguish initial from final consonants. kwami 12:09, 2005 July 14 (UTC)

 * Kanji is not logosyllabic. Many kanji have multisyllabic readings (especially kun-yomi, although there are two-syllable on-yomi like "shutsu"). And how logographic are hieroglyphics? I was under the impression that most symbols were strictly sound-based, with only some logographic glyphs.   — Gwalla | Talk 01:12, 16 July 2005 (UTC)

 * There are historical traces of logosyllabicity in Japanese kanji, though they are neither completely logographic (since they do not have a one-to-one correspondance with either a word or a morpheme), nor are they ideographic. So they break the mold in both the logo and the syllabic arenas. I don't think we have a good term for what they are, but "logographic" in a broad usage of the word is hopefully not too misleading.

 * However, Chinese characters as conceived for Chinese are logosyllabic: the vast majority of characters consist of both a radical (equivalent to hieroglyphic determiners) and a phonetic element, and the phonetic element is syllabic. When phonetic characters were created for a disyllabic word, two phonetics (A+B) were chosen, and the radical (x) was applied to both (x[A+B] => xA+xB), so that neither (xA, xB) had any meaning on its own.

Egyptian hieroglyphs
The vast majority of hieroglyphs are logographic. There are thousands of glyphs (5000 by some counts, some limited to a single temple), but only a few hundred phonetic elements - and many of these were logograms used for their phonetic values, rather than dedicated phonetic glyphs. Of course, you'll see the same phonograms used over and over, but that's because there were relatively few to choose from. Egyptian was written with three main elements: logograms, ideograms ("determiners"), and phonograms. Of the three, the logogram was the most important, and the others optional to varying degrees. Egyptian was not written phonetically except for loan words and foreign names, which had no logograms (rather like Chinese today, or katakana). kwami 02:00, 2005 July 16 (UTC)
 * Um, no. No and no. Determinatives are not logographs. -- Evertype·✆ 22:51, 24 September 2006 (UTC)

Scripts used in Georgia
So in Georgia we use "Georgian and Cyrillic" scripts, and in Armenia they use "Armenian and Latin"

That is a bullshit. Thats why people dont trust wikipedia. Keep up like this. --GeoRugby (talk) 09:30, 5 February 2017 (UTC)

Redundancies, Inconsistencies - how to resolve?
The Abugida list here, that one in Abugida, and Category:Abugida writing systems will tend to be different. Is there a Bot taking care of such issues? Should the category system hold the only list? I tend to prefer the category solution. Pjacobi 21:17, 10 Jul 2004 (UTC)

It is also a bit odd to have "Meroitic (an abugida)" with the undeciphered and "Old Italic alphabet" with linear nonfeatural alphabets, Etruscan being as unknown to us as is Meroitic. Wikipeditor 10:58, 24 October 2005 (UTC)
 * We know Meroitic is an abugida. We just don't know Meroitic very well. We know the Etruscan alphabet perfectly well too. But we don't know the Etruscan language. -- Evertype·✆ 00:43, 25 September 2006 (UTC)

Possible merge
I have proposed at List of alphabets that that page be merged into this one. Please discuss here. - dcljr (talk) 01:12, 18 July 2005 (UTC) [changed over there to here on Oct 25th -- sorry, didn't realize the mergeto template directs discussion to the target page]

I don't like the idea of a merge because that would be a VERY large article. Cameron Nedland

Discussion copied from Talk:List of alphabets:

Given the nuances of what constitutes a "true" alphabet (versus other things like syllabaries), I propose that this list be merged and redirected to the more general List of writing systems. - dcljr (talk) 01:09, 18 July 2005 (UTC)

 * I'm inclined to agree that there would be benefits in merging this listing into list of writing systems. The former is a subset of the latter, and having the same list maintained in two separate places will lead to inconsistency of treatment and synchronisation. Personally I also find it more useful to have all the differing scripts documented on the same page, tho' as the list becomes more extensive this could lead to its own difficulties.
 * The alternative would be to replace the current alphabetic writing systems on the overall list with just a link to this list of alphabets, and thus avoid duplication and inconsistency. However, since in certain cases classification into a single type may be problematic, removing these to another page would mean that one could not easily check on whether an entry had already been recorded, or mis-classified. However, I'd be open to other suggestions. --cjllw | TALK  04:57, 2005 July 19 (UTC)
 * I thought that such a merge would be a good thing just for the same reason. So, I support.--Imz 17:23, 24 October 2005 (UTC)

well,I didn't agree such proposal,because some writing systems like Chinese do not fall into the catogery of alphabets.Ksyrie 23:17, 17 August 2006 (UTC)

 * I have a mixed opinion. "Writing system" is a broader concept that "alphabet". On the other hand, if the "list of alphabets" were trimmed — like getting rid of all the "English alphabet", "French alphabet", "Ruritanian alphabet" nonsense, and just listing the "Latin alphabet" —, then a merger might be feasible. FilipeS 15:56, 23 November 2006 (UTC)

 * Almost all items on List of alphabets were already categorized (correctly) on List of writing systems, so I took the "other" writing systems and merged them in. The merger is now complete; cleanup is requested. theProject 17:53, 11 December 2006 (UTC)

Descriptions
I've included brief descriptions in each section because most people won't know what all these things are and it's ridiculous to make the reader check 5 other pages just to understand this one. - dcljr (talk) 01:38, 18 July 2005 (UTC)

Thaana
I think we might want to discuss Thaana. It gets to the heart of what the difference is supposed to be between an alphabet and an abugida: like devanagari, it has letters for consonants, and diacritics for vowels. However, a letter does not presume the vowel a: All vowels are marked explicitly. However, the absence of a vowel is also marked explicitly, just as in Devanagari. So we might be able to argue that it is an abugida or an alphabet with equal justification. As far as whether a letter in of itself represents a consonant or a syllable, the question would seem unanswerable, because isolated letters do not occur. Both possibilities are explicitly indicated by a diacritic.

So, is is an abugida, or an alphabet? kwami 21:37, 2005 July 18 (UTC)

all alphabets?
It looks almost all the writing systems on this page link to pages titled "something alphabet". Most of the writing systems aren't alphabets! I think they should all be renamed to "something (script)" or "something (writing system)" or something like that. --Yodakii 15:59, 28 September 2005 (UTC)

 * The word 'alphabet' is used for two rather different things: 1. a segmental script, which would include every major writing system on Earth today except Chinese and Japanese, and 2. a segmental script that indicates consonants and vowels equally, which is much more restricted. (Tone, of course, is not considered.)

 * The problem is that the word 'alphabet' has a rather racially loaded history: "Everyone" knows the alphabet is one of humanity's greatest inventions; the Greeks invented the alphabet; ergo, Europeans/the inheritors of Greek civilization are mentally superior/more civilized than the rest of the world. Other segmental scripts were called 'syllabics' (sometimes 'alphasyllabics' or 'defective scripts'). Part of this attitude was specifically anti-Semitic, but it was also used to justify European colonialism in Asia, as silly as that sounds today. As a reaction to this, people started insisting that the word 'alphabet' (which after all was the name for the Semitic abjad) be used for abjads and abugidas as well, and therefore it is extremely common to speak of the 'Arabic alphabet', 'Hindi alphabet', etc. The use in English of the words abjad and abugida for specific types of script, and the simultaneous restriction of alphabet back to Greek-type scripts, has only been around for about ten years now. While this is useful for a linguistic analysis of scripts, it hasn't filtered down into non-academic usage.

 * I think it might be appropriate to use the word 'script' instead of 'alphabet'. However, 'alphabet' is the more common usage, and I imagine that a lot of people might insist on keeping it. kwami 19:04, 28 September 2005 (UTC)

In school, I was taught the Phoenicians had invented the alphabet. That's what the Greeks believed, too. Now, I come to Wikipedia, and see that the "experts" call the Phoenician alphabet an abugida. Perhaps they're the racists. FilipeS 15:57, 23 November 2006 (UTC)

The "experts" are right, it is widely accepted in the linguistic world that systems with separate symbols for vowels and consonants, which neither were diacritics, are called alphabets. Abjad are purely consonants with optional vowel marking and abugidas have vowels marked as diacritics. Experts are not racist, they are right in stating what is technically what, and you are ignorant and misinformed in calling them racist Mr anonymous username (talk) 18:52, 10 February 2021 (UTC)

The list for Cyrillic
I've made the list for Cyrillic longer and more descriptive. Would you mind removing the list of individual Slavic languages to make it shorter? Suggested change:
 * a number of Slavic languages (Belarusian, Bulgarian, Macedonian, Russian, Serbian, Ukrainian language, etc.) and...

to
 * a number of Slavic languages and...

--Imz 17:16, 24 October 2005 (UTC)

I like it how it is because it allows more movement between articles quite easily.

map with colours named

If one would like the colour nmaes included in the legend, feel free to replace by the version below. I personally don't thnik they are necessary. --Donar Reiskoffer 11:31, 4 July 2006 (UTC)
 * By the way, why Korean is called syllabary? It is clearly alphabet.--Dojarca (talk) 18:31, 18 January 2008 (UTC)

It's clearer when expanded, but you're right, the small version is wrong. kwami (talk) 00:50, 19 January 2008 (UTC)

Abjad
Hebrew alphabet called aleph-bet an Hebrew, why it is designated with the Arabic word "abjad" in the image?--Dojarca (talk) 15:42, 18 January 2008 (UTC)

 * Because the Arabic term is used in English. Hebrew is functionally equivalent to Arabic. kwami (talk) 18:06, 18 January 2008 (UTC)

Inherent vowel but full vowel letters?
''Generally, however, if a single letter is understood to have an inherent unwritten vowel, and only vowels other than this are written, then the system is classified as an abugida regardless of whether the vowels look like diacritics or full letters. ''

Are there any actual writing systems with a nonzero vowel graphically represented by zero sign and/or zero vowel graphically represented by nonzero sign, but where the vowels with nonzero graphic representation are all full letters, i.e. are signs that appear in linear sequence the same way as the consonants and mixed with the consonants?

--JWB (talk) 10:01, 13 June 2008 (UTC)

 * Old Persian Cuneiform. Phagspa. Pahawh Hmong (though that's a little odd, in that CV is written VC (it's actually onset & rime, not C & V). Zhuyin (with the "buzzing" vowels in zi, si, ri, etc). That's all I can think of off the top of my head.

 * I think that statement is probably false. I hope I didn't write it! (it feels suspiciously familiar...) Well, it's true for phagspa, while OPC, zhuyin, and and pahawh are semi-syllabic and thus don't qualify as abugidas because they're not fully segmental. kwami (talk) 11:03, 13 June 2008 (UTC)

 * Actually, phagspa is questionable. True, the vowels are all written in-line and in temporal order. But they still have distinct initial forms, suggesting they aren't quite full letters after a consonant. (The are also joined to the consonant, but so all all segments in a syllable.) kwami (talk) 11:18, 13 June 2008 (UTC)

Is there really any functional differences?
I'm asking is there any functional difference between north and south Indic except the fact that North Indic is squarer and Devanagari written with a line strike through. | ខែមារាតសប្បាយ | Talk 21:35, 8 September 2008 (UTC)

South Korea still uses Hanja. The map is not showing that in South Korea, many publications still use hanja(Chinese characters) mixed in with hangul. Only North Korea uses just hangul,they got rid of the hanja. The map needs to get changed. —Preceding unsigned comment added by 69.140.72.224 (talk) 04:13, 11 January 2009 (UTC)

List of scripts by number of users
I don't think this article exists, but if anyone has the sources to get it started, I'd be happy to work on it. 74.78.116.93 (talk) 02:06, 17 January 2009 (UTC)

Greek and Italic alphabets
Really italic alphabets were a lot, since before Roman conquest in Italy lived many peoples: Ligurians, Rhetians, Venetians, Celtic, Etruscans, Picenians, Umbrians, Samnites, Oscans, Faliscans, Daunians, Japigians, Messapians, Brutii, Sicanians, Siculi and so on, and every people had its own alphabet. They similar, but not identical. The same way each Greek city had its own alphabet. The standardized Greek alphabet is the Athenian alphabet adopted in Hellenistic age. And Athenian alphabet was borrowed by Miletus, increased with the two "spirits" from Taranto. Lele giannoni (talk) 20:44, 15 June 2011 (UTC)

Further Reading Section needed
A section of Further (notable) books on the subject is required to assist users. --Inayity (talk) 10:37, 19 September 2012 (UTC)

Table of rare writing systems
In the table http://en.wikipedia.org/wiki/List_of_writing_systems#Less_than_20_million_users there appears to be a copy-paste-error under Khmer. Also, Tibetan could be added. In both cases, I do not know enough to make correct entries, so I am posting it here. — Preceding unsigned comment added by 109.40.58.207 (talk) 12:21, 8 April 2013 (UTC)

WritingSystemsoftheWorld4.png updated to v6
I noticed that has been updated/corrected to. So I was going to change that, until I noticed that v4 has 11 different versions (language etc.) of it (scroll down to "Other versions" at previous link). The information that those are available is likely to disappear, so I have left the current (but incorrect) v4 in place. If you have any good ideas, then proceed, but in any case, there is some "documentation" in this post. -- Katana (talk) 18:32, 6 November 2013 (UTC)

Cyrillic and Georgia
Georgia does not have a Cyrillic-based language as an official. That is true, but so do the almost unrecognized breakaway republics, which are in fact independent (South Ossetia and Abkhazia). Also, note the color Moldova uses: Moldova's only official language is Romanian, but breakaway unrecognized, but independent and existent Transnistria has Cyrillic languages.

It would probably be fair to have two pictures: one on official languages and one on really used (Serbian is officially Cyrillic, but in fact also uses Latin). But for now, why is then Kosovo shown independent? It is not a UN member either and a UN member claims that territory.--R8R (talk) 10:10, 5 August 2014 (UTC)

Examples
Currently the names of the scripts are only given in their native script in the List of scripts by adoption section. It would be informative to give the same information for other scripts. It wouldn't look very good to do this for only a few examples. We could do it comprehensively throughout the article, I suppose. I'm not sure what the best approach is. Does anyone have any thoughts as to where the best place for this to be might be? Waitak (talk) 14:40, 4 September 2014 (UTC)

Arabic script usage by population
this article lists that Arabic script is used by 380 million but it's wrong. Arabic script is used solely in Arab countries, Iran, Pakistan, Afghanistan. Some countries have also languages which uses Arabic script like India, China and Malaysia. If we count only those countries population where Arabic script is solely used then it will exceed this wrong 380 million. remember all lancuages in these countries written in Arabic script Arab countries (356 Million)+Pakistan(196 Million)+Iran(78 Million)+Afghanistan(31 Million) it becomes 661 Million. and so far we are not including those languages in India, China and Malaysia written in Arabic script. Please correct that list. — Preceding unsigned comment added by 107.167.103.128 (talk) 19:36, 27 March 2015 (UTC)

Odia script (name) to be added on a map
Please look at the second map of this article — that's the map "Writing systems of the world today." There is an example of text written in Odia script (old name Oriya) here, but the English name for this writing system is missing. It would be useful to add the inscription "Odia script" on the map.

Inuktitut > Canadian Aboriginal Syllabics
The table at List of writing systems lists "Inuktitut"; this is sort of like listing Cyrillic but calling it "Russian" and ignoring the fact that it is used for other languages, too. The writing system in question was originally developed for Ojibwe, though it is rarely used for that now, but it remains in vigorous use in Cree (and is used in portions of the Cree Wikipedia), particularly Plains Cree, and this is not Inuktitut, nor is it entirely "north of the tree line". It has also been used for other Algonquian and Athabascan languages of Canada, though whether any of them currently use it I'm not sure. --Haruo (talk) 16:02, 20 November 2015 (UTC)

 * This correction should also add to the population figure, though not enough to affect the order of the list. --Haruo (talk) 16:04, 20 November 2015 (UTC)

 * Corrected, though it is missing the Anishinaabemowin name for syllabics. Also, the names used for the script are simplified: I am certain various Dene languages call them differently as do the various Cree Nations. I also made sure to note the territories/countries where syllabics are used within the Canadian State. Danachos (talk) 19:30, 26 April 2020 (UTC)

Yiddish script is NOT an abjad
As far as I learned, plain Yiddish script is NOT an abjad, but a true alphabet: all vowel sounds are represented by full letters and occational diacritics may not be omitted. This said, it certainly has numerous Hebrew loanwords, which usually (but not always) adjust to Hebrew orthography. Since I am not a speaker of Yiddish, I dare not change the article page, but it should probably be revised. --Heinrich Puschmann (talk) 13:44, 7 September 2016 (UTC)

 * Well yes, but you should be very careful with your terminology. I usually follow Unicode practice: The Yiddish writing system uses the Hebrew script as an alphabet, except in the case of Hebrew (and to a lesser degree Aramaic) loanwords which retain their original orthography of the Holy Scriptures where the Hebrew script is used as an abjad. Love —LiliCharlie (talk) 08:03, 9 September 2016 (UTC)

Map needs to be updated
I realized the old map had light green for Arabic/Latin and Medium green for Hebrew. However, the new map lacks the light green colour, so a small corner of southern Sudan and Brunei (Arabic and Jawi/Malay, respectively) is incorrectly listed as "Hebrew". Can someone correct this? 23.233.82.54 (talk) 01:20, 23 December 2016 (UTC)

 * Indeed, so too does the Canadian part. It shows only Nunavut, where syllabics are employed in Denendeh as well as Nunavik and much of northern Quebec, much of Ontario (definitely Nishnawbe Aski) and most of the Prairies. Even if it only political designations, Northwest Territories should at minimum also be included. Danachos (talk) 19:40, 26 April 2020 (UTC)

Pinyin users
Hey, I was just looking at this page and I realized to myself that Pinyin users should probably be included under Latin script users. I'm a newbie on wikipedia, so I don't really know what I'm doing, but I'm pretty sure something like the edits I made needs to be done. Everyone I meet that uses Chinese uses the Latin alpahbet to type out the sounds of Chinese. There is only a very small minority of people that have the ability to avoid the Latin alphabet, and they are experts that could use pinyin if they wanted to. Geographyinitiative (talk) 11:41, 14 March 2018 (UTC)

 * On a related note, the second map lists "Zhuyin" (Zhuyin Fuhao, the Mandarin Phonetic Alphabet or "bo po mo fo") as the writing system associated with Taiwan. This is misleading. Taiwanese normally write with traditional Chinese characters (as opposed to the simplified ones used in the PRC). Zhuyun Fuhao is studied by Taiwan schoolchildren learning to read, much as Hanyu Pinyin is in the PRC. Both groups gradually transition to characters (which are harder). In the PRC, it may be that some people are literate only in Hanyu Pinyin (which was in fact originally intended to replace characters, a plan which Mao deemed unfeasible). Outside of young schoolchildren, I don't think there could be anybody in Taiwan who only reads Zhuyin Fuhao. At any rate, we need to decide whether auxiliary scripts ought to be represented in the maps. (Cf. Mongolia, which has two official scripts.) --Dawud — Preceding unsigned comment added by 1.164.172.214 (talk) 00:53, 31 May 2019 (UTC)

Undeciphered systems that may be writing
The heading and intro of this section has been changed. Has anyone bothered to check afterwards whether it was still appropriate for the languages listed there? I'm asking because Khitan small script definitively is a script, it has even been partially deciphered, and no longer belongs in this section, but I don't know enough about the others to tell. — Preceding unsigned comment added by 77.61.180.106 (talk) 12:53, 10 December 2020 (UTC)

Preposterous number of Latin script users
False and preposterous number of Latin users - 4900+ millions, was edited by me to more accurate (but still exaggerated) number of 3900+ millions. user:95.180.78.189 10:44, 21 January 2021 (UTC)
 * I agree with IP editor and have edited it further to say simply "unknown". The numbers in the table are all uncited and the notes clearly indicate that it was constructed by making an inference from population and literacy estimates, contrary to WP:OR and WP:SYNTH. Is there a convincing reason why that whole column should not be deleted? --John Maynard Friedman (talk) 11:16, 21 January 2021 (UTC)
 * I don't see one. Love —LiliCharlie (talk) 12:48, 21 January 2021 (UTC)
 * I don't know if the "number of 3900+ millions" is "still exaggerated" as half+ of the world population speak more than one language, and many master more than one script. Even those who do not speak a "Latin-script language" may have been taught the basics of one and/or use Latin letters every day for maths, email addresses and lots of other stuff in our modern globalised world. — We shouldn't give estimates unless we have a very reliable source that is based on a clear definition of "script user", and though it is beyond doubt that Latin is currently the most-used script world-wide we don't need a table to say so. Love —LiliCharlie (talk) 12:48, 21 January 2021 (UTC)
 * While I'm not against original research, even if that is not by the rules of Wikipedia, but only if it is honestly and thoroughly documented. In this table at least two different criteria are employed. One for Latin, the other for most of the other scripts. If for other scripts we take into the account only native users of the script (first language, or other official language of a country), then we have to use the same criterium for Latin, not this ambiguous "half+ of the world population". I would be more than willing to do the count if some reasonable and consistent criteria could be established and agreed upon.MačakPodŠlemom (talk) 21:52, 21 January 2021 (UTC) 12:48, 21 January 2021 (UTC)

 * We are an encyclopaedia that summarises reliable published sources. If you are "not against original research" you can visit a zillion blogs out there, but this is not the place for publishing what someone thinks are research results worth to share.
 * This is an article dealing with scripts, not one that "take[s] into the account only native users of the script (first language, or other official language of a country)", and presents that as the ultimate given fact that characterises some script, especially if we dont't have a source that uses the same well-defined criteria for all scripts we deal with. You are free to write a List of writing systems by first and official language, though. If correctly categorised, it'll be on my watchlist, too. Love —LiliCharlie (talk) 22:40, 21 January 2021 (UTC)
 * I know what's Wikipedia, and it's basic rules, that's why I say I'm not against original research EVEN if it is against the rule, if it is well documented. I'm new here as an editor, but I've been using Wikipedia almost 2 decades, and I saw a plethora of worthless and unreliable published sources used as a reference.
 * I know what about is this article, I'm talking only about poorly made table (without references) with number of users. It is an original research, and very poor one. One criterium for Latin script, other criteria for other scripts. That won't do. We can agree on one criterium and follow it through the table. It can be, but not necessarily, the number of native users I defined above. It can't be the number of "supposed" everyday casual users (" every day for maths, email addresses and lots of other stuff"), because it is hard/impossible to establish with any certainty. It can be some other measurable criterium.MačakPodŠlemom (talk) 06:51, 22 January 2021 (UTC)

Adlam Alphabet 𞤢𞤤𞤳𞤵𞤤𞤫 𞤀𞤣𞤤𞤢𞤥
I haven't seen Adlam on your map of world writing systems. Could you please edit and add it up? 102.143.9.164 (talk) 16:12, 17 August 2023 (UTC)

Perhaps we reformat this?
Okay so, I find this article super confusing and awkward to navigate. I might not be as experienced as others here, but I think I would be better to reformat this list into something more easily readable. Ashertheaxolotl (talk) 19:09, 14 September 2023 (UTC)

title: Talk:List of writing systems


text:
Aichach

Aichach (Central Bavarian: Oacha) is a town in Germany, located in the Bundesland of Bavaria and situated just northeast of Augsburg. It is the capital of the district of Aichach-Friedberg. The municipality of Aichach counts some 20,000 inhabitants. It is not far from the motorway that connects Munich and Stuttgart, the A8. The local river is called Paar.

History
Aichach's history dates back nearly 1000 years.

A prison for women was established in Aichach in 1909. In 1967, Ilse Koch, also known as The Witch of Buchenwald, committed suicide here.

Parish parts
There are 40 parts of the municipality:

 * Aich (chapel)
 * Aichach (main town)
 * Aichach (railway station)
 * Algertshausen (Kirchdorf)
 * Andersbach (wasteland)
 * Blumenthal (hamlet)
 * Ecknach (Pfarrdorf)
 * Edenried (Kirchdorf)
 * Eitershofen (wasteland)
 * Frogham (Wasteland)
 * Gallenbach (parish village)
 * Gansbach (wasteland)
 * Griesbeckerzell (parish village)
 * Hiesling (village)
 * courtyard garden (village)
 * Ippertshausen (hamlet)
 * Blades (parish village)
 * Knottenried (wasteland)
 * Matzenberg (wasteland)
 * Neuhausen (hamlet)
 * Neulhof (upper) (hamlet)
 * Neulhof (lower) (wasteland)
 * Neumuehle (hamlet)
 * Nisselsbach (wasteland)
 * Oberbernbach (parish village)
 * Obermauerbach (parish village)
 * Oberschneitbach (Kirchdorf)
 * Oberwittelsbach (parish village)
 * Röckerszell (wasteland)
 * Sulzbach (parish village)
 * Potting Mill (Wasteland)
 * Untergriesbach (village)
 * Untermauerbach (Kirchdorf)
 * Untermühle (Wasteland)
 * Unterschneitbach (Kirchdorf)
 * Unterwittelsbach (village)
 * Walchshofen
 * Wilpersberg (wasteland)
 * Windten (Wasteland)
 * Wöresbach (wasteland)

Mayor
Since 1996: Klaus Habermann (born 1953) (SPD)

Twin towns – sister cities
Aichach is twinned with:
 * 🇦🇹 Brixlegg, Austria
 * 🇭🇺 Gödöllő, Hungary
 * 🇩🇪 Schifferstadt, Germany

Notable people

 * Erhard Bühler (born 1956), major general
 * Christoph Burkhard (born 1984), footballer
 * Johannes Engel (1453–1512), doctor, astronomer and astrologer
 * Matthias Greitter (1495–1550), cantor and composer
 * Chrislo Haas (1956–2004), musician
 * Vincenz Müller (1894–1961), officer of Reichswehr, Wehrmacht and NVA

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Aichach


text:
Talk:Levinson recursion

--Tbackstr 13:39, 12 Mar 2004 (UTC) I copied a description of the algorithm directly from my own PhD-thesis, i.e., with permission from the author.

This page has been plagiarized by NetHelper
Nethelper.com.au appears to have an exact copy of this page: http://www.nethelper.com.au/article/Levinson_recursion

Since the math formatting isn't done correctly, and the format of the page is in a wiki-style, and since Nethelper.com.au appears to be largely or perhaps entirely copies of wiki pages without any visible citations, I'd say it seems pretty clear that Nethelper just trawls Wikipedia (and possibly other sites) for content. Their "about us" page (http://www.nethelper.com.au/aboutus.php) is also fairly suspicious. — Preceding unsigned comment added by 199.3.246.231 (talk) 21:18, 4 February 2014 (UTC)
 * That's ok! That's the point of Wikipedia. Anyone should be able to grab stuff from here and do what they want with it. Hamsterlopithecus (talk) 05:12, 18 January 2016 (UTC)

External links modified
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title: Talk:Levinson recursion


text:
River Itchen, Hampshire

The River Itchen in Hampshire, England, rises to the south of New Alresford and flows 26 mi to meet Southampton Water below the Itchen Bridge. The Itchen Navigation was constructed in the late 17th and early 18th centuries to enable barges to reach Winchester from Southampton Docks, but ceased to operate in the mid-19th century and is largely abandoned today.

The river is one of the world's premier chalk streams for fly fishing, amenable to dry fly or nymphing. The local chalk aquifer has excellent storage and filtration and the river has long been used for drinking water. Watercress thrives in its upper reaches.

Much of the river from its source to Swaythling is classified as a 748.5 ha biological Site of Special Scientific Interest (SSSI), and a Special Area of Conservation, of which the 9 ha Hockley Meadows nature reserve is a part. The Itchen estuary is part of the separate Lee-on-The Solent to Itchen Estuary SSSI.

Etymology and other name
The name is likely from a Brittonic language and in meaning unknown. The settlement of Itchen Abbas, before its abbot's ownership, is transcribed by early Middle English orthography which used the letter c followed by e or i to denote the sound and an e invariably after an n to avoid those down strokes being ambiguous thus, as Icene in the Domesday Book of 1086. The name, in upper parts was once seen as interchangeable with the tributary Alre.

Course
The source is north of Kilmeston, in that civil and ecclesiastical parish. The river flows north, through Cheriton village and east of overwhelmingly agricultural Tichborne village, before joining with tributaries the River Alre and Candover Brook immediately north-west of the town of New Alresford. The river then carves its way west passing the villages of Avington, Itchen Stoke, Itchen Abbas, Martyr Worthy, Easton, and Abbots Worthy; to enter the cathedral city of Winchester it crosses Winnall Moors to turn south for its remainder.

It flows in anabranches through the city, some close enough to Winchester Cathedral to cause problems to the foundations in the past. The main channel flows through Winchester City Mill and to the east of the city's Roman walls, along a promenaded reach known as "The Weirs".

It waters can overflow into water-meadows, passing: the Hospital of St Cross; the villages of Twyford; Shawford; between the town of Eastleigh and the village of Bishopstoke; and through Itchen Valley Country Park before reaching the northern suburbs of Southampton at Mansbridge. Between Winchester and Mansbridge, sections were deepened or widened for the long-disused Itchen Navigation, and the former towpath forms part of the Itchen Way.

Monks Brook flows into the Itchen at Swaythling, and the river then passes under Woodmill Bridge and becomes tidal. Four further bridges cross it before the Test estuary in Southampton Water:
 * Cobden Bridge, a road bridge connecting Bitterne Park and St Denys.
 * Cobden railway bridge carrying the Southampton – Portsmouth railway line. Opened in 1866, the bridge is supported by two sets of cast iron cylinders resulting in three 115-foot openings. The bridge's girders are made from wrought iron. In order to try to prevent the Northam Bridge Company from trying to block the bill authorising the railway, it was agreed that pedestrians would not be allowed to use the bridge. Even so the opening of the viaduct diverted traffic from other crossings with the Woolston Floating Bridge seeing a significant fall in income.
 * Northam Bridge, a road bridge carrying the A3024 road from Bitterne Manor to Northam, which was originally opened in 1799.
 * The Itchen Bridge, a high-level toll road bridge connecting the docks area with Woolston. This replaced the Woolston Floating Bridge (a chain ferry) which had previously crossed the river at this point.

Between the latter two, the river passes St Mary's Stadium, the home of Southampton F.C. As the river meets Southampton Water it passes several rowing clubs, sailing clubs and the major mixed-development on the left bank in Woolston, called Centenary Quay.

Watermills
Surviving mills on the Itchen include Winchester City Mill (restored to working order by the National Trust), Abbey Mill, Winchester; converted to a restaurant, Wharf Mill, Winchester; converted to apartments, St Cross Mill, Winchester; converted to private dwellings, College Mill, Winchester; converted to a private dwelling, Hockley Mill, Winchester; converted to private dwellings, Gaters Mill, Southampton; converted to offices, and Woodmill, Southampton; now a recreational water activities centre and the limit of navigation. Sites of demolished mills include those at Durngate, Winchester; at the southern entrance to Winnall Moors Nature Reserve, St Catherine's lock, Winchester and Shears Mill in Bishopstoke.

Marinas and boatyards
The lower part of the river is an important yachting centre and contains several marinas, sailing centres and boatyards. From seaward they are:

 * Ocean Village Marina (MDL), on the western shore just below the Itchen Bridge and close to the city centre
 * Southampton Water Activities Centre, almost underneath the bridge on the western shore
 * Itchen Marine, just above the bridge, principally a towage business but with some berths for yachts and the only fuel berth in the river
 * Merlin Boatyard, opposite Itchen Marine, on the eastern shore
 * Lauren Marine Services, a small marina on the eastern side
 * Ocean Quay and Solent Breeze Yacht Charter on the west side
 * Shamrock Quay (MDL), the biggest marina in the river, also on the west
 * Saxon Wharf (MDL) is almost adjacent to Shamrock Quay, containing the biggest boat lift in Britain (200t)
 * Kemps Quay Marina on the eastern (now northern) shore, a drying marina and boatyard
 * Quayside Marina, a single long pontoon next to Kemps
 * Drivers Wharf, another single long pontoon, parallel to the shore, with a crane and boatyard

Above Northam Bridge, the limit of navigation for masted craft, are the Vespasian Road boatyard and numerous small establishments.

Ecology
The SSSI covers the river and its banks, with fen, flood meadows, wet woodland and swamp. It has populations of the nationally rare southern damselfly and white-clawed crayfish. Other fauna include otters, water voles, Atlantic salmon, shovelers and Cetti's warblers.

Water quality
The Environment Agency measure water quality of the river systems in England. Each is given an overall ecological status, which may be one of five levels: high, good, moderate, poor and bad. There are several components that are used to determine this, including biological status, which looks at the quantity and varieties of invertebrates, angiosperms and fish, and chemical status, which compares the concentrations of various chemicals against known safe concentrations. Chemical status is rated good or fail.

Water quality of the River Itchen in 2022:

In popular culture
Death and the Maiden, a 1947 novel by Gladys Mitchell features the river.

title: River Itchen, Hampshire


text:
Charles VII

Charles VII may refer to:

 * Charles VII of Sweden (1130–1167), actually Charles I of Sweden (1161–1167)
 * Charles VII of France (1403–1461), "the Victorious"
 * Charles VII, Holy Roman Emperor (1697–1745)
 * Charles III of Spain (1716–1788), and Charles VII of Naples
 * Carlos María de los Dolores (1848–1909), pretender to the throne of Spain, styled "Charles VII" by Carlists
 * Charles VII, Archduke of Austria

title: Charles VII


text:
Faisal

Faisal, Faisel, Fayçal or Faysal (فيصل) is an Arabic given name. Faisal, Fayçal or Faysal may also refer to:

People

 * King Faisal (disambiguation)
 * Faisal I of Iraq and Syria (1885–1933), leader during the Arab Revolt
 * Faisal II of Iraq (1935–1958), last King of the Kingdom of Iraq
 * Faisal of Saudi Arabia (1906–1975), third King of Saudi Arabia
 * Faisal Al-Fayez (born 1952), Jordanian Prime Minister.
 * Faisal al-Duwaish (1882–1931), Arabian tribe sheik
 * Faisal Amin Abu-Rass, Yemeni diplomat
 * Faisal Buressli (born 1961), Kuwaiti basketball player and coach
 * Faisal Karami (born 1971), Lebanese politician
 * Faisal bin Abdullah Al Saud (born 1950), Saudi royal
 * Faisal bin Bandar Al Saud (born 1945), Saudi government official
 * Faisal bin Bandar Al Saud, Saudi royal and businessman
 * Faisal bin Khalid Al Saud (born 1973), Saudi government official
 * Faisal bin Mishaal Al Saud (born 1959), Saudi government official
 * Faisal bin Musaid Al Saud, Saudi royal
 * Faisal bin Sattam Al Saud (born 1970), Saudi ambassador to Italy
 * Faisal bin Turki Al Saud, Saudi royal
 * Faisal bin Turki I Al Saud (1920–1968), Saudi royal
 * Faisal Saeed Al Mutar (1991–Present), Iraqi Human Rights Activist
 * Faisal bin Turki Al Saud (1785–1865), Emir of Nejd
 * Faissal Sam Shaib, Lebanese-American filmmaker

Places

 * Faisal Cantonment, a cantonment town in Karachi, Pakistan
 * Faisal Colony, a neighbourhood of New Karachi Town in Karachi, Pakistan.
 * Faisal Equestrian Club, an equestrian club and restaurant in Gaza
 * Faisal Town, a residential area of Lahore
 * King Faisal Mosque (disambiguation)
 * Faisal Mosque in Islamabad, Pakistan
 * King Faisal Mosque, Sharjah, United Arab Emirates

Other uses

 * Faisal (film), a 2017 Indian film retitled Rubaai
 * Faysal Bank, a bank based in Pakistan

title: Faisal


text:
Altenburg

Altenburg is a city in Thuringia, Germany, located 40 km south of Leipzig, 90 km west of Dresden and 100 km east of Erfurt. It is the capital of the Altenburger Land district and part of a polycentric old-industrial textile and metal production region between Gera, Zwickau and Chemnitz with more than 1 million inhabitants, while the city itself has a population of 33,000. Today, the city and its rural county is part of the Central German Metropolitan Region.

Altenburg was first mentioned in 976 and later became one of the first German cities within former Slavic area, east of the Saale river (as part of the medieval Ostsiedlung movement). The emperor Frederick Barbarossa visited Altenburg several times between 1165 and 1188, hence the town is named a Barbarossa town today. Since the 17th century, Altenburg was the residence of different Ernestine duchies, of whom the Saxe-Altenburg persisted until the end of monarchy in Germany in 1918. Industrialization reached Altenburg and the region quite early in the first half of the 19th century and flourished until the Great Depression around 1930. Economic malaise set in while Altenburg was in East Germany and continued after German reunification in 1990, evidenced by a decline in population, high unemployment and house vacancy rates.

The main sights of Altenburg are the castle, the Lindenau-Museum, the historic city center (most buildings are from early-modern origin) and the Gründerzeit architecture around the center. The popular German card game Skat was developed in Altenburg during the 1810s and the founder of the famous Brockhaus Enzyklopädie, Friedrich Arnold Brockhaus, lived and worked in Altenburg between 1810 and 1817.

Altenburg lies in the flat and fertile landscape of Osterland on the Pleiße river in the very east of Thuringia, next to the neighboring federal state of Saxony.

Middle Ages
The town (civitas Altenburg) was first mentioned in a deed to the Bishop of Zeitz in 976. Remains of a Slavic castle on the Schloßberg ("Castle Hill") demonstrate that the town was probably a Slavic foundation, the capital of the shire of Plisni, taken over during the conquest of Meissen by Henry I. As shown by placenames, the surrounding area (Osterland) was mainly settled by Slavs.

The town's location on the imperial road 'Via Imperii' between Halle and Cheb in Bohemia gave Altenburg economic importance in the salt trade.

The first castle, located under the present day church St. Bartholomäi, was destroyed after the Battle of Hohenmölsen between Henry IV and Rudolph of Swabia. It was rebuilt on the Schloßberg outside of the town. The 11th century Mantelturm tower is still preserved. The castle later became an imperial palatinate and played an important part in the German takeover and settlement of the area between the Harz-mountains and the Elbe.

In the middle of the 12th century, the Hohenstaufen emperors patronized Altenburg as one of their Kaiserpfalzes, allowing the town to become a market and a mint. Together with the Royal forests Leina, Pahna, Kammerforst and Luckauer Forst, lands of the Groitzsch family bought by Frederick Barbarossa, Altenburg, Colditz, Zwickau and Chemnitz were turned into the Terra Plisnensis. Altenburg and Chemnitz as Imperial towns were intended to reduce the importance of Leipzig held by the Margrave of Meissen. Under Frederick Barbarossa much building took place, especially in the market area, and the town grew rapidly. A priory of canons regular was founded and the parish church was finished in 1172. The twin towers of the 12th century Augustinian monastery (Rote Spitzen) are still preserved. A town wall with 5 gates was constructed at the end of the 12th century. Altenburg got its charter around 1200, in 1256 the Wettins confirmed it again. The law structure was transposed from Goslar municipal law.

During the Interregnum, the Terra Plisnensis was impounded, but bought back by Rudolph I of Germany, who desired the crown of Thuringia. Together with Zwickau and Chemnitz, Altenburg was part of the anti-Meissen Pleiße-city Union of 1290. After the Battle of Lucka in 1307 against Frederick the Brave of Meissen and his brother Diezmann, King Albert I lost Altenburg and the Pleiße-lands to the Wettin margraves of Meissen, who held the city until 1918.

In 1455, Altenburg saw the division (Altenburger Teilung) of the Meissen lands between Elector Frederick II (the Gentle) and Duke William that led, after a failed attempt at reconciliation (Hallescher Machtspruch) to a war (1446–1451) between the two brothers (Bruderkrieg). In the second division of the Wettin lands between Ernest and Albert at Leipzig in 1485, Altenburg fell to Ernest, together with the Electorate (Kurland), Grimma, the Mutschener Pflege, Leisnig, Thuringia and the Vogtland. From this time on, Altenburg was historically connected with Thuringia and its dynasty, the Ernestine Wettins.

Early modern period
The Reformation was introduced in Altenburg quite early, in 1522, by George Spalatin, Wenzeslaus Linck and Gabriel Zwilling. During the German Peasants' War of 1525, the Altenburg Augustinian monastery was attacked. In the summer, four peasant rebels were executed at the marketplace. After the Schmalkaldic War brought defeat for the Ernestines, Altenburg belonged to the Albertines for short time (1547–1554) before coming back to the Ernestines after the Naumburg Treaty.

From 1603 to 1672, Altenburg was the residence of an Ernestine line, after that, it fell to Saxe-Gotha-Altenburg. The Thirty Years' War brought heavy damage to the city and more than half of the population died.

From 1705 onwards, the Magdalenenstift was a Protestant boarding school for young noble ladies.

During the Napoleonic wars it was a scene of a brief Allied raid by the Saxon General Johann von Thielmann.

Since 1815
When the Ernestine lands were re-divided in 1826, Altenburg became the capital of Saxe-Altenburg, successor state to the dissolved Saxe-Hildburghausen. Around 1830, the city walls and gates were knocked down and the old suburbia in front of the former wall were incorporated. Industrialization began around this time and the economy and population both saw rapid growth, strengthened by the connection to the railway in 1842 (as Thuringia's first connected city). The Revolution of 1848 led to the abdication of the conservative duke Joseph, who was replaced by his more liberal brother George. The last duke abdicated during the Revolution of 1918 on 13 November 1918 after being promised 12 million Marks and the ownership of numerous castles. The free-state of Saxe-Altenburg was merged with Thuringia in 1920.

Altenburg was a working-class city during the Weimar Republic, ruled by SPD and KPD, which led to heavy conflicts between left- and right-wing forces after 1933. The Jewish community was destroyed during the Kristallnacht in 1938, many Jews emigrated or were killed in the concentration camps. Furthermore, communists and invalids from Altenburg were murdered. During World War II, several subcamps of the Buchenwald concentration camp were located here. They provided 13,000 forced labourers for HASAG, the third largest German company to use concentration camp labour. The US Army reached Altenburg on 15 April 1945 and was replaced by the Soviet Army on 1 July 1945.

In 1952, Thuringia was dissolved and replaced by administrative divisions of East Germany (Bezirks). Altenburg became part of the Leipzig administrative district, in which it was the second largest city. After reunification, previously extant states were re-established in the former east Germany as federal states in the reunified Germany. Although a majority of 54% in the district voted for Saxony in a plebiscite, the district council decided to join Thuringia together with the Schmölln district, partly because a vast majority of 80% in the neighboring Schmölln district had voted for Thuringia. These districts had formed the eastern part of Saxe-Altenburg until 1920 and were reunified as the modern-day Altenburger Land district in 1994.

Topography
Altenburg is located on the southern edge of the Leipzig Bay, a flat and fertile area, with intensive agricultural use and lignite surface mining 10 km north of the town. The nearest mountains are the Ore Mountains approximately 50 km to the south-east. The Pleiße valley runs east of the city, whereas the center itself is located at Blaue Flut (blue flood) river, a small tributary of Pleiße river. The Blaue Flut feeds the Großer Teich (big pond), an inner-city pond with an island in the middle, on which the Altenburg zoo is located. There is relatively little forest within the town's municipal territory: the small Stadtwald (town wood) and Herzog-Ernst-Wald (Duke Ernest wood) south of the center and the Fasanerieholz (Pheasantry wood) at Ehrenberg district. The elevation of the city center is 200 m, to the north it declines down to 162 m and in the south next to Mockzig district it rises up to 261 m.

Administrative division
Altenburg is bounded by Windischleuba, Nobitz, Altkirchen, Göhren, Lödla, Rositz, Meuselwitz, Treben and Gerstenberg.

Because of the quite densely settlement of the region, many small villages and ancient former suburbs are situated within the municipal territory, which is officially divided into 4 districts:
 * Altenburg (including the villages Drescha, Kauerndorf, Poschwitz, Rasephas, Steinwitz and Zschernitzsch)
 * Ehrenberg, incorporated in 1993 (with Greipzig, Lehnitzsch, Mockzig, Modelwitz, Paditz, Stünzhain, Zschaiga and Zschechwitz)
 * Kosma, incorporated in 1996 (with Altendorf and Kürbitz)
 * Zetzscha, incorporated in 1994 (with Knau, Oberzetzscha, Rautenberg and Unterzetzscha)

For a detailed view of Altenburg's administrative division, see the list in German Wikipedia.

Demographics
Altenburg has been a relatively big town (compared to others in the wider region) down the centuries, because of its importance as a Kaiserpfalz and later as a ducal residence. During the Middle Ages, it had a population of 3,000 to 4,000 inhabitants, rising to 6,000 around 1700, 10,000 around 1800 and 20,000 around 1870, making it Thuringia's second-largest city after Erfurt for a short time in the mid-19th century. The population grew further to 40,000 in 1910, 44,000 in 1940 and peaked at about 56,000 around 1980 before starting to decline. In 1988, before reunification, Altenburg had 53,000 inhabitants, shrinking to 41,000 by 2000 and 33,000 in 2012. With a decline of nearly 40% since 1988, Altenburg is among the most rapidly declining urban areas in Germany.

The average decrease of population between 2009 and 2012 was approximately 1.27% p.a., whereas the population in bordering rural regions is shrinking with accelerating tendency. Suburbanization played only a small role in Altenburg. It occurred after the reunification for a short time in the 1990s, but most of the suburban areas were situated within the administrative city borders, others are Windischleuba and Nobitz. During the 1990s and the 2000s, many inhabitants left Altenburg to search a better life in western Germany or other major east German cities like Leipzig, Dresden or Jena. The birth deficit, caused by the high average age of the population, is becoming a bigger problem because there is no immigration to compensate it. Despite urban planning activities to tear down unused flats, vacancy is still a problem with rates around 16% (according to 2011 EU census), which is the largest amount among Thuringia's major towns. A positive side effect for the inhabitants is that Altenburg has one of the lowest rent levels in Germany.

The birth deficit was 282 in 2012, this is -8.5 per 1,000 inhabitants (Thuringian average: -4.5; national average: -2.4). The net migration rate was -1.7 per 1,000 inhabitants in 2012 (Thuringian average: -0.8; national average: +4.6). The most important target regions of Altenburg migrants are the large cities around like Leipzig, Dresden, Halle, Jena and Erfurt, as well as the western German conurbations.

Like most other small to middling eastern German towns, Altenburg has only a small amount of foreign population: circa 1.7% are non-Germans by citizenship and overall 1.6% are migrants (according to 2011 EU census). Differing from the national average, the biggest groups of migrants in Altenburg are Russians and Vietnamese people. During recent years, the economic situation of the city has improved a bit: the unemployment rate in the district declined from 24% in 2005 to 10% in 2013 with little bit higher rates in the city than in the other municipalities of the Altenburger Land district. Nevertheless, this is still one of the highest rates in Thuringia.

Partly due to official atheism in former GDR, most of the population is non-religious. 12.1% are members of the Evangelical Church in Central Germany and 2.5% are Catholics (according to 2011 EU census).

 * Source (since 1994): Thüringer Landesamt für Statistik

Culture
Altenburg is nicknamed playing cards town. The game of skat is said to have originated here, based on the Bavarian tarock. Because of the influence Emperor Frederick Barbarossa had on the town, it is nicknamed a "Barbarossa town".

Museums
There are some museums in Altenburg:
 * The Schloss- und Spielkartenmuseum (castle and playing card museum) inside the castle hosts an exhibition about Altenburg as a ducal residence, and a historical playing cards collection.
 * The Lindenau-Museum (established in 1876) shows Bernhard von Lindenau's art collection including Italian paintings of the 13th–15th centuries, a collection of classical antiquities and cast and modern art.
 * The Mauritianum (opened in 1908) is Altenburg's museum of natural history with the largest well-known mummified rat king, which was found in 1828 in a miller's fireplace at Buchheim.
 * The Brauereimuseum inside the Altenburger brewery shows an exhibition about beer and its history with a local focus.

Old town
The town center of Altenburg within the former wall has a planned grid of 12th/13th century origin, whereas the two old settlement cores were located around Brühl in the north-east and around St. Nicholas' Church in the south-west. The area around the castle and southward on the eastern bank of Blaue Flut river has also been urban since the Middle Ages, nevertheless, it was located outside the city walls.

During the 19th and early 20th century, Altenburg saw a construction boom and the town enlarged to all directions, particularly north and east towards the new railway station (opened in 1878). Its time as a state capital until 1918 led to many interesting public and private buildings in Gründerzeit style, same as in other historist styles. Later, the town lost some of its importance and became a simple district capital in Thuringia. World War II left Altenburg unscathed, so all historic buildings are preserved. Nevertheless, the buildings' maintenance was neglected during the East German period, instead, some big Plattenbau settlements were set up on the northern and south-eastern periphery of Altenburg.

After reunification, most of the main sights and historic buildings were refurbished. On the other hand, the city lost much of its population which led to a very high vacancy rate, especially in the old inner-city buildings, which today endangers the historic city structure because of many vacant old private houses.

Churches

 * St. Bartholomew's Church, built between 1428 and 1443 in Gothic style at Burgstraße, is Altenburg's main evangelical parish church.
 * The Fraternity Church at Marktplatz is another evangelical parish church, founded as a Franciscans monastery during the 12th century. It was abandoned after the Reformation in 1529 and reused as evangelical church. Today's building was established in 1905 in Neo-Gothic style.
 * St. Nicholas' Church at Nikolaikirchhof was the first church of Altenburg, mentioned in 1140. It was demolished during the 16th century, only the steeple remains.
 * St. George's Chapel is the castle's church, built in picturesque Gothic style during the mid-15th century.
 * St. Mary's Monastery (Augustinians) is a former monastery on a hill east of the center. It was founded during the 12th century and abandoned in 1543. Only the two Romanesque towers and some walls remain, being the city's landmark called "Rote Spitzen" (red spires).
 * The Duchess Agnes Memorial Church at Hausweg is an evangelical church, built between 1903 and 1906 in historist style.
 * The Cemetery Church at Hospitalplatz was built between 1639 and 1651 in Renaissance style.

Ducal buildings

 * Altenburg Castle is the big ducal residence on a hill above the city center. It was in use first as fortification and later as residence since the 9th century until the end of monarchy in Germany in 1918. Today's castle is of 17th and 18th century origin and was built in Renaissance, later in Baroque style. The chapel is of older origin (see above). The castle is the scene of the famous Prinzenraub, related by Carlyle in his "Miscellanies".
 * Further buildings within the castle complex are the Junkerei in the north, built in the 16th century, the Flasche, an old castle tower from the 11th century, the Hausmannturm, built in the 12th century as the castle's keep as well as the Prinzenpalais and the Hofmarschallamt in the south, built after a fire in 1868.
 * Next to the castle lies the Schlosspark (castle park) with some more ducal buildings: the Baroque Teehaus (tea house) (1712), the Baroque Orangerie (1712) and the historist Marstall (1851).
 * The Amtshaus was built between 1725 and 1728 in Baroque style at Burgstraße and is now used as district court.
 * The Kanzlei was the seat of some ducal authorities, built in 1476 at Brühl.
 * The Neues Ministerium at Lindenaustraße was built in 1895 in Classicist style and is now the seat of the district government.
 * The former Landgericht at Pauritzer Platz was built in 1859 in Neo-Renaissance style and hosts the social court now.
 * The Landesbank (federal state bank) at Burgstraße was built in 1865 in Neo-Renaissance style. Currently, it is vacant.
 * The Ducal Mausoleum in Altenburg Cemetery (burial site of several Dukes and Duchesses), now in ruins (those buried there removed to other parts of the cemetery in 1974).

Other sights

 * The town hall is one of the most important Renaissance buildings in Germany. It was built between 1562 and 1564 by the architect Nikolaus Gromann.
 * The Landestheater is the theater of Altenburg, built in Neo-Baroque style in 1871.
 * The city wall remained in the north behind Johannisstraße and in the south at Langengasse. Two towers are preserved at Langengasse and Kunstgasse.
 * The Palais Seckendorff at Brühl is a Baroque style nobility palace, built in 1724.
 * The Wasserkunst at Kunstgasse is an old waterwork, built in 1844 in Classicist style.
 * The Pohlhof at Pohlhofgasse is a small Renaissance palace.
 * The Freemasons' Lodge at Johannisgraben was built in 1804 in Classicist style.

Agriculture, industry and services
Agriculture plays an important role in the region, because the Loess soil around the city is very fertile. 69% of the municipal territory is in agricultural use, both for cultivation and cattle farming. A famous product of the region is Altenburger Ziegenkäse, a soft cheese of goat milk with some caraway seed inside with protected Geographical indication. Another typical dish of Altenburg is Mutzbraten, a flame-grilled pork speciality. Altenburg is also known for its mustard, which is produced by Altenburger Senf und Feinkost. Altenburger beer also enjoys some popularity.

The industry of Altenburg was based on the production of textiles and hardware during the 19th and 20th century, but most of the companies did not survive the transition to capitalism after German reunification. Altenburg was especially known for its sewing machine production from companies like L. O. Dietrich (machines Vesta, Saxonia, Löwe, Minerva, Juno), Hermann Köhler AG (machines Köhler, Diva, Clementine, Hermann, Dora, Orion, Globus), Textima Wittenberge (Columba, Altin), Gustav Winselmann GmbH (Titan, Vera, Bera, Hera, Heraldus, Omnia, Marvel, Noris, Regia, Saxony, the Wellington).

Today, the main branches are food industries (with a factory of Vion NV amongst others) and engineering with car component production in focus. The Gumpert Sportwagenmanufaktur produced the Gumpert Apollo supercar between 2005 and its bankruptcy in 2013. Altenburg is also noted for produced playing cards. The Altenburger Spielkartenfabrik (playing card factory) was founded in 1831; today it is a subsidiary company of Cartamundi and market leader in Germany. In 2012, there were 19 companies with more than 20 workers in industrial sector, employing 1,400 people and generating an overall turnover of €451 million.

Altenburg is a regional service hub for retail, hospitals, government, culture etc. Tourism doesn't play a big role, although the city hosts many historic sights and is a local center of culture with theater and museums. In 2012, 37,000 hotel guests had 87,000 overnight stays in Altenburg.

Transport
Altenburg was the first city in Thuringia with connection to the railway, established in 1842 with the Leipzig–Hof line. Initially, Altenburg station was a terminus station, which became impractical as traffic grew, so a new through station was built in 1878 in the north-east of the city. In addition to the connections to Leipzig (1842) and Hof/Zwickau (1844), some more lines were established to Glauchau (1858, via Gößnitz), to Gera (1865, via Gößnitz), to Zeitz (1872, abandoned in 2002) and to Narsdorf (1901, abandoned in 1998). Today, the Altenburg station is the only station within the municipality, after the Paditz station (on the Hof line) and the North Altenburg station (on the Zeitz line) were closed. There are some regional trains to Leipzig, Zwickau and Erfurt (via Gera and Jena) today. Glauchau can be reached by changing trains in the near Gößnitz station. With the rollout of the Leipzig City Tunnel in December 2013, the services on the Leipzig–Hof/Zwickau line were reorganized and integrated in the S-Bahn Mitteldeutschland with connections to Leipzig twice an hour. The connection to long-distance trains is carried out via Leipzig (north and east), Jena (south) and Erfurt (west), since the last long-distance trains on the Leipzig–Hof line ceased operating in 2006.

Altenburg is situated between two Autobahns: the A4 (Erfurt–Dresden) passes approximately 20 km south and the A72 (Leipzig–Chemnitz) passes approximately 20 km east of the city. Regional traffic is carried by the three national roadsn in Altenburg. The B7 is the connection to Gera (and A4 in Erfurt direction) in the south-west and to Rochlitz in the north-east, the B93 runs to Borna (and A72 in Leipzig direction) in the north and to Zwickau in the south and the B180 links Altenburg with Zeitz in the west and Hohenstein-Ernstthal (and A72/A4 in Chemnitz and Dresden direction via secondary road 1357) in the south-east. An important secondary road connects Altenburg with Gera north to the B7 via Lumpzig. The B7 and B93 bypass the town center on a ring road to absorb the transit traffic between Leipzig and Zwickau.

The Leipzig-Altenburg Airport, a former Soviet military airport, is situated 5 km east of the city and was used by Ryanair for flights to London Stansted, Barcelona-Girona and Edinburgh until 2011. Currently, there are no public flights at this airport. The next regional airport is located in Leipzig/Halle (60 km to the north).

Biking is becoming more popular since the construction of long-distance cycle tracks began in the 1990s. Both the Pleiße track, along the Pleiße valley from Werdau near Zwickau to Leipzig, and the Thuringian city track (Radweg Thüringer Städtekette) from Eisenach via Erfurt, Weimar, Jena and Gera to Altenburg, connect points of tourist interest.

Local public transport within Altenburg is based exclusively on buses. Six lines connect the outlying quarters to the inner city. Furthermore, there are some regional bus services to the villages in the district. The Altenburg Tramway was in operation from 1895 until 1920.

Education
There are three Gymnasiums in Altenburg: two state-owned and one Christian (ecumenical).

Mayor and city council
The first freely elected mayor after German reunification was Johannes Ungvari of the Christian Democratic Union (CDU), who served from 1990 to 2000. He was succeeded by Michael Wolf of the Social Democratic Party (SPD), who served until 2018. Since 2018, André Neumann of the CDU has been mayor. The most recent mayoral election was held on 15 April 2018, and the results were as follows:

! colspan=2| Candidate ! Party ! Votes ! % ! colspan=3| Valid votes ! 13,705 ! 98.7 ! colspan=3| Invalid votes ! 177 ! 1.3 ! colspan=3| Total ! 13,882 ! 100.0 ! colspan=3| Electorate/voter turnout ! 27,725 ! 50.1
 * bgcolor=|
 * align=left| André Neumann
 * align=left| Christian Democratic Union
 * 7,614
 * 55.6
 * bgcolor=|
 * align=left| Frank Schütze
 * align=left| Independent
 * 3,296
 * 24.0
 * bgcolor=|
 * align=left| Katharina Schenk
 * align=left| Social Democratic Party
 * 2,795
 * 20.4
 * align=left| Social Democratic Party
 * 2,795
 * 20.4
 * colspan=5| Source: Wahlen in Thüringen
 * }
 * }

The most recent city council election was held on 26 May 2019, and the results were as follows:

! colspan=2| Party ! Lead candidate ! Votes ! % ! +/- ! Seats ! +/- ! colspan=3| Valid votes ! 12,718 ! 95.7 ! ! ! ! colspan=3| Invalid votes ! 566 ! 4.3 ! ! ! ! colspan=3| Total ! 13,284 ! 100.0 ! ! 36 ! ±0 ! colspan=3| Electorate/voter turnout ! 27,296 ! 48.7 ! 7.0 ! !
 * bgcolor=|
 * align=left| Christian Democratic Union (CDU)
 * align=left| André Neumann
 * 11,510
 * 30.4
 * 5.0
 * 11
 * 2
 * align=left| Pro Alternburg
 * align=left| Peter Müller
 * 9,326
 * 24.7
 * 6.4
 * 9
 * 3
 * align=left| City Forum Altenburg
 * align=left| Johannes Schaefer
 * 5,227
 * 13.8
 * New
 * 5
 * New
 * bgcolor=|
 * align=left| The Left (Die Linke)
 * align=left| Kati Klaubert
 * 5,187
 * 13.7
 * 8.3
 * 5
 * 3
 * bgcolor=|
 * align=left| Social Democratic Party (SPD)
 * align=left| Nikolaus Dorsch
 * 4,356
 * 11.5
 * 13.3
 * 4
 * 5
 * bgcolor=|
 * align=left| Free Democratic Party (FDP)
 * align=left| Detlef Zschiegner
 * 1,274
 * 3.4
 * 1.7
 * 1
 * ±0
 * bgcolor=|
 * align=left| Alliance 90/The Greens (Grüne)
 * align=left| Claudia Große
 * 942
 * 2.5
 * New
 * 1
 * New
 * bgcolor=|
 * align=left| Alliance 90/The Greens (Grüne)
 * align=left| Claudia Große
 * 942
 * 2.5
 * New
 * 1
 * New
 * New
 * colspan=8| Source: Wahlen in Thüringen
 * }
 * }

Twin towns – sister cities
Altenburg is twinned with:
 * 🇩🇪 Offenburg, Germany
 * Olten, Switzerland
 * 🇨🇿 Zlín, Czech Republic
 * 🇺🇸 Hickory, North Carolina, United States of America

Notable people

 * Günter Beier (born 1942), gymnast
 * Friedrich von Beust (1817–1899), Privy Councillor, Chamberlain and Lord Marshal, Major General and Adjutant General
 * Perry Bräutigam (born 1963), footballer
 * Joachim Büchner (1905–1983), sprinter athlete
 * Friedrich Arnold Brockhaus (1772–1823), encyclopedia publisher
 * Hermann Askan Demme (1802–1867), physician and surgeon
 * Adolph Douai (1819–1888), German-American journalist, social reformer and founder of the first kindergartens of the US
 * Bernhard Erdmannsdörffer (1833–1901), historian
 * Walther Flemming (1843–1905), biologist
 * Frederick I (1122–1190), Holy Roman Emperor
 * Erhard Frommhold (1928–2007), art historian and publicist
 * Hans Conon von der Gabelentz (1807–1874)
 * Georg von der Gabelentz (1840–1893), linguist and sinologist
 * Johann Georg August Galletti (1750–1828)
 * Hanns Bruno Geinitz (1814–1900), geologist and paleontologist
 * Ralf Haber (born 1962), hammer thrower
 * Dieter Kalka (born 1957), author, musician and songwriter
 * Volker Kluge (born 1944), journalist
 * Johann Ludwig Krebs (1713–1780), Baroque musician and composer
 * Bernhard von Lindenau (1780–1854), lawyer, astronomer and politician
 * Sophie Mereau (1770–1806), writer of German Romanticism
 * Charles-Henri Petersen (1792-1859), paysagist architect who designed famous parks in Belgium, notably Les Orangeries de Bierbais
 * Uwe Rösler (born 1968), football player and manager
 * Hermann Schlegel (1804–1884), ornithologist, herpetologist and ichthyologist
 * Ingo Schulze (born 1962), writer
 * George Spalatin (1484–1545), humanist
 * Heinrich Schmidt, Hauptsturmführer and camp doctor in the concentration camps
 * Johann Severin Vater, theologian and linguist
 * Jürgen Thiele, rowers, Olympic champion in 1980 in coxless four
 * Kerstin Walther, athlete, with the GDR relay won the gold medal in the 4x400-meter race at the 1983 World Championship
 * Christian Friedrich Witt, church musician and composer

title: Altenburg


text:
User:Bcorr/Wikiwisdom

Wikiwisdom I've run across...

User:Mark (aka Mark Ryan) (9/0); ends 23:07, 17 Mar 2004 (UTC)
Last night, during a 3-hour attack of a disgruntled user, Mark helped to revert mass graffiti posts. However, he could not help in blocking all the IPs being thrown at us. Mark cares deeply for the welfare of Wikipedia, and he should be given the kinds of powers to help fight vandalism. He should not be judged by his quanity (about 1000 lifetime edits, about 200 in 2004), but on his servitude (registered 8 Jan 2002) and character (never abusive, always caring). Kingturtle 23:07, 10 Mar 2004 (UTC)

 * I have spent some time considering whether I should accept this nomination. On one hand, it is very fashionable at the moment within Wikipedia to be promoted to administrator status, and it is very handy in quickly dealing with vandalism and performing the "chores" of wikipedia (like clearing out Votes for Deletion). On the other hand, in my opinion being an administrator places a very heavy burden of responsibility. Administrators become the brunt of all the misguided antagonism from disgruntled and problem users, and yet must always act with the utmost dignity and level-headedness. Whilst Kingturtle's words are quite kind, I do not believe that I would be a suitable Wikipedia administrator. I love Wikipedia; I've been coming back to it for so long now that it's almost in my blood. I have lost count of the times I have stayed up until 3am and 4am on university nights, watching for vandals. But I do not feel that I am worthy of adminship, or indeed ready for it. Plus, I have just returned to uni, so my time is more restricted than it was in the last few months. Wikipedia already has a great team of administrators who do their job with valour - I would add nothing good to this group. I would rather wait between 3 and 6 months before becoming an administrator. - Mark 13:38, 12 Mar 2004 (UTC)

title: User:Bcorr/Wikiwisdom


text:
Schur's lemma

In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N, i.e. φ is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on M. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.

Representation theory of groups
Representation theory is the study of homomorphisms from a group, G, into the general linear group GL(V) of a vector space V; i.e., into the group of automorphisms of V. (Let us here restrict ourselves to the case when the underlying field of V is $$\mathbb{C}$$, the field of complex numbers.) Such a homomorphism is called a representation of G on V. A representation on V is a special case of a group action on V, but rather than permit any arbitrary bijections (permutations) of the underlying set of V, we restrict ourselves to invertible linear transformations.

Let ρ be a representation of G on V. It may be the case that V has a subspace, W, such that for every element g of G, the invertible linear map ρ(g) preserves or fixes W, so that (ρ(g))(w) is in W for every w in W, and (ρ(g))(v) is not in W for any v not in W. In other words, every linear map ρ(g): V→V is also an automorphism of W, ρ(g): W→W, when its domain is restricted to W. We say W is stable under G, or stable under the action of G. It is clear that if we consider W on its own as a vector space, then there is an obvious representation of G on W—the representation we get by restricting each map ρ(g) to W. When W has this property, we call W with the given representation a subrepresentation of V. Every representation of G has itself and the zero vector space as trivial subrepresentations. A representation of G with no non-trivial subrepresentations is called an irreducible representation. Irreducible representations – like the prime numbers, or like the simple groups in group theory – are the building blocks of representation theory. Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations.

Just as we are interested in homomorphisms between groups, and in continuous maps between topological spaces, we are also interested in certain functions between representations of G. Let V and W be vector spaces, and let $$\rho_V$$ and $$\rho_W$$ be representations of G on V and W respectively. Then we define a G-linear map f from V to W to be a linear map from V to W that is equivariant under the action of G; that is, for every g in G, $$ \rho_W(g) \circ f = f \circ \rho_V(g)$$. In other words, we require that f commutes with the action of G. G-linear maps are the morphisms in the category of representations of G.

Schur's Lemma is a theorem that describes what G-linear maps can exist between two irreducible representations of G.

Statement and Proof of the Lemma
Theorem (Schur's Lemma): Let V and W be vector spaces; and let $$\rho_V$$ and $$\rho_W$$ be irreducible representations of G on V and W respectively.
 * 1) If $$V$$ and $$W$$ are not isomorphic, then there are no nontrivial G-linear maps between them.
 * 2) If $$V=W$$ finite-dimensional over an algebraically closed field (e.g. $$\mathbb{C}$$); and if $$\rho_V = \rho_W$$, then the only nontrivial G-linear maps are the identity, and scalar multiples of the identity. (A scalar multiple of the identity is sometimes called a homothety.)

Proof: Suppose $$f$$ is a nonzero G-linear map from $$V$$ to $$W$$. We will prove that $$V$$ and $$W$$ are isomorphic. Let $$V'$$ be the kernel, or null space, of $$f$$ in $$V$$, the subspace of all $$x$$ in $$V$$ for which $$f(x)=0$$. (It is easy to check that this is a subspace.) By the assumption that $$f$$ is G-linear, for every $$g$$ in $$G$$ and choice of $$x$$ in $$V'$$, $$f((\rho_V(g))(x)) = (\rho_W(g))(f(x))=(\rho_W(g))(0) = 0$$. But saying that $$f(\rho_V(g)(x))=0$$ is the same as saying that $$\rho_V(g)(x)$$ is in the null space of $$f:V\rightarrow W$$. So $$V'$$ is stable under the action of G; it is a subrepresentation. Since by assumption $$V$$ is irreducible, $$V'$$ must be zero; so $$f$$ is injective.

By an identical argument we will show $$f$$ is also surjective; since $$f((\rho_V(g))(x)) = (\rho_W(g))(f(x))$$, we can conclude that for arbitrary choice of $$f(x)$$ in the image of $$f$$, $$\rho_W(g)$$ sends $$f(x)$$ somewhere else in the image of $$f$$; in particular it sends it to the image of $$\rho_V(g)x$$. So the image of $$f(x)$$ is a subspace $$W'$$ of $$W$$ stable under the action of $$G$$, so it is a subrepresentation and $$f$$ must be zero or surjective. By assumption it is not zero, so it is surjective, in which case it is an isomorphism.

In the event that $$V=W$$ finite-dimensional over an algebraically closed field and they have the same representation, let $$\lambda$$ be an eigenvalue of $$f$$. (An eigenvalue exists for every linear transformation on a finite-dimensional vector space over an algebraically closed field.) Let $$f' = f-\lambda I$$. Then if $$x$$ is an eigenvector of $$f$$ corresponding to $$\lambda, f'(x)=0$$. It is clear that $$f'$$ is a G-linear map, because the sum or difference of G-linear maps is also G-linear. Then we return to the above argument, where we used the fact that a map was G-linear to conclude that the kernel is a subrepresentation, and is thus either zero or equal to all of $$V$$; because it is not zero (it contains $$x$$) it must be all of V and so $$f'$$ is trivial, so $$f = \lambda I$$.

Corollary of Schur's Lemma
An important corollary of Schur's lemma follows from the observation that we can often build explicitly $$G$$-linear maps between representations by "averaging" over the action of individual group elements on some fixed linear operator. In particular, given any irreducible representation, such objects will satisfy the assumptions of Schur's lemma, hence be scalar multiples of the identity. More precisely:

Corollary: Using the same notation from the previous theorem, let $$h$$ be a linear mapping of V into W, and set$$ h_0 = \frac{1}{|G|} \sum_{g\in G} (\rho_W(g))^{-1} h \rho_V(g). $$Then,

 * 1) If $$V$$ and $$W$$ are not isomorphic, then $$h_0 = 0$$.
 * 2) If $$V=W$$ is finite-dimensional over an algebraically closed field (e.g. $$\mathbb{C}$$); and if $$\rho_V = \rho_W$$, then $$h_0 = I\, \mathrm{Tr}[h]/ n$$, where n is the dimension of V. That is, $$h_0$$ is a homothety of ratio $$\mathrm{Tr}[h]/n$$.

Proof: Let us first show that $$h_0$$ is a G-linear map, i.e., $$\rho_W(g) \circ h_0 = h_0 \circ \rho_V(g)$$ for all $$g \in G$$. Indeed, consider that

$$ \begin{align} (\rho_W(g'))^{-1} h_0 \rho_V(g') & =  \frac{1}{|G|} \sum_{g\in G} (\rho_W(g'))^{-1}(\rho_W(g))^{-1} h \rho_V(g) \rho_V(g') \\ & = \frac{1}{|G|} \sum_{g\in G} (\rho_W(g\circ g'))^{-1} h \rho_V(g\circ g') \\ & = h_0 \end{align} $$

Now applying the previous theorem, for case 1, it follows that $$h_0 = 0$$, and for case 2, it follows that $$h_0$$ is a scalar multiple of the identity matrix (i.e., $$h_0 = \mu I$$). To determine the scalar multiple $$\mu$$, consider that

$$ \mathrm{Tr}[h_0] = \frac{1}{|G|} \sum_{g\in G} \mathrm{Tr}[(\rho_V(g))^{-1} h \rho_V(g)] = \mathrm{Tr}[ h ] $$

It then follows that $$\mu = \mathrm{Tr}[h] / n$$.

This result has numerous applications. For example, in the context of quantum information science, it is used to derive results about complex projective t-designs.

Formulation in the language of modules
Theorem: If M and N are two simple modules over a ring R, then any homomorphism f: M → N of R-modules is either invertible or zero. In particular, the endomorphism ring of a simple module is a division ring.

The condition that f is a module homomorphism means that

 * $$ f(rm) = rf(m)\text{ for all }m \in M\text{ and }r \in R. $$

Proof: It suffices to show that $$f$$ is either zero or surjective and injective. We first show that both $$\ker(f)$$ and $$\operatorname{im}(f)$$ are $$R$$-modules. If $$m\in\ker(f), r\in R$$ we have $$f(rm)=rf(m)=0$$, hence $$rm\in\ker(f)$$. Similarly, if $$n=f(m)\in\operatorname{im}(f)$$, then $$rn=rf(m)=f(rm)\in\operatorname{im}(f)$$ for all $$r\in R$$. Now, since $$\ker(f)\subseteq M$$ and $$\operatorname{im}(f)\subseteq N$$ are submodules of simple modules, they are either trivial or equal $$M, N$$, respectively. If $$f\ne0$$, its kernel cannot equal $$M$$ and must therefore be trivial (hence $$f$$ is injective), and its image cannot be trivial and must therefore equal $$N$$ (hence $$f$$ is surjective). Then $$f$$ is bijective, and hence an isomorphism. Consequently, every homomorphism $$M\to N$$ is either zero or invertible, which makes $$\operatorname{Hom}_R(M,N)$$ into a division ring.

The group version is a special case of the module version, since any representation of a group G can equivalently be viewed as a module over the group ring of G.

Schur's lemma is frequently applied in the following particular case. Suppose that R is an algebra over a field k and the vector space M = N is a simple module of R. Then Schur's lemma says that the endomorphism ring of the module M is a division algebra over k. If M is finite-dimensional, this division algebra is finite-dimensional. If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Thus the endomorphism ring of the module M is "as small as possible". In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.

More generally, if $$R$$ is an algebra over an algebraically closed field $$k$$ and $$M$$ is a simple $$R$$-module satisfying $$\dim_k(M) < \# k$$ (the cardinality of $$k$$), then $$\operatorname{End}_R(M) = k$$. So in particular, if $$R$$ is an algebra over an uncountable algebraically closed field $$k$$ and $$M$$ is a simple module that is at most countably-dimensional, the only linear transformations of $$M$$ that commute with all transformations coming from $$R$$ are scalar multiples of the identity.

When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A simple module over a $$k$$-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to $$k$$. This is in general stronger than being irreducible over the field $$k$$, and implies the module is irreducible even over the algebraic closure of $$k$$.

Application to central characters
Definition: Let $$R$$ be a $$k$$-algebra. An $$R$$-module $$M$$ is said to have central character $$\chi: Z(R)\to k$$ (here, $$Z(R)$$ is the center of $$R$$) if for every $$m\in M, z\in Z(R)$$ there is $$n\in\N$$ such that $$(z-\chi(z))^nm=0$$, i.e. if every $$m\in M$$ is a generalized eigenvector of $$z$$ with eigenvalue $$\chi(z)$$.

If $$\operatorname{End}_R(M)=k$$, say in the case sketched above, every element of $$Z(R)$$ acts on $$M$$ as an $$R$$-endomorphism and hence as a scalar. Thus, there is a ring homomorphism $$\chi: Z(R)\to k$$ such that $$(z-\chi(z))m =0$$ for all $$z\in Z(R), m\in M$$. In particular, $$M$$ has central character $$\chi$$.

If $$R=U(\mathfrak{g}), k=\C$$ is the universal enveloping algebra of a Lie algebra, a central character is also referred to as an infinitesimal character and the previous considerations show that if $$\mathfrak{g}$$ is finite-dimensional (so that $$R=U(\mathfrak{g})$$ is countable-dimensional), then every simple $$\mathfrak{g}$$-module has an infinitesimal character.

In the case where $$k=\C, R=\C[G]$$ is the group algebra of a finite group $$G$$, the same conclusion follows. Here, the center of $$R$$ consists of elements of the shape $$\sum_{g\in G} a(g) g$$ where $$a: G\to\C$$ is a class function, i.e. invariant under conjugation. Since the set of class functions is spanned by the characters $$\chi_\pi$$ of the irreducible representations $$\pi\in\hat{G}$$, the central character is determined by what it maps $$u_\pi:= \frac{1}{\#G}\sum_{g\in G} \chi_\pi(g) g$$ to (for all $$\pi\in\hat{G}$$). Since all $$u_\pi$$ are idempotent, they are each mapped either to 0 or to 1, and since $$u_\pi u_{\pi'}=0$$ for two different irreducible representations, only one $$u_\pi$$ can be mapped to 1: the one corresponding to the module $$M$$.

Representations of Lie groups and Lie algebras
We now describe Schur's lemma as it is usually stated in the context of representations of Lie groups and Lie algebras. There are three parts to the result.

First, suppose that $$V_1$$ and $$V_2$$ are irreducible representations of a Lie group or Lie algebra over any field and that $$\phi:V_1\rightarrow V_2$$ is an intertwining map. Then $$\phi$$ is either zero or an isomorphism.

Second, if $$V$$ is an irreducible representation of a Lie group or Lie algebra over an algebraically closed field and $$\phi:V\rightarrow V$$ is an intertwining map, then $$\phi$$ is a scalar multiple of the identity map.

Third, suppose $$V_1$$ and $$V_2$$ are irreducible representations of a Lie group or Lie algebra over an algebraically closed field and $$\phi_1, \phi_2:V_1\rightarrow V_2$$ are nonzero intertwining maps. Then $$\phi_1=\lambda\phi_2$$ for some scalar $$\lambda$$.

A simple corollary of the second statement is that every complex irreducible representation of an abelian group is one-dimensional.

Application to the Casimir element
Suppose $$\mathfrak{g}$$ is a Lie algebra and $$U(\mathfrak{g})$$ is the universal enveloping algebra of $$\mathfrak{g}$$. Let $$\pi:\mathfrak{g}\rightarrow\mathrm{End}(V)$$ be an irreducible representation of $$\mathfrak{g}$$ over an algebraically closed field. The universal property of the universal enveloping algebra ensures that $$\pi$$ extends to a representation of $$U(\mathfrak{g})$$ acting on the same vector space. It follows from the second part of Schur's lemma that if $$x$$ belongs to the center of $$U(\mathfrak{g})$$, then $$\pi(x)$$ must be a multiple of the identity operator. In the case when $$\mathfrak{g}$$ is a complex semisimple Lie algebra, an important example of the preceding construction is the one in which $$x$$ is the (quadratic) Casimir element $$C$$. In this case, $$\pi(C)=\lambda_\pi I$$, where $$\lambda_\pi$$ is a constant that can be computed explicitly in terms of the highest weight of $$\pi$$. The action of the Casimir element plays an important role in the proof of complete reducibility for finite-dimensional representations of semisimple Lie algebras.

Generalization to non-simple modules
The one module version of Schur's lemma admits generalizations involving modules M that are not necessarily simple. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

A module is said to be strongly indecomposable if its endomorphism ring is a local ring. For the important class of modules of finite length, the following properties are equivalent :
 * A module M is indecomposable;
 * M is strongly indecomposable;
 * Every endomorphism of M is either nilpotent or invertible.

In general, Schur's lemma cannot be reversed: there exist modules that are not simple, yet their endomorphism algebra is a division ring. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers. Even for group rings, there are examples when the characteristic of the field divides the order of the group: the Jacobson radical of the projective cover of the one-dimensional representation of the alternating group A5 over the finite field with three elements F3 has F3 as its endomorphism ring.

title: Schur's lemma


text:
Ferdinand III

Ferdinand III may refer to:

 * Ferdinand III of Castile (died 1252), the Saint (1199–1252, king from 1217)
 * Ferdinand III of Naples, the Catholic (1452–1516, king from 1504) (Ferdinand V of Castile and Ferdinand II of Aragon and of Sicily), husband of Isabella of Castile
 * Ferdinand III, Holy Roman Emperor (1608–1657, emperor from 1637)
 * Ferdinand III of Sicily (1751–1825, king 1759–1816), (Ferdinand IV of Naples 1759–1799; 1799–1806; 1815–1816 and Ferdinand I of the Two Sicilies 1816–1825)
 * Ferdinand III, Grand Duke of Tuscany (1769–1824, grand-duke 1791–1799 and 1814–1824)
 * Ferdinand III of Navarre (1784–1833), (Ferdinand VII of Castille)

Ferdinando (regantoj)

title: Ferdinand III