By Enzo R. Gentile

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The most objective of those lectures is first to in short survey the basic con nection among the illustration idea of the symmetric workforce Sn and the idea of symmetric features and moment to teach how combinatorial tools that come up evidently within the concept of symmetric features result in effective algorithms to precise a variety of prod ucts of representations of Sn by way of sums of irreducible representations.

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Many other varieties of algebras exist, often named for some mathematician: Jordan, Hopf, etc. 6 Ordered Structures Ordered sets often show up and some important algebraic structures are closely related to orders. , we have x Ä x for every x 2 S and x Ä y and y Ä z implies x Ä z. x; y/ has one element if x Ä y and is empty otherwise. Conversely, the objects of any category in which there is at most one arrow between two objects can be preordered. 6. 2 A partially ordered set or poset is a preordered set S such that x Ä y and y Ä x implies x D y.

2 Rings The most common structures with two operations are commutative groups (whose operation is usually called “addition”) on which a second operation has been defined. 1 A ring R is a set with two operations, the first denoted by C, called “addition,” and the second denoted by juxtaposition, called “multiplication,” such that 1. R with addition is a commutative group whose identity element is called 0; 2. R with multiplication is a monoid whose identity element is called 1; 3. b C c/a D ba C ca: A ring with a commutative multiplication is called a commutative ring.

Standard examples are addition, multiplication, and composition of functions. Elementary texts often emphasize the “closure” property of an operation (or, sometimes, of an algebraic structure): the product of two elements in S must be an element of S . We have, instead, built this into the definition. An algebraic structure (Bourbaki says a magma) is a set equipped with one or more operations. Such structures sometimes come with distinguished elements (such as identity elements) or functions associated with the operation (such as taking inverses).