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| In mathematics, '''Schur algebras''', named after [[Issai Schur]], are certain finite-dimensional [[associative algebra|algebras]] closely associated with [[Schur–Weyl duality]] between [[general linear group|general linear]] and [[symmetric group|symmetric]] groups. They are used to relate the [[representation theory|representation theories]] of those two [[group (mathematics)|groups]]. Their use was promoted by the influential monograph of [[J. A. Green]] first published in 1980.<ref>[[J. A. Green]], ''Polynomial Representations of GL<sub>n</sub>'', Springer Lecture Notes 830, Springer-Verlag 1980. {{MR|2349209}}, ISBN 978-3-540-46944-5, ISBN 3-540-46944-3</ref> The name "Schur algebra" is due to Green. In the modular case (over infinite [[field (mathematics)|fields]] of positive characteristic) Schur algebras were used by Gordon James and Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent.<ref>Karin Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules. ''Journal of Algebra'' 180 (1996), 316–320. {{doi|10.1006/jabr.1996.0067}} {{MR|1375581}}</ref> Schur algebras were used by Friedlander and [[Andrei Suslin|Suslin]] to prove finite generation of [[cohomology]] of finite [[group scheme]]s.<ref>[[Eric Friedlander]] and [[Andrei Suslin]], Cohomology of finite group schemes over a field. ''Inventiones Mathematicae'' 127 (1997), 209--270. {{MR|1427618}} {{doi|10.1007/s002220050119}}</ref> | | In Candy Crush Saga, the player have to score factors by matching game pieces (stylized as candy) of the same kind horizontally or vertically, by swapping adjacent pieces. If you play [http://candycrushsagacheatsx.wordpress.com/ Candy Crush Saga Cheats] you will know how utterly addictive this sweet, colorful Swedish game is. From licorice swirls to wrapped candy to chocolate fountains and some others, this mouth-watering, sugary sweet game amuses and will take our minds off stressful points. When you match three, four and five candies up in a row, the explosion of fruity candy is scintillating and other candy drops down. For instance: a piece of candy with vertical stripes will clear the row vertically, best to bottom.<br><br> |
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| == Construction ==
| | Often you may well also be essential to swap two electrical power candies with just about every other as a requisite to clear the stage, other Candy Crush Saga phases call for you to clear the way for cherries and hazelnuts to filter down to the bottom. In addition to electrical power candies, you also have electrical power-ups for sale to enable you, such as a +five moves when you have reached the finish of the level and is brief of moves to finish it. Resist the urge to unleash the energy candy as and when you have one particular.<br><br>This guide will give fantastic Candy Crush Saga ideas and procedures and techniques which will help you comprehensive the game. Browse by means of video walkthroughs, game reviews, and see a list of tweets about Candy Crush Saga in a chatroom. With Candy Crush Saga Hints Cheats, consumers can come across suggestions on how to make [http://en.search.wordpress.com/?q=sweet+combinations sweet combinations] and quit chocolate and pause timed amounts with out making use of boosters. Level 425 appears to be a difficult 1. |
| The Schur algebra <math>S_k(n, r)</math> can be defined for any [[commutative ring]] <math>k</math> and integers <math>n, r \geq 0</math>. Consider the [[associative algebra|algebra]] <math>k[x_{ij}]</math> of [[polynomials]] (with coefficients in <math>k</math>) in <math>n^2</math> commuting variables <math>x_{ij}</math>, 1 ≤ ''i'', ''j'' ≤ <math>n</math>. Denote by <math>A_k(n, r)</math> the homogeneous polynomials of degree <math>r</math>. Elements of <math>A_k(n, r)</math> are ''k''-linear combinations of monomials formed by multiplying together <math>r</math> of the generators <math>x_{ij}</math> (allowing repetition). Thus
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| : <math>k[x_{ij}] = \bigoplus_{r\ge 0} A_k(n, r).</math>
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| Now, <math>k[x_{ij}]</math> has a natural [[coalgebra]] structure with comultiplication <math>\Delta</math> and counit <math>\varepsilon</math> the algebra homomorphisms given on generators by
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| : <math> \Delta(x_{ij}) = \textstyle\sum_l x_{il} \otimes x_{lj}, \quad \varepsilon(x_{ij}) = \delta_{ij}\quad\ </math> ([[Kronecker delta|Kronecker’s delta]]).
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| Since comultiplication is an algebra homomorphism, <math>k[x_{ij}]</math> is a [[bialgebra]]. One easily
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| checks that <math>A_k(n, r)</math> is a subcoalgebra of the bialgebra <math>k[x_{ij}]</math>, for every ''r'' ≥ 0.
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| '''Definition.''' The Schur algebra (in degree <math>r</math>) is the algebra <math>S_k (n, r) = \mathrm{Hom}_k( A_k (n, r), k)</math>. That is, <math>S_k(n,r)</math> is the linear dual of <math>A_k(n,r)</math>.
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| It is a general fact that the linear [[dual space|dual]] of a coalgebra <math>A</math> is an algebra in a natural way, where the multiplication in the algebra is induced by dualizing the comultiplication in the coalgebra. To see this, let
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| : <math>\Delta(a) = \textstyle \sum a_i \otimes b_i</math>
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| and, given linear functionals <math>f</math>, <math>g</math> on <math>A</math>, define their product to be the linear functional given by | |
| : <math>\textstyle a \mapsto \sum f(a_i) g(b_i).</math>
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| The identity element for this multiplication of functionals is the counit in <math>A</math>.
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| == Main properties ==
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| * One of the most basic properties expresses <math>S_k(n,r)</math> as a centralizer algebra. Let <math>V = k^n</math> be the space of rank <math>n</math> column vectors over <math>k</math>, and form the [[tensor product|tensor]] power
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| : <math>V^{\otimes r} = V \otimes \cdots \otimes V \quad (r\text{ factors}). \, </math>
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| Then the [[symmetric group]] <math>\mathfrak{S}_r</math> on <math>r</math> letters acts naturally on the tensor space by place permutation, and one has an isomorphism
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| : <math>S_k(n,r) \cong \mathrm{End}_{\mathfrak{S}_r} (V^{\otimes r}).</math>
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| In other words, <math>S_k(n,r)</math> may be viewed as the algebra of [[endomorphisms]] of tensor space commuting with the action of the [[symmetric group]].
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| * <math>S_k(n,r)</math> is free over <math>k</math> of rank given by the [[binomial coefficient]] <math>\tbinom{n^2+r-1}{r}</math>.
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| * Various bases of <math>S_k(n,r)</math> are known, many of which are indexed by pairs of semistandard [[Young tableau]]x of shape <math>\lambda</math>, as <math>\lambda</math> varies over the set of [[partition (number theory)|partitions]] of <math>r</math> into no more than <math>n</math> parts.
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| * In case ''k'' is an infinite field, <math>S_k(n,r)</math> may also be identified with the enveloping algebra (in the sense of H. Weyl) for the action of the [[general linear group]] <math>\mathrm{GL}_n(k)</math> acting on tensor space (via the diagonal action on tensors, induced from the natural action of <math>\mathrm{GL}_n(k)</math> on <math>V = k^n</math> given by matrix multiplication).
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| * Schur algebras are "defined over the integers". This means that they satisfy the following change of scalars property:
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| : <math>S_k(n,r) \cong S_{\mathbb{Z}}(n,r) \otimes _{\mathbb{Z}} k</math>
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| :for any commutative ring <math>k</math>.
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| * Schur algebras provide natural examples of quasihereditary algebras<ref>Edward Cline, Brian Parshall, and Leonard Scott, Finite-dimensional algebras and highest weight categories. ''Journal für die Reine und Angewandte Mathematik'' [Crelle's Journal] 391 (1988), 85–99. {{MR|0961165}}</ref> (as defined by Cline, Parshall, and Scott), and thus have nice [[homological algebra|homological]] properties. In particular, Schur algebras have finite [[global dimension]].
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| == Generalizations ==
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| * '''Generalized Schur algebras''' (associated to any reductive [[algebraic group]]) were introduced by Donkin in the 1980s.<ref>Stephen Donkin, On Schur algebras and related algebras, I. ''Journal of Algebra'' 104 (1986), 310–328. {{doi|10.1016/0021-8693(86)90218-8}} {{MR|0866778}}</ref> These are also quasihereditary.
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| * Around the same time, Dipper and James<ref>Richard Dipper and Gordon James, The q-Schur algebra. ''Proceedings of the London Math. Society'' (3) 59 (1989), 23–50. {{doi|10.1112/plms/s3-59.1.23}} {{MR|0997250}}</ref> introduced the '''quantized Schur algebras''' (or '''q-Schur algebras''' for short), which are a type of q-deformation of the classical Schur algebras described above, in which the symmetric group is replaced by the corresponding [[Hecke algebra]] and the general linear group by an appropriate [[quantum group]].
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| * There are also '''generalized q-Schur algebras''', which are obtained by generalizing the work of Dipper and James in the same way that Donkin generalized the classical Schur algebras.<ref>Stephen Doty, Presenting generalized q-Schur algebras. ''Representation Theory'' 7 (2003), 196--213 (electronic). {{doi|10.1090/S1088-4165-03-00176-6}}</ref>
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| * There are further generalizations, such as the '''affine q-Schur algebras'''<ref>R. M. Green, The affine q-Schur algebra. ''Journal of Algebra'' 215 (1999), 379--411. {{doi|10.1006/jabr.1998.7753}}</ref> related to affine [[Kac-Moody algebra|Kac-Moody]] [[Lie algebra]]s and other generalizations, such as the '''cyclotomic q-Schur algebras'''<ref>Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic q-Schur algebras. ''Math. Zeitschrift'' 229 (1998), 385--416. {{doi|10.1007/PL00004665}} {{MR|1658581}}</ref> related to Ariki-Koike algebras (which are q-deformations of certain [[complex reflection group]]s).
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| The study of these various classes of generalizations forms an active area of contemporary research.
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| == References ==
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| {{Reflist}}
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| == Further reading ==
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| * Stuart Martin, ''Schur Algebras and Representation Theory'', Cambridge University Press 1993. {{MR|2482481}}, ISBN 978-0-521-10046-5
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| * Andrew Mathas, [http://www.ams.org/bookstore-getitem/item=ULECT-15 Iwahori-Hecke algebras and Schur algebras of the symmetric group], University Lecture Series, vol.15, American Mathematical Society, 1999. {{MR|1711316}}, ISBN 0-8218-1926-7
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| * [[Hermann Weyl]], ''The Classical Groups. Their Invariants and Representations''. Princeton University Press, Princeton, N.J., 1939. {{MR|0000255}}, ISBN 0-691-05756-7
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| [[Category:Algebra]]
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| [[Category:Representation theory]]
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In Candy Crush Saga, the player have to score factors by matching game pieces (stylized as candy) of the same kind horizontally or vertically, by swapping adjacent pieces. If you play Candy Crush Saga Cheats you will know how utterly addictive this sweet, colorful Swedish game is. From licorice swirls to wrapped candy to chocolate fountains and some others, this mouth-watering, sugary sweet game amuses and will take our minds off stressful points. When you match three, four and five candies up in a row, the explosion of fruity candy is scintillating and other candy drops down. For instance: a piece of candy with vertical stripes will clear the row vertically, best to bottom.
Often you may well also be essential to swap two electrical power candies with just about every other as a requisite to clear the stage, other Candy Crush Saga phases call for you to clear the way for cherries and hazelnuts to filter down to the bottom. In addition to electrical power candies, you also have electrical power-ups for sale to enable you, such as a +five moves when you have reached the finish of the level and is brief of moves to finish it. Resist the urge to unleash the energy candy as and when you have one particular.
This guide will give fantastic Candy Crush Saga ideas and procedures and techniques which will help you comprehensive the game. Browse by means of video walkthroughs, game reviews, and see a list of tweets about Candy Crush Saga in a chatroom. With Candy Crush Saga Hints Cheats, consumers can come across suggestions on how to make sweet combinations and quit chocolate and pause timed amounts with out making use of boosters. Level 425 appears to be a difficult 1.