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Volume 7 Algebras John Cannon Wieb Bosma Claus Fieker Allan Steel HANDBOOK OF MAGMA FUNCTIONS (PROVISIONAL) Volume 7 Algebras John Cannon Wieb Bosma Claus Fieker Allan Steel Editors Version 2.18 Sydney December 16, 2011 ii MAGMA COMPUTER • ALGEBRA HANDBOOK OF MAGMA FUNCTIONS Editors: John Cannon Wieb Bosma Claus Fieker Allan Steel Handbook Contributors: Geoff Bailey, Wieb Bosma, Gavin Brown, Nils Bruin, John Cannon, Jon Carlson, Scott Contini, Bruce Cox, Bren- dan Creutz, Steve Donnelly, Tim Dokchitser, Willem de Graaf, Claus Fieker, Damien Fisher, Volker Gebhardt, Sergei Haller, Michael Harrison, Florian Hess, Derek Holt, Al Kasprzyk, Markus Kirschmer, David Kohel, Axel Kohnert, Dimitri Leemans, Paulette Lieby, Graham Matthews, Scott Murray, Eamonn O’Brien, Dan Roozemond, Ben Smith, Bernd Souvignier, William Stein, Allan Steel, Damien Stehl´e, Nicole Sutherland, Don Taylor, Bill Unger, Alexa van der Waall, Paul van Wamelen, Helena Verrill, John Voight, Mark Watkins, Greg White Production Editors: Wieb Bosma Claus Fieker Allan Steel Nicole Sutherland HTML Production: Claus Fieker Allan Steel VOLUME 7: OVERVIEW X ALGEBRAS . 2327 77 ALGEBRAS 2329 78 STRUCTURE CONSTANT ALGEBRAS 2341 79 ASSOCIATIVE ALGEBRAS 2351 80 FINITELY PRESENTED ALGEBRAS 2377 81 MATRIX ALGEBRAS 2415 82 GROUP ALGEBRAS 2455 83 BASIC ALGEBRAS 2469 84 QUATERNION ALGEBRAS 2515 85 ALGEBRAS WITH INVOLUTION 2559 86 CLIFFORD ALGEBRAS 2575 XI REPRESENTATION THEORY . 2579 87 MODULES OVER AN ALGEBRA 2581 88 K[G]-MODULES AND GROUP REPRESENTATIONS 2617 89 CHARACTERS OF FINITE GROUPS 2647 90 REPRESENTATIONS OF SYMMETRIC GROUPS 2669 vi VOLUME 7: CONTENTS VOLUME 7: CONTENTS X ALGEBRAS 2327 77 ALGEBRAS . 2329 77.1 Introduction 2331 77.1.1 The Categories of Algebras 2331 77.2 Construction of General Algebras and their Elements 2331 77.2.1 Construction of a General Algebra 2332 77.2.2 Construction of an Element of a General Algebra 2333 77.3 Construction of Subalgebras, Ideals and Quotient Algebras 2333 77.3.1 Subalgebras and Ideals 2333 77.3.2 Quotient Algebras 2334 77.4 Operations on Algebras and Subalgebras 2334 77.4.1 Invariants of an Algebra 2334 77.4.2 Changing Rings 2335 77.4.3 Bases 2335 77.4.4 Decomposition of an Algebra 2336 77.4.5 Operations on Subalgebras 2338 77.5 Operations on Elements of an Algebra 2339 77.5.1 Operations on Elements 2339 77.5.2 Comparisons and Membership 2340 77.5.3 Predicates on Elements 2340 78 STRUCTURE CONSTANT ALGEBRAS . 2341 78.1 Introduction 2343 78.2 Construction of Structure Constant Algebras and Elements 2343 78.2.1 Construction of a Structure Constant Algebra 2343 78.2.2 Construction of Elements of a Structure Constant Algebra 2344 78.3 Operations on Structure Constant Algebras and Elements 2345 78.3.1 Operations on Structure Constant Algebras 2345 78.3.2 Indexing Elements 2346 78.3.3 The Module Structure of a Structure Constant Algebra 2347 78.3.4 Homomorphisms 2347 79 ASSOCIATIVE ALGEBRAS . 2351 79.1 Introduction 2353 79.2 Construction of Associative Algebras 2353 79.2.1 Construction of an Associative Structure Constant Algebra 2353 79.2.2 Associative Structure Constant Algebras from other Algebras 2354 79.3 Operations on Algebras and their Elements 2355 79.3.1 Operations on Algebras 2355 79.3.2 Operations on Elements 2357 79.3.3 Representations 2358 79.3.4 Decomposition of an Algebra 2358 79.4 Orders 2360 79.4.1 Creation of Orders 2361 79.4.2 Attributes 2364 79.4.3 Bases of Orders 2365 79.4.4 Predicates 2366 VOLUME 7: CONTENTS vii 79.4.5 Operations with Orders 2367 79.5 Elements of Orders 2368 79.5.1 Creation of Elements 2368 79.5.2 Arithmetic of Elements 2368 79.5.3 Predicates on Elements 2369 79.5.4 Other Operations with Elements 2369 79.6 Ideals of Orders 2370 79.6.1 Creation of Ideals 2370 79.6.2 Attributes of Ideals 2371 79.6.3 Arithmetic for Ideals 2372 79.6.4 Predicates on Ideals 2372 79.6.5 Other Operations on Ideals 2373 79.7 Quaternionic Orders 2375 79.8 Bibliography 2376 80 FINITELY PRESENTED ALGEBRAS . 2377 80.1 Introduction 2379 80.2 Representation and Monomial Orders 2379 80.3 Exterior Algebras 2380 80.4 Creation of Free Algebras and Elements 2380 80.4.1 Creation of Free Algebras 2380 80.4.2 Print Names 2380 80.4.3 Creation of Polynomials 2381 80.5 Structure Operations 2381 80.5.1 Related Structures 2381 80.5.2 Numerical Invariants 2381 80.5.3 Homomorphisms 2382 80.6 Element Operations 2383 80.6.1 Arithmetic Operators 2383 80.6.2 Equality and Membership 2383 80.6.3 Predicates on Algebra Elements 2383 80.6.4 Coefficients, Monomials, Terms and Degree 2384 80.6.5 Evaluation 2386 80.7 Ideals and Gr¨obnerBases 2387 80.7.1 Creation of Ideals 2387 80.7.2 Gr¨obnerBases 2388 80.7.3 Verbosity 2389 80.7.4 Related Functions 2390 80.8 Basic Operations on Ideals 2392 80.8.1 Construction of New Ideals 2393 80.8.2 Ideal Predicates 2393 80.8.3 Operations on Elements of Ideals 2394 80.9 Changing Coefficient Ring 2395 80.10 Finitely Presented Algebras 2395 80.11 Creation of FP-Algebras 2395 80.12 Operations on FP-Algebras 2397 80.13 Finite Dimensional FP-Algebras 2398 80.14 Vector Enumeration 2402 80.14.1 Finitely Presented Modules 2402 80.14.2 S-algebras 2402 80.14.3 Finitely Presented Algebras 2403 80.14.4 Vector Enumeration 2403 80.14.5 The Isomorphism 2404 80.14.6 Sketch of the Algorithm 2405 80.14.7 Weights 2405 viii VOLUME 7: CONTENTS 80.14.8 Setup Functions 2406 80.14.9 The Quotient Module Function 2406 80.14.10 Structuring Presentations 2406 80.14.11 Options and Controls 2407 80.14.12 Weights 2407 80.14.13 Limits 2408 80.14.14 Logging 2409 80.14.15 Miscellaneous 2410 80.15 Bibliography 2413 81 MATRIX ALGEBRAS . 2415 81.1 Introduction 2419 81.2 Construction of Matrix Algebras and their Elements 2419 81.2.1 Construction of the Complete Matrix Algebra 2419 81.2.2 Construction of a Matrix 2419 81.2.3 Constructing a General Matrix Algebra 2421 81.2.4 The Invariants of a Matrix Algebra 2422 81.3 Construction of Subalgebras, Ideals and Quotient Rings 2423 81.4 The Construction of Extensions and their Elements 2425 81.4.1 The Construction of Direct Sums and Tensor Products 2425 81.4.2 Construction of Direct Sums and Tensor Products of Elements 2427 81.5 Operations on Matrix Algebras 2428 81.6 Changing Rings 2428 81.7 Elementary Operations on Elements 2428 81.7.1 Arithmetic 2428 81.7.2 Predicates 2429 81.8 Elements of Mn as Homomorphisms 2433 81.9 Elementary Operations on Subalgebras and Ideals 2434 81.9.1 Bases 2434 81.9.2 Intersection of Subalgebras 2434 81.9.3 Membership and Equality 2434 81.10 Accessing and Modifying a Matrix 2435 81.10.1 Indexing 2435 81.10.2 Extracting and Inserting Blocks 2436 81.10.3 Joining Matrices 2436 81.10.4 Row and Column Operations 2437 81.11 Canonical Forms 2437 81.11.1 Canonical Forms for Matrices over Euclidean Domains 2437 81.11.2 Canonical Forms for Matrices over a Field 2439 81.12 Diagonalising Commutative Algebras over a Field 2442 81.13 Solutions of Systems of Linear Equations 2444 81.14 Presentations for Matrix Algebras 2445 81.14.1 Quotients and Idempotents 2445 81.14.2 Generators and Presentations 2448 81.14.3 Solving the Word Problem 2452 81.15 Bibliography 2454 VOLUME 7: CONTENTS ix 82 GROUP ALGEBRAS . 2455 82.1 Introduction 2457 82.2 Construction of Group Algebras and their Elements 2457 82.2.1 Construction of a Group Algebra 2457 82.2.2 Construction of a Group Algebra Element 2459 82.3 Construction of Subalgebras, Ideals and Quotient Algebras 2460 82.4 Operations on Group Algebras and their Subalgebras 2462 82.4.1 Operations on Group Algebras 2462 82.4.2 Operations on Subalgebras of Group Algebras 2463 82.5 Operations on Elements 2465 83 BASIC ALGEBRAS . 2469 83.1 Introduction 2471 83.2 Basic Algebras 2471 83.2.1 Creation 2471 83.2.2 Access Functions 2472 83.2.3 Elementary Operations 2474 83.3 Modules over Basic Algebras 2477 83.3.1 Indecomposable Projective Modules 2477 83.3.2 Creation 2478 83.3.3 Access Functions 2478 83.3.4 Predicates 2479 83.3.5 Elementary Operations 2479 83.4 Homomorphisms 2481 83.4.1 Creation 2482 83.4.2 Access Functions 2483 83.4.3 Projective Covers 2483 83.5 Opposite Algebras 2488 83.5.1 Creation 2488 83.5.2 Injective Modules 2488 83.6 Cohomology 2492 83.6.1 Ext Algebras 2497 83.7 Group Algebras of p-groups 2499 83.7.1 Access Functions 2499 83.7.2 Projective Resolutions 2500 83.7.3 Cohomology Generators 2501 83.7.4 Cohomology Rings 2501 83.7.5 Restrictions and Inflations 2502 83.8 A-infinity Algebra Structures on Group Cohomology 2506 83.8.1 Homological Algebra Toolkit 2507 83.8.2 Special Basic Algebras 2510 83.9 Bibliography 2514 84 QUATERNION ALGEBRAS . 2515 84.1 Introduction 2517 84.2 Creation of Quaternion Algebras 2518 84.3 Creation of Quaternion Orders 2522 84.3.1 Creation of Orders from Elements 2523 84.3.2 Creation of Maximal Orders 2524 84.3.3 Creation of Orders with given Discriminant 2526 84.3.4 Creation of Orders with given Discriminant over the Integers 2527 84.4 Elements of Quaternion Algebras 2528 84.4.1 Creation of Elements 2528 x VOLUME 7: CONTENTS 84.4.2 Arithmetic of Elements 2528 84.5 Attributes of Quaternion Algebras 2530 84.6 Hilbert Symbols and Embeddings 2532 84.7 Predicates on Algebras 2535 84.8 Recognition Functions 2536 84.9 Attributes of Orders 2538 84.10 Predicates of Orders 2539 84.11 Operations with Orders 2540 84.12 Ideal Theory of Orders 2540 84.12.1 Creation and Access Functions 2541 84.12.2 Enumeration of Ideal Classes 2543 84.12.3 Operations on Ideals 2546 84.13 Norm Spaces and Basis Reduction 2547 84.14 Isomorphisms 2550 84.14.1 Isomorphisms of Algebras 2550 84.14.2 Isomorphisms of Orders 2550 84.14.3 Isomorphisms of Ideals 2551 84.14.4 Examples 2552 84.15 Units and Unit Groups 2555 84.16 Bibliography 2557 85 ALGEBRAS WITH INVOLUTION .
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