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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>
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| == Construction ==
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| 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>
| | These fuzzies may be a small annoyance, but they can make your gym seem unkempt. Normal pedaling motion disengages the brake cane, allowing for normal pedaling. As a result of this various protective gear manufacturers have also ventured into production of improved design of padded shorts. Alison Addy is the author of many articles on subjects like mountain bike crashes and published at. When shopping for different bike accessories, parts, or whatever it is you are looking for, being a smart shopper is always in order. <br><br>If you are using linear pull brakes with drop levers, you must use a lever that is designed to function with the increased mechanical advantage that linear pull brakes afford. By just taking a few minutes to make sure your bike is well-tuned before you set out, you can keep from spending hours on issues later. So how exactly would you go about making this dream a reality. If biking is your passion or your favorite sport, then you know the worth of good accessories while riding. If you are riding through a rock garden, for instance, you don't want a fork that is bouncing you all over the place. <br><br>Technological know-how is advancing very rapidly and manufactures have observed ways to make batteries much more efficient and Eco-pleasant at the very same time along with photo voltaic PV cells getting additional and a lot more effortless and economical to make. Equally expensive can be the tag attached to a lighter-weighted bike. People are buying bicycles to take care of their fitness as many go riding every morning or on weekends instead of jogging. Explore, eat, and shop in places you typicaly wouldn't. However, be careful that any loose ends (from straps or laces) and buckles don't have over, as they can pose a safety hazard if you elect to use toe clips. <br><br>Decide how and where you want to ride, then choose a mountain bike that fits your personal style, rather than that of the sales staff. Find more information about Hardtail mountain bikes here. But older children may have more adventurous tastes, and want for toys that are - well, still miniatures, but are flashier, more expensive, and more functional. Mountain bikes are equipped with knobby tires, large number of gears and handle bars that are wider; they also include shocks and a much stronger suspension. But that wasn't the bad part 'cause, see, those two almost invisible grey hairs were only a distraction from the real culprit. <br><br>When you have almost any concerns concerning exactly where in addition to the way to utilize [http://www.tipsofthedays.com/profile/stlus Biking for modern life mountain bike sizing.], you possibly can e mail us in our webpage. Getting stuck in a trap is one way beginners kill their scores. The circumstances will dictate which method to use. Mountain biking amongst the majestic backdrop of the great mountains of Kerala is an unforgettable and thrilling experience. Hence, XC suspension forks tend to have a short travel and are light weight. The bike retailer staff will likely be more then glad to expand on and point out the advantages and disadvantages of each design. |
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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
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| : <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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These fuzzies may be a small annoyance, but they can make your gym seem unkempt. Normal pedaling motion disengages the brake cane, allowing for normal pedaling. As a result of this various protective gear manufacturers have also ventured into production of improved design of padded shorts. Alison Addy is the author of many articles on subjects like mountain bike crashes and published at. When shopping for different bike accessories, parts, or whatever it is you are looking for, being a smart shopper is always in order.
If you are using linear pull brakes with drop levers, you must use a lever that is designed to function with the increased mechanical advantage that linear pull brakes afford. By just taking a few minutes to make sure your bike is well-tuned before you set out, you can keep from spending hours on issues later. So how exactly would you go about making this dream a reality. If biking is your passion or your favorite sport, then you know the worth of good accessories while riding. If you are riding through a rock garden, for instance, you don't want a fork that is bouncing you all over the place.
Technological know-how is advancing very rapidly and manufactures have observed ways to make batteries much more efficient and Eco-pleasant at the very same time along with photo voltaic PV cells getting additional and a lot more effortless and economical to make. Equally expensive can be the tag attached to a lighter-weighted bike. People are buying bicycles to take care of their fitness as many go riding every morning or on weekends instead of jogging. Explore, eat, and shop in places you typicaly wouldn't. However, be careful that any loose ends (from straps or laces) and buckles don't have over, as they can pose a safety hazard if you elect to use toe clips.
Decide how and where you want to ride, then choose a mountain bike that fits your personal style, rather than that of the sales staff. Find more information about Hardtail mountain bikes here. But older children may have more adventurous tastes, and want for toys that are - well, still miniatures, but are flashier, more expensive, and more functional. Mountain bikes are equipped with knobby tires, large number of gears and handle bars that are wider; they also include shocks and a much stronger suspension. But that wasn't the bad part 'cause, see, those two almost invisible grey hairs were only a distraction from the real culprit.
When you have almost any concerns concerning exactly where in addition to the way to utilize Biking for modern life mountain bike sizing., you possibly can e mail us in our webpage. Getting stuck in a trap is one way beginners kill their scores. The circumstances will dictate which method to use. Mountain biking amongst the majestic backdrop of the great mountains of Kerala is an unforgettable and thrilling experience. Hence, XC suspension forks tend to have a short travel and are light weight. The bike retailer staff will likely be more then glad to expand on and point out the advantages and disadvantages of each design.