Topics 4: Algebras. Groups. Representation

1. Topics 4: Algebras. Groups. Representation Lecture 9

2. Algorithms for Polynomial Ideals and Their Varieties • A central problem of algorithmic commutative algebra and algebraic geometry is solving of systems of polynomial equations in several variables. • The exact algebraic computation of the finitely many solutions requires the transformation of the ystem to some manageble form. • The following approaches can accomplish this: • computation of the Groebner basis with respect to lexicographical ordering; • conversion of the Groebner basis with respect to total degree ordering to one with respect to lexicographical ordering or by a Groebner walk; • combination of a Groebner basis computation with factorization of all intermediate polynomials; • reduction to the computation of the solutions of a single univariate polynomial by computation of a shape basis or rational univariate form; • computation of a characteristic set; • computation of a triangular set from a Groebner basis; • iterative computation of resultant systems together with suitable transformations of the variables (approach of classical elimination theory).

3. Algorithms for Polynomial Ideals and Their Varieties • Most of these methods above have been implemented in several computer algebra systems. • Another important topic of algorithmic commutative algebra are quantifier elimination methods for algebraically closed fields. • Here the problem is as follows: given a system of polynomial equations and inequalities, depending on certain parameters, compute a Boolean combination of polynomial equations in those parameters that it is a necessary and sufficient condition for the solvability of the system in any algebraically closed field. • In principle, the problem can be solved by elementary methods of classical elimination theory. • Those algorithms however, are not practical. • Their worst case asymptotic complexity is doubly exponential in the number of variables. • There are implementations only for quantifier elimination algorithm for linear and quadratic problems.

4. Singularities of Varieties • Many algebraic systems of equations features singularities, i.e. points or higher dimensional subvarieties, on which the functional matrix of the defining polynomials locally does not have constant rank. • For this case, the implicit function theorem does not hold, and the solution set can therefore locally not be parameterized. • The structure of these singularities is encoded in the local ring of the variety at the singular point. • Many invariants can be computed using standard bases. The fundamental algorithm for computing standard bases in local rings is a variant of Buchberger’s algorithms; • implementations of this algorithm (e.g. in Singular) already contributed interesting results to the theory of singularities.

5. Real Algebraic Geometry • In real commutative algebra, basic concepts of classical commutative algebra are enriched by imports from the theory of ordered rings and fields. • Model theory has a natural relationship to real algebraic geometry, e.g., the projection theorem for semi-algebraic sets is a geometric version of quantifier elimination in the theory of real algebraic fields.

6. Algorithmic Aspects of the Theory of Algebras • Mathematical structures with addition, a scalar multiplication for scalars from a commutative ring of field and an algebra multiplication satisfying the usual distribution laws, play an important role in many areas. • Well-known examples of such algebras are • commutative polynomial algebras, or • the associative matrix algebras or function algebras. • However, the concept of an algebra is of a more general nature. • As examples one can mention • group algebras, • path algebras, • quantum groups, i.e. noncommutative algebras in quantum mechanics which are often given by generators and transformation relations, • Lie and Jordan algebras, as well as • genetic algebras for modelling non-associative rules of heredity.

7. Algorithmic Aspects of the Theory of Algebras There are various ways to represent algebras on a computer depending on the nature of available computational information of the algebra: • finite-dimensional algebras of dimension n with multiplication given by n3 structure constants; • algebras given by generators and relations; • finite-dimensional associative algebras over fields are subalgebras of endomorphism algebras of a finite-dimensional vector space, and so can be given by means of a finite set of generating matrices; • nilpotent algebras can be given by so-called power commutator presentation; • monad algebras like group or path algebras.

8. Algorithmic Aspects of the Theory of Algebras • For a finite-dimensional algebra the most straightforward presentation is the abstract one, which assumes the knowledge of a basis {x_i} on which the multiplication is given by the so-called structure constants c^(k)_ij which are defined by the relation xi · xj = sum_k c^(k)_ijx_k. • Another way to represent an algebra is by generators and relations. In the case of free Lie-algebras several bases are known, the famous one being Hall basis. • If the algebra is associative, the algebra can directly be viewed as a quotient of the free associative algebra on the generators by the ideal generated by the relations. • The Weyl algebra is generated by {x,Dx} with the rule Dx * x = x ·Dx+1. • General implementations can be found in Felix and MAS. • A natural generalization of the concept of a group algebra RG for a group G, where the group elements are used as an R-basis of RG and the multiplication of the algebra is given by extending the multiplication of the basis elements to arbitrary structures is a monad. • Monad algebras can easily be implemented in particular if the monad is finite. • Slight generalizations are path algebras, where in addition 0 is also allowed as the result of the multiplication of two basic elements.

9. Algorithmic Aspects of the Theory of Algebras • Another topic in general algebras and rings is the study of identities. • The most popular example is the associative identity x(yz) − (xy)z = 0, or the Jacobi identity x(yz) + (yz)x + (zx)y = 0, or the alternative identities (xx)y −x(xy) = 0 and y(xx)−(yx)x = 0. • The left-hand sides can be considered as objects in the free algebra generated by x, y, z, which e.g. can be constructed in a computer algebra system like Axiom as the monad rig over the monad of all binary trees. • The system Albert is devoted to identities exclusively; • it verifies whether a given identity holds in an algebra by constructing the free algebra determined by the given set of defining identities of the algebra and testing whether the polynomial in question is zero. • The concept of path algebras may be considered as a natural generalization of the concept of free algebras in non-commuting variables. • A suitable adaption of the Groebner basis technique was established for path algebras and implemented in a package of C programs called Groebner.

10. Describing Groups • Computational group theory is divided into subfields based on the way the group G is specified. • The questions asked and the techniques used to answer them depend critically on the nature of the specification of G. • There are four ways to specify a group G: 1. define a finite subset S of some previously specified group M and say that G is the subgroup of M generated by S; 2. define by a finite presentation; 3. define to be the group of all automorphisms of some algebraic or combinatorial structure (e.g. graphs). 4. give a black box description of G - an abstraction of the minimal information one would need to begin computing in a finite group; • it involves a positive integer N, an encoding of elements of G by bit strings of length N, the bit strings corresponding to a generating set for G, and two oracles (let U and V be the bit strings corresponding to two elements g and h of G; • given U, the first oracle returns the bit string for g−1; • given U and V , the second oracle returns the bit string for gh).

11. Describing Groups • Except in groups given by finite representations, questions about individual elements of a group tend not to be particularly challenging. • Frequently one asks questions involving large sets of elements, either in the statement of the question or in its answer (e.g determine all elements that commute with a give element). • The subject of computational group theory about groups is generally considered to have originated with three questions: 1. the word problem: does a given word represent the identity element of the group; 2. the conjugacy problem: do two given words represent conjugate elements of the group; 3. the isomorphism problem: are two given groups isomorphic groups. • All three questions turned out not the have algorithmic solutions.

12. Permutation Groups • The algorithmic theory of permutation is one of the most developed areas of computational group theory. • The classical example of permutation group is the Rubik Cube group. • One of the first lessons learned about machine computation with permutation groups is that, although cycle notation is useful for hand computation, it is not a good representation for elements of permutation groups in a computer. • In most cases, an element g in Sum_n (n is the size of permutation) is represented by a vector of length n whose ith component is the image i^g of i under g. • The first question to ask about a permutation group is whether or not is transitive or, more generally, what the orbits of G are (if G has only one orbit, then G is transitive). • This question can be answered in time O(n^2).

13. Permutation Groups • There are some algorithmic questions about permutation groups which are known to have polynomial time solutions. • For example, the problem of finding generators for the intersection of two permutation groups is equivalent to graph isomorphism and may not have a polynomial time solution. • It is easy to determine the order of the Rubik Cube group (43250033274489856000) an thus to compute the number of possible configurations the cube can have; • however, one cannot yet know how to find the minimum number of moves which is needed to return an arbitrary configuration to the starting configuration.

14. Matrix Groups • It is natural to consider a group generated by a set of invertible matrices. • There are several questions that can be posed: • does the group act reducibly on the underlying vector space; • is the group finite; • If so, what is its order and what are its composition factors; • is solvable or nilpotent. • An algorithm for deciding whether the group is reducible and if so, for finding an invariant subspace, is implemented in Meat-Axe.

15. Black Box Groups • Most algorithms related to black box groups are randomized. • An important class of such algorithms are algorithms for recognizing simple groups. • The idea is to decide whether the black box group is isomorphic to a given simple group and, if it is, to exhibit an isomorphism.

16. Abelian Groups • One class of groups in which computation is relatively easy is the class of finitely generated abelian groups. • Any such group is isomorphic to a quotient group of a free abelian group M = Z_n, the direct sum of copies of the additive group of integers. • A subgroup is finitely generated and thus may be described as the group generated by the rows of an m × n integer matrix A. • Given A, the first question, as usual, is to decide the membership in the subgroup. • This is usually done by replacing A by its Hermite normal form by applying integer row operations to A.

17. Polycyclic Groups • A group G is polycyclic iff it is solvable and all subgroups are finitely generated. • The rewriting approach to computing with elements of polycyclic groups is usually called collection. • Subgroups can be represented also by integer matrices. • Algorithms foe many constructions in finite solvable groups are available in both of the packages Magma and Gap.

18. Group-Theoretic Software • There are three software packages which offer support for group theoretic computation: Magma, Gap, Magnus. • In addition, Maple provides some tools for group theory and there are many smaller packages which are either intended to be run by themselves or with one or more of the larger systems. • The system Magmus is much newer and is less developed than Magma and Gap, but it support infinite groups. • While Magma and Gap only attempt to find presentations of subgroups of finite index in finitely presented groups, Magnus is able to produce partial presentations for subgroups of infinite index. • Among the more specialized packages are Anuqa, C-Meataxe, MAT, MOC, Quotpic.

19. Ordinary Representation Theory • Soon as the first applications on computers in group theory algorithmic methods have been introduced in representation theory, beginning with the character theory of finite groups. • The classification of the finite simple groups lead to the demand for computing character tables of specific finite groups from incomplete knowledge of some of its characters. • For this purpose several interactive methods were implemented. • GAP includes such methods, as well their enhancement by an arithmetic for cyclotomic fields, routines to compute tensor products, inner products, and symmetrizations of characters, induced characters and fusion of subgroups, that have lead to the computation of numerous character tables, some of them included in the Atlas of finite groups (this contains the character tables of the sporadic groups, their covering groups and automorphism groups). • An algorithm for computing the character table of a finite group from its class multiplication coefficients is also included in Gap and Magma. • Moreover there are algorithms for computing the entries of the character tables in terms of the labels for the rows and columns; such algorithms are known for all Weyl groups of classical types and have implemented in GAP.

20. Ordinary Representation Theory • The irreducible matrix representations of finite group over a field of characteristic 0 are considerably more difficult to construct than the corresponding characters. • Nevertheless, some methods are available. • The GAP share package Arep computes, symbolically, with structured representations of finite groups; • examples for structured representations are induced representations or tensor products of representations • Applications of Arep include the automatic construction of fast algorithms for discrete linear signal transforms. • The Carat package contains tables and implementations of various algorithms for enumeration and recognition problems for crystallographic groups.

21. Modular Representation Theory • Explicit modular representations of finite groups were used in a substantial way in the construction of perfect groups. • The package Meat-Axe is dealing with modular representation theory and investigation of module structures. • The modular character tables for the symmetric groups can also be computed with Specht, a Gap share package.

22. Generic Character Tables • It is often possible to encode the character tables of an infinite series of groups in a single table or in a program. • Such a table or program is then called a generic character table. • Many computations with characters can be performed symbolically on such a generic table. • One can calculate scalar products of character types, calculate tensor products of characters or compute class multiplication coefficient. • The Chevie system contains a library of generic tables for Chevalley groups and a collection of Maple routines to perform such computations. • The tables for the symmetric groups are also available in Ace, a Maple share package, and in Symmetrica.