en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q5282582

2013-08-13T02:57:36Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q5282582

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	~~Wilber Berryhill~~ is ~~what his wife loves~~ to ~~call him and he completely loves this title~~. ~~North Carolina~~ is where we'~~ve been living for many years~~ and ~~will never move~~. ~~I am presently~~ a ~~travel agent~~. ~~To play lacross~~ is 1 of the ~~issues she enjoys most.~~<br><br>~~Also visit my web page ::~~ [http://~~isaworld~~.pe.kr/?~~document_srl~~=~~392088 clairvoyants~~]		{{for\|the more general Hall algebra of a category\|Ringel–Hall algebra}}
			In [[mathematics]], the '''Hall algebra''' is an [[associative algebra]] with a basis corresponding to isomorphism classes of finite abelian ''p''-groups. It was first discussed by E. {{harvtxt\|Steinitz\|1901}} but forgotten until it was rediscovered by {{harvs\|txt\|authorlink=Philip Hall\|first= Philip\|last= Hall\|year=1959}}, both of whom published no more than brief summaries of their work. The '''Hall polynomials''' are the [[structure constants]] of the '''Hall algebra'''. The Hall algebra plays an important role in the theory of [[Masaki Kashiwara\|Kashiwara]]–[[George Lusztig\|Lusztig]]'s [[crystal basis\|canonical bases]] in [[quantum group]]s. {{harvtxt\|Ringel\|1990}} generalized Hall algebras to more general categories, such as the category of representations of a [[quiver (mathematics)\|quiver]].

			== Construction ==
			A [[finite set\|finite]] [[abelian group\|abelian]] [[p-group\|''p''-group]] ''M'' is a direct sum of [[cyclic group\|cyclic]] ''p''-power components <math>C_{p^{\lambda_i}},</math> where
			<math>\lambda=(\lambda_1,\lambda_2,\ldots)</math> is a [[Partition (number theory)\|partition]] of <math>n</math> called the ''type'' of ''M''. Let <math>g^\lambda_{\mu,\nu}(p)</math> be the number of subgroups ''N'' of ''M'' such that ''N'' has type <math>\nu</math> and the quotient ''M/N'' has type <math>\mu</math>. Hall proved that the functions ''g'' are [[polynomial]] functions of ''p'' with integer coefficients. Thus we may replace ''p'' with an indeterminate ''q'', which results in the '''Hall polynomials'''

			: <math>g^\lambda_{\mu,\nu}(q)\in\mathbb{Z}[q]. \, </math>

			Hall next constructs an [[associative ring]] <math>H</math> over <math>\mathbb{Z}[q]</math>, now called the '''Hall algebra'''. This ring has a basis consisting of the symbols <math>u_\lambda</math> and the structure constants of the multiplication in this basis are given by the Hall polynomials:

			:<math> u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu}(q) u_\lambda. \, </math>

			It turns out that ''H'' is a commutative ring, freely generated by the elements <math>u_{\mathbf1^n}</math> corresponding to the [[elementary abelian group\|elementary ''p''-groups]]. The linear map from ''H'' to the algebra of [[symmetric function]]s defined on the generators by the formula

			: <math>u_{\mathbf 1^n} \mapsto q^{-n(n-1)}e_n \, </math>

			(where ''e''<sub>''n''</sub> is the ''n''th [[elementary symmetric function]]) uniquely extends to a [[ring homomorphism]] and the images of the basis elements <math>u_\lambda</math> may be interpreted via the [[Hall–Littlewood polynomial\|Hall–Littlewood symmetric functions]]. Specializing ''q'' to 1, these symmetric functions become [[Schur function]]s, which are thus closely connected with the theory of Hall polynomials.

			==References==
			*{{citation\|first=Philip\|last=Hall\|authorlink=Philip Hall\|
			year=1959\|chapter=The algebra of partitions\|title=Proceedings of the 4th Canadian mathematical congress, Banff\|pages=147–159}}
			* [[George Lusztig]], ''Quivers, perverse sheaves, and quantized enveloping algebras.'' J. Amer. Math. Soc. 4 (1991), no. 2, 365–421.
			* {{Citation \| last1=Macdonald \| first1=I. G. \| author1-link=Ian G. Macdonald \| title=Symmetric functions and Hall polynomials \| url=http://www.oup.com/uk/catalogue/?ci=9780198504504 \| publisher=The Clarendon Press Oxford University Press \| edition=2nd \| series=Oxford Mathematical Monographs \| isbn=978-0-19-853489-1 \| mr=1354144 \| year=1995}}
			* {{Citation \| last1=Ringel \| first1=Claus Michael \| title=Hall algebras and quantum groups \| doi=10.1007/BF01231516 \| mr=1062796 \| year=1990 \| journal=[[Inventiones Mathematicae]] \| volume=101 \| issue=3 \| pages=583–591}}
			*{{citation\|arxiv=math/0611617\|title=Lectures on Hall algebras\|last=Schiffmann\|first=O\|year=2006}}
			*{{citation\|authorlink=Ernst Steinitz\|last=Steinitz\|first=E.\|title=Zur Theorie der Abel'schen Gruppen\|journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]]\|year=1901\|volume=9\|pages=80–85}}

			[[Category:Algebras]]
			[[Category:Invariant theory]]
			[[Category:Symmetric functions]]

en>David Eppstein: explain terminological confusion

2009-04-21T01:22:07Z

explain terminological confusion

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Wilber Berryhill is what his wife loves to call him and he completely loves this title. North Carolina is where we've been living for many years and will never move. I am presently a travel agent. To play lacross is 1 of the issues she enjoys most.<br><br>Also visit my web page :: [http://isaworld.pe.kr/?document_srl=392088 clairvoyants]

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en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q5282582

en>David Eppstein: explain terminological confusion