Ebook: Topics in Computational Algebra

The main purpose of these lectures is first to briefly survey the fundamental con­ nection between the representation theory of the symmetric group Sn and the theory of symmetric functions and second to show how combinatorial methods that arise naturally in the theory of symmetric functions lead to efficient algorithms to express various prod­ ucts of representations of Sn in terms of sums of irreducible representations. That is, there is a basic isometry which maps the center of the group algebra of Sn, Z(Sn), to the space of homogeneous symmetric functions of degree n, An. This basic isometry is known as the Frobenius map, F. The Frobenius map allows us to reduce calculations involving characters of the symmetric group to calculations involving Schur functions. Now there is a very rich and beautiful theory of the combinatorics of symmetric functions that has been developed in recent years. The combinatorics of symmetric functions, then leads to a number of very efficient algorithms for expanding various products of Schur functions into a sum of Schur functions. Such expansions of products of Schur functions correspond via the Frobenius map to decomposing various products of irreducible representations of Sn into their irreducible components. In addition, the Schur functions are also the characters of the irreducible polynomial representations of the general linear group over the complex numbers GLn(C).

---

---

Content:
Front Matter....Pages iii-v
Branching functions for winding subalgebras and tensor products....Pages 3-39
Computing with characters of finite groups....Pages 41-56
Some remarks on the computation of complements and normalizers in soluble groups....Pages 57-76
Methods for computing in algebraic geometry and commutative algebra Rome, March 1990....Pages 77-103
Combinatorial Algorithms for the Expansion of Various Products of Schur Functions....Pages 105-135
Polynomial Identities for 2 ? 2 Matrices....Pages 137-161
Cayley Factorization and a Straightening Algorithm....Pages 163-184
The Nagata-Higman Theorem....Pages 185-192
Supersymmetric Bracket Algebra and Invariant Theory....Pages 193-246
Aspects of characteristic-free representation theory of GLn, and some applications to intertwining numbers....Pages 247-261

---

Content:
Front Matter....Pages iii-v
Branching functions for winding subalgebras and tensor products....Pages 3-39
Computing with characters of finite groups....Pages 41-56
Some remarks on the computation of complements and normalizers in soluble groups....Pages 57-76
Methods for computing in algebraic geometry and commutative algebra Rome, March 1990....Pages 77-103
Combinatorial Algorithms for the Expansion of Various Products of Schur Functions....Pages 105-135
Polynomial Identities for 2 ? 2 Matrices....Pages 137-161
Cayley Factorization and a Straightening Algorithm....Pages 163-184
The Nagata-Higman Theorem....Pages 185-192
Supersymmetric Bracket Algebra and Invariant Theory....Pages 193-246
Aspects of characteristic-free representation theory of GLn, and some applications to intertwining numbers....Pages 247-261
....