Multicriteria classification: Difference between revisions

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The '''Isotypic component''' of weight <math>\lambda</math> of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight <math>\lambda</math>.


==Definition==


* A finite dimensional [[Module (mathematics)|module]] <math>V</math> of a [[reductive Lie algebra]] <math>\mathfrak{g}</math> (or of the corresponding [[Lie group]]) can be decomposed into [[Irreducible representation|irreducible]] submodules
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have been a staple of [http://www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=North+American North American] living. 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These cook in less than 20 minutes and lead to no-mess, lip-smacking results. Children will love the surprise element and can easily switch packets. You can even ask your family for combo suggestions before the barbecue, or draw ingredient names out of a hat to create new meal ideas. <br><br>3) A little variety goes a long way. If you tend to stick to the same old beef burgers and pork wieners, why not try grilled pizza or salmon steaks? Similarly, try to mix it up with condiments and seasoning. Sick of [http://tinyurl.com/kecvhhb ugg boots sale] ketchup? Try salsa instead. Look online where you'll find hundreds of easy recipes for barbecue rubs. These give your food that extra pizzazz and prevent the mess and drip that comes with barbecue sauces. <br><br>4) Barbecues don't have to be rushed. In fact, barbecue devotees will tell you that today's quick broiling is an insult to the term, which should represent a process of slow cooking on low heat. 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:<math> V = \oplus_{i=1}^N V_i </math>.
* Each finite dimensional irreducible representation of <math>\mathfrak{g}</math> is uniquely identified (up to isomorphism) by its [[Weight_(representation_theory)#Highest_weight|highest weight]]
:<math>\forall i \in \{1,\ldots,N\} \exists \lambda \in P(\mathfrak{g}) : V_i \simeq M_\lambda</math>, where <math>M_\lambda</math> denotes the [[Weight_(representation_theory)#Highest_weight_module|highest weight module]] with highest weight <math>\lambda</math>.
* In the decomposition of <math> V </math>, a certain isomorphism class might appear more than once, hence
:<math> V \simeq \oplus_{\lambda \in  P(\mathfrak{g})} ( \oplus_{i=1}^{d_\lambda} M_{\lambda}) </math>.
 
This defines the isotypic component of weight <math>\lambda</math> of V:
<math>\lambda(V) := \oplus_{i=1}^{d_\lambda} V_i \simeq \mathbb{C}^{d_\lambda} \otimes  M_{\lambda}</math>  where <math>d_\lambda</math> is maximal.
 
==See also==
* [[Lie algebra representation]]
* [[Weight (representation theory)]]
 
== References ==
* {{Cite journal
| doi = 10.1016/j.jalgebra.2010.10.031
| issn = 0021-8693
| volume = 328
| issue = 1
| pages = 322–329
| last = Bürgisser
| first = Peter
| coauthors = Matthias Christandl, Christian Ikenmeyer
| title = Even partitions in plethysms
| journal = Journal of Algebra
| accessdate = 2011-03-26
| date = 2011-02-15
| url = http://www.sciencedirect.com/science/article/B6WH2-51JTYBD-1/2/4f609599c6fff11261c5d5272c78e274
}}
 
* {{Cite journal
| doi = 10.1007/s00220-005-1330-9
| last = Heinzner
| first = P.
| coauthors = A. Huckleberry, M. R Zirnbauer
| title = Symmetry classes of disordered fermions
| date = 2004-11-10
| arxiv = math-ph/0411040
}}
 
<!--- Categories --->
 
[[Category:Representation theory of Lie algebras]]
[[Category:Articles created via the Article Wizard]]

Latest revision as of 13:49, 18 July 2014


Title:
5 Ways to Make Your BBQ a Memorable Family Event

Word Count:
517

Summary:
Every barbecue can and should be an exciting family event. Here are 5 ways to make your routine evening barbecue an experience you'll look back on with a grin.


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Article Body:
Since Americans in the South began roasting pigs publicly, Barbecues have been a staple of North American living. For many, grilling becomes a routine mealtime activity, while for the most devoted grillers, summer is ruled by massive barbecue parties. But grillers do not have to choose between boring repetition and outrageous get-togethers: every barbecue can and should be an exciting family event. Here are 5 ways to make your routine evening barbecue a more unique experience:

1) Who said you have to stick to meat? Much can be done with bread and fruits over an open flame. To make this into a fun activity with your kids, use cookie cutters to make the bread and fruit into shapes. You can decide on themes like "animals" and "letters" or ask that each shape be different from the others! Then, let your kids decide which toppings they would like to put on. Add cheese, onion and various spices to bread and cinnamon and honey to fruit.

2) Try foil-wrapped surprise meals. Place different combinations of chicken, fish and vegetables with sauces and spices in foil packets. These cook in less than 20 minutes and lead to no-mess, lip-smacking results. Children will love the surprise element and can easily switch packets. You can even ask your family for combo suggestions before the barbecue, or draw ingredient names out of a hat to create new meal ideas.

3) A little variety goes a long way. If you tend to stick to the same old beef burgers and pork wieners, why not try grilled pizza or salmon steaks? Similarly, try to mix it up with condiments and seasoning. Sick of ugg boots sale ketchup? Try salsa instead. Look online where you'll find hundreds of easy recipes for barbecue rubs. These give your food that extra pizzazz and prevent the mess and drip that comes with barbecue sauces.

4) Barbecues don't have to be rushed. In fact, barbecue devotees will tell you that today's quick broiling is an insult to the term, which should represent a process of slow cooking on low heat. You can find many slow cooker recipes online, and the wait is a great opportunity to pass some quality time with your family and friends. Entertaining backyard activities include creating an outdoor canvas to paint on or playing backyard bowling with household items.

5) Mom said not to play with your food, but there's always an exception! Barbecue games are great fun and tasty at the same time. In the game Glutton, each player gets a certain amount of mashed potatoes, macaroni and cheese or any other side dish. Various kitchen utensils are put in a box and the players blindly pick a few. On the go, they begin to eat their dish with the items that they took out. The first to finish his or her food wins the prize!

Finally, don't fret! Far too many people get lost in the details of planning the perfect barbecue party and forget to enjoy the social atmosphere and delicious food. So kick back, bask in the relaxing sun and prepare for a summer of unforgettable barbecues!

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