Applications of Combinatorial Mathematics (Institute of by C. Mitchell

By C. Mitchell

Within the final thirty years, combinatorial arithmetic has came upon itself on the center of many technological functions. The goals of the convention on which this publication is predicated have been to stimulate combinatorial mathematicians to pursue new traces of study of capability and functional significance, and to discover the breadth of purposes to the topic. subject matters coated contain neural networks, cryptography, radio frequency task for cellular telecommunications, coding idea, sequences for communications purposes, interconnection networks, info forms, knot conception, radar, parallel processing, community reliability, formal specification of courses and protocols, and combinatorial optimization.

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We want to answer with a bijection 1−1 ←→ nice combinatorial objects λ irreducible representations V λ. (b) What are their dimensions? We should answer with a formula of the form dim(V λ ) = # of nice combinatorial objects. (c) What are their characters? We want a character formula of the type wta (T ), χ(a) = T where the sum runs over all T in a set of nice combinatorial objects and wta is a weight on these objects which depends on the element a ∈ A where we are evaluating the character. C. How do we construct the irreducible modules?

We give the answers to the main questions for the case of Sn . The precise definitions of all the objects used below can be found in Appendix A2. As always, by a representation of the symmetric group we mean a representation of its group algebra A = CSn . I. What are the irreducible Sn -modules? COMBINATORIAL REPRESENTATION THEORY 31 (a) How do we index/count them? There is a bijection 1−1 ←→ partitions λ of n irreducible representations S λ . (b) What are their dimensions? The dimension of the irreducible representation S λ is n!

One should note that tensoring of representations works for group algebras (and Hopf algebras) but not for general algebras. 3. Answers for GL(n, C), the General Linear Group The results for the general linear group are just as beautiful and just as fundamental as those for the symmetric group. The results are surprisingly similar and yet different in many crucial ways. We shall see that the results for GL(n, C) have been generalized to a very wide class of groups whereas the results for Sn have only been generalized successfully to groups that look very COMBINATORIAL REPRESENTATION THEORY 35 similar to symmetric groups.

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