Robotics conventions - Revision history

2001:638:208:3603:18A:9BDB:64E7:C9B6 at 08:02, 1 April 2014

2014-04-01T08:02:39Z

← Older revision		Revision as of 10:02, 1 April 2014
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	~~I would like to introduce myself to you~~, ~~I am Jayson Simcox but I don't like when individuals use my full~~ title~~. Kentucky~~ is ~~exactly [http://ltreme.com/index.php?do=/profile-127790/info/ love psychic] where I've usually been residing~~. The preferred ~~hobby~~ for him and ~~[http://www.taehyuna.net/xe/?document_srl=78721 real psychics]~~ his ~~children~~ is ~~to perform lacross~~ and he'll be ~~starting some thing~~ else alongside with it. ~~He is an order clerk and it~~'~~s some thing he truly enjoy~~.<br><br>~~Feel free to visit~~ my web-site ~~real psychics~~ ([http://~~www~~.~~publicpledge~~.com/~~blogs~~/~~post~~/~~7034 updated blog post~~])		Hi there. Allow me begin by introducing the author, her title is Sophia. The preferred pastime for him and his kids is style and he'll be beginning something else alongside with it. I am currently a journey agent. I've usually [http://cartoonkorea.com/ce002/1093612 phone psychic] loved living in Kentucky but now I'm considering other options.<br><br>my web site :: psychic love [http://165.132.39.93/xe/visitors/372912 free tarot readings] ([http://formalarmour.com/index.php?do=/profile-26947/info/ http://formalarmour.com/index.php?do=/profile-26947/info])

en>Myasuda: /* Plücker coordinates */ sp

2014-03-05T02:13:38Z

Plücker coordinates: sp

← Older revision		Revision as of 04:13, 5 March 2014
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	'''Brauer's main theorems''' are three theorems in [[representation theory of finite groups]] linking the [[modular representation theory\|blocks]] of a [[finite group]] (in characteristic ''p'') with those of its [[p-local subgroup\|''p''-local subgroups]], that is to ~~say, the [[normalizer]]s of its non-trivial ''p''-subgroups.~~		I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my full title. Kentucky is exactly [http://ltreme.com/index.php?do=/profile-127790/info/ love psychic] where I've usually been residing. The preferred hobby for him and [http://www.taehyuna.net/xe/?document_srl=78721 real psychics] his children is to perform lacross and he'll be starting some thing else alongside with it. He is an order clerk and it's some thing he truly enjoy.<br><br>Feel free to visit my web-site real psychics ([http://www.publicpledge.com/blogs/post/7034 updated blog post])

	The second and third main theorems allow refinements of orthogonality relations for [[character theory\|ordinary character]]s which may be applied in finite [[group theory]]. These do not presently admit a proof purely in terms of ordinary characters.
	~~All three main theorems are stated in terms of the '''Brauer correspondence'''.~~

	~~==Brauer correspondence==~~
	~~There are many ways~~ to ~~extend the definition which follows~~, but ~~this is close to the early treatments~~
	~~by Brauer. Let ''G'' be a finite group, ''p'' be a prime, ''F'' be a ''field'' of characteristic ''p'~~'.
	~~Let ''H'' be a subgroup of ''G'' which contains~~

	~~:<math>QC_G(Q)</math>~~

	~~for some ''p''-subgroup ''Q''~~
	~~of ''G,'' and~~ is ~~contained in the normalizer~~

	:~~<math>N_G(Q)<~~/~~math>.~~

	The '''Brauer homomorphism''' (with respect to ''H'') is a linear map from the center of the group algebra of ''G'' over ''F'' to the corresponding algebra for ''H''. Specifically, it is the restriction to
	~~<math>Z(FG)</math> of the (linear) projection from <math>FG<~~/~~math> to <math>FC_G(Q)</math> whose~~
	~~kernel is spanned by the elements of ''G'' outside <math>C_G(Q)</math>~~. ~~The image of this map is contained in~~
	~~<math>Z(FH)<~~/~~math>, and it transpires that the map is also a ring homomorphism~~.

	~~Since it is a [[ring homomorphism]], for any block ''B'' of ''FG'', the Brauer homomorphism~~
	~~sends the identity element of ''B'' either to ''0'' or to an idempotent element. In the latter case,~~
	~~the idempotent may be decomposed as a sum of (mutually orthogonal) [[primitive idempotent]]s of ''Z(FH).''~~
	~~Each of these primitive idempotents is the multiplicative identity of some block of ''FH.'' The block ''b'' of ''FH'' is said to be a '''Brauer correspondent''' of ''B'' if its identity element occurs~~
	~~in this decomposition of the image of the identity of ''B'' under the Brauer homomorphism.~~

	~~==Brauer's first main theorem==~~

	~~Brauer's first main theorem {{harvs\|last=Brauer\|year1=1944\|year2=1956\|year3~~=~~1970}} states that if <math>G<~~/~~math> is a finite group a <math>D</math> is a <math>p<~~/~~math>-subgroup of <math>G<~~/~~math>, then there is a [[bijection~~]~~] between the set of~~
	~~(characteristic~~ '~~'p'') blocks of <math>G</math> with defect group <math>D</math>~~ and ~~blocks of the normalizer <math>N_G(D)<~~/~~math> with~~
	~~defect group ''D''. This bijection arises because when <math>H = N_G(D)<~~/~~math>, each block of ''G''~~
	~~with defect group ''D'' has a unique Brauer correspondent block of ''H'', which also has defect~~
	~~group ''D''~~.

	~~==Brauer's second main theorem==~~

	Brauer's second main theorem {{harvs\|last=Brauer\|year1=1944\|year2=1959}} gives, for an element ''t'' whose order is a power of a prime ''p'', a criterion for a (characteristic ''p'') block of <math>C_G(t)</math> to correspond to a given block of <math>G</math>, via ''generalized decomposition numbers''. ~~These are the coefficients which occur when the restrictions of ordinary characters of <math>G<~~/~~math> (from the given block) to elements of the form ''tu'', where ''u'' ranges over elements of order prime to ''p'' in <math>C_G(t)<~~/~~math>, are written as linear combinations of the irreducible [[modular representation theory\|Brauer character]~~]~~s of <math>C_G(t)</math>. The content of the theorem~~ is ~~that it is only necessary~~ to ~~use Brauer characters from blocks of <math>C_G(t)</math> which are Brauer correspondents of the chosen block of ''G'~~'.

	~~==Brauer's third main theorem==~~

	~~Brauer's third main theorem {{harv\|Brauer\|1964\|loc=theorem3}} states that when ''Q''~~ is ~~a ''p''-subgroup of the finite group ''G'',~~
	and '~~'H'' is a subgroup of ''G,'' containing <math>QC_G(Q)</math>, and contained in~~ <~~math~~>~~N_G(Q)~~<~~/math~~>,
	~~then the [[modular representation theory\|principal block]] of ''H'' is the only Brauer correspondent of the principal block of ''G'' (where the blocks referred~~ to ~~are calculated in characteristic ''p'').~~

	~~==References==~~

	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On the arithmetic in a group ring \| jstor=87919 \| mr=0010547 \| year=1944 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=30 \| pages=109–114}}
	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On blocks of characters of groups of finite order I \| jstor=87578 \| mr=0016418 \| year=1946 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=32 \| pages=182–186}}
	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On blocks of characters of groups of finite order. II \| jstor=87838 \| mr=0017280 \| year=1946 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=32 \| pages=215–219}}
	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=Zur Darstellungstheorie der Gruppen endlicher Ordnung \| doi=10.1007/BF01187950 \| mr=0075953 \| year=1956 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=63 \| pages=406–444}}
	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=Zur Darstellungstheorie der Gruppen endlicher Ordnung. II \| doi=10.1007/BF01162934 \| mr=0108542 \| year=1959 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=72 \| pages=25–46}}
	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=Some applications of the theory of blocks of characters of finite groups. I \| doi=10.1016/0021-8693(~~64)90031-6 \| mr=0168662 \| year=1964 \| journal=~~[~~[Journal of Algebra]] \| issn=0021-8693 \| volume=1 \| pages=152–167}}~~
	*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On the first main theorem on blocks of characters of finite groups. \| url=http://~~projecteuclid.org/euclid.ijm/1256053174 \| mr=0267010 \| year=1970 \| journal=Illinois Journal of Mathematics \| issn=0019-2082 \| volume=14 \| pages=183–187}}~~
	*{{Citation \| last1=Dade \| first1=Everett C. \| author1-link=Everett C. Dade \| editor1-last=Powell \| editor1-first=M. B. \| editor2-last=Higman \| editor2-first=Graham \| editor2-link=Graham Higman \| title=Finite simple groups. ~~Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969~~. \| publisher=[[Academic Press]] \| location=Boston, MA \| isbn=978-0-12-563850-0 \| mr=0360785 \| year=1971 \| chapter=Character theory pertaining to finite simple groups \| pages=249–327}} gives a detailed proof of the Brauer's main theorems.
	*{{eom\|id=b/~~b120440\|first=H.\|last= Ellers\|title=Brauer's first main theorem}}~~
	*{{eom\|id=b/~~b120450\|first=H.\|last= Ellers\|title=Brauer height-zero conjecture}}~~
	*{{eom\|id=b/b120460\|first=H.\|last= Ellers\|title=Brauer's second main theorem}}
	*{{eom\|id=b/~~b120470\|first=H.\|last= Ellers\|title=Brauer's third main theorem}}~~

	* [[Walter Feit]], ''The representation theory of finite groups.'' North-Holland Mathematical Library, 25. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+502 pp. ISBN 0-444-86155-6

	~~[[Category:Representation theory of finite groups]]~~
	~~[[Category:Theorems in representation theory]~~]

en>Hiaslee: add Link to Plücker Coodinates

2013-11-09T20:32:47Z

add Link to Plücker Coodinates

← Older revision		Revision as of 22:32, 9 November 2013
Line 1:		Line 1:
	The ~~author~~ is ~~known as Wilber Pegues~~. ~~Her family members life~~ in ~~Alaska but her husband desires them~~ to ~~move~~. ~~To perform lacross~~ is ~~something he would~~ by ~~no means give up~~. ~~Distributing production~~ is ~~how he tends~~ to ~~make~~ a ~~living~~.<br><br>~~Also visit my blog: spirit messages~~ ([http://~~appin~~.co.kr/~~board_Zqtv22~~/~~688025 view site…~~])		'''Brauer's main theorems''' are three theorems in [[representation theory of finite groups]] linking the [[modular representation theory\|blocks]] of a [[finite group]] (in characteristic ''p'') with those of its [[p-local subgroup\|''p''-local subgroups]], that is to say, the [[normalizer]]s of its non-trivial ''p''-subgroups.

			The second and third main theorems allow refinements of orthogonality relations for [[character theory\|ordinary character]]s which may be applied in finite [[group theory]]. These do not presently admit a proof purely in terms of ordinary characters.
			All three main theorems are stated in terms of the '''Brauer correspondence'''.

			==Brauer correspondence==
			There are many ways to extend the definition which follows, but this is close to the early treatments
			by Brauer. Let ''G'' be a finite group, ''p'' be a prime, ''F'' be a ''field'' of characteristic ''p''.
			Let ''H'' be a subgroup of ''G'' which contains

			:<math>QC_G(Q)</math>

			for some ''p''-subgroup ''Q''
			of ''G,'' and is contained in the normalizer

			:<math>N_G(Q)</math>.

			The '''Brauer homomorphism''' (with respect to ''H'') is a linear map from the center of the group algebra of ''G'' over ''F'' to the corresponding algebra for ''H''. Specifically, it is the restriction to
			<math>Z(FG)</math> of the (linear) projection from <math>FG</math> to <math>FC_G(Q)</math> whose
			kernel is spanned by the elements of ''G'' outside <math>C_G(Q)</math>. The image of this map is contained in
			<math>Z(FH)</math>, and it transpires that the map is also a ring homomorphism.

			Since it is a [[ring homomorphism]], for any block ''B'' of ''FG'', the Brauer homomorphism
			sends the identity element of ''B'' either to ''0'' or to an idempotent element. In the latter case,
			the idempotent may be decomposed as a sum of (mutually orthogonal) [[primitive idempotent]]s of ''Z(FH).''
			Each of these primitive idempotents is the multiplicative identity of some block of ''FH.'' The block ''b'' of ''FH'' is said to be a '''Brauer correspondent''' of ''B'' if its identity element occurs
			in this decomposition of the image of the identity of ''B'' under the Brauer homomorphism.

			==Brauer's first main theorem==

			Brauer's first main theorem {{harvs\|last=Brauer\|year1=1944\|year2=1956\|year3=1970}} states that if <math>G</math> is a finite group a <math>D</math> is a <math>p</math>-subgroup of <math>G</math>, then there is a [[bijection]] between the set of
			(characteristic ''p'') blocks of <math>G</math> with defect group <math>D</math> and blocks of the normalizer <math>N_G(D)</math> with
			defect group ''D''. This bijection arises because when <math>H = N_G(D)</math>, each block of ''G''
			with defect group ''D'' has a unique Brauer correspondent block of ''H'', which also has defect
			group ''D''.

			==Brauer's second main theorem==

			Brauer's second main theorem {{harvs\|last=Brauer\|year1=1944\|year2=1959}} gives, for an element ''t'' whose order is a power of a prime ''p'', a criterion for a (characteristic ''p'') block of <math>C_G(t)</math> to correspond to a given block of <math>G</math>, via ''generalized decomposition numbers''. These are the coefficients which occur when the restrictions of ordinary characters of <math>G</math> (from the given block) to elements of the form ''tu'', where ''u'' ranges over elements of order prime to ''p'' in <math>C_G(t)</math>, are written as linear combinations of the irreducible [[modular representation theory\|Brauer character]]s of <math>C_G(t)</math>. The content of the theorem is that it is only necessary to use Brauer characters from blocks of <math>C_G(t)</math> which are Brauer correspondents of the chosen block of ''G''.

			==Brauer's third main theorem==

			Brauer's third main theorem {{harv\|Brauer\|1964\|loc=theorem3}} states that when ''Q'' is a ''p''-subgroup of the finite group ''G'',
			and ''H'' is a subgroup of ''G,'' containing <math>QC_G(Q)</math>, and contained in <math>N_G(Q)</math>,
			then the [[modular representation theory\|principal block]] of ''H'' is the only Brauer correspondent of the principal block of ''G'' (where the blocks referred to are calculated in characteristic ''p'').

			==References==

			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On the arithmetic in a group ring \| jstor=87919 \| mr=0010547 \| year=1944 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=30 \| pages=109–114}}
			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On blocks of characters of groups of finite order I \| jstor=87578 \| mr=0016418 \| year=1946 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=32 \| pages=182–186}}
			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On blocks of characters of groups of finite order. II \| jstor=87838 \| mr=0017280 \| year=1946 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| issn=0027-8424 \| volume=32 \| pages=215–219}}
			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=Zur Darstellungstheorie der Gruppen endlicher Ordnung \| doi=10.1007/BF01187950 \| mr=0075953 \| year=1956 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=63 \| pages=406–444}}
			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=Zur Darstellungstheorie der Gruppen endlicher Ordnung. II \| doi=10.1007/BF01162934 \| mr=0108542 \| year=1959 \| journal=[[Mathematische Zeitschrift]] \| issn=0025-5874 \| volume=72 \| pages=25–46}}
			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=Some applications of the theory of blocks of characters of finite groups. I \| doi=10.1016/0021-8693(64)90031-6 \| mr=0168662 \| year=1964 \| journal=[[Journal of Algebra]] \| issn=0021-8693 \| volume=1 \| pages=152–167}}
			*{{Citation \| last1=Brauer \| first1=R. \| author1-link=Richard Brauer \| title=On the first main theorem on blocks of characters of finite groups. \| url=http://projecteuclid.org/euclid.ijm/1256053174 \| mr=0267010 \| year=1970 \| journal=Illinois Journal of Mathematics \| issn=0019-2082 \| volume=14 \| pages=183–187}}
			*{{Citation \| last1=Dade \| first1=Everett C. \| author1-link=Everett C. Dade \| editor1-last=Powell \| editor1-first=M. B. \| editor2-last=Higman \| editor2-first=Graham \| editor2-link=Graham Higman \| title=Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. \| publisher=[[Academic Press]] \| location=Boston, MA \| isbn=978-0-12-563850-0 \| mr=0360785 \| year=1971 \| chapter=Character theory pertaining to finite simple groups \| pages=249–327}} gives a detailed proof of the Brauer's main theorems.
			*{{eom\|id=b/b120440\|first=H.\|last= Ellers\|title=Brauer's first main theorem}}
			*{{eom\|id=b/b120450\|first=H.\|last= Ellers\|title=Brauer height-zero conjecture}}
			*{{eom\|id=b/b120460\|first=H.\|last= Ellers\|title=Brauer's second main theorem}}
			*{{eom\|id=b/b120470\|first=H.\|last= Ellers\|title=Brauer's third main theorem}}

			* [[Walter Feit]], ''The representation theory of finite groups.'' North-Holland Mathematical Library, 25. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+502 pp. ISBN 0-444-86155-6

			[[Category:Representation theory of finite groups]]
			[[Category:Theorems in representation theory]]

en>Michael Hardy: /* Denavit–Hartenberg line coordinates */

2011-10-16T15:32:18Z

Denavit–Hartenberg line coordinates

New page

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