The theory of group characters and matrix representations of groups
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The book starts with necessary information about matrices, algebras, and groups.
Character of a representation of a group
Then the author proceeds to representations of finite groups. Of particular interest in this part of the book are several chapters devoted to representations and characters of symmetric groups and the closely related theory of symmetric polynomials. The concluding chapters present the representation theory of classical Lie groups, including a detailed description of representations of the unitary and orthogonal groups. The book, which can be read with minimal prerequisites an undergraduate algebra course , allows the reader to get a good understanding of beautiful classical results about group representations.
By the way, on p. It should be stressed, however, that the reader would be ill-advised to come to the book under review without the background referred to above, given that Littlewood not only wastes no time, but arranges his presentation in what is today an alien style: the organization of the book is about as far away from, say, a Landau-type telegraph-style as can be imagined. One reads the book, pencil in hand, ready to supply details where needed — as with all good books, this should occur with great frequency.
Possibility 2. Generic character tables —as addressed by 3. Several occurrences of 4.
The last of the above possibilities is currently not supported and will be described in a chapter of its own when it becomes available. The operation CharacterTable Called with first argument a group G or an ordinary character table ordtbl , and second argument a prime p , CharacterTable calls the operation BrauerTable Called with a string name and perhaps optional parameters param , CharacterTable tries to access a character table from the GAP Character Table Library.
An error is signalled if this GAP package is not loaded in this case. Probably the most interesting information about the character table is its list of irreducibles, which can be accessed as the value of the attribute Irr If the argument of CharacterTable is a string name then the irreducibles are just read from the library file, therefore the returned table stores them already. However, if CharacterTable is called with a group G or with an ordinary character table ordtbl , the irreducible characters are not computed by CharacterTable.
Theory Group Characters Matrix Representations Groups
They are only computed when the Irr This means for example that CharacterTable returns its result very quickly, and the first call of Display 6. The value of the filter HasIrr indicates whether the irreducible characters have been computed already. The reason why CharacterTable does not compute the irreducible characters is that there are situations where one only needs the "table head", that is, the information about class lengths, power maps etc. For example, if one wants to inspect permutation characters of a group then all one has to do is to induce the trivial characters of subgroups one is interested in; for that, only class lengths and the class fusion are needed.
If the group G is given as an argument, CharacterTable accesses the conjugacy classes of G and therefore causes that these classes are computed if they were not yet stored see Called with an ordinary character table ordtbl or a group G , BrauerTable returns its p -modular character table if GAP can compute this table, and fail otherwise. The p -modular table can be computed for p -solvable groups using the Fong-Swan Theorem and in the case that ordtbl is a table from the GAP character table library for which also the p -modular table is contained in the table library.
The default method for a group and a prime delegates to BrauerTable for the ordinary character table of this group.
The default method for ordtbl uses the attribute ComputedBrauerTables for storing the computed Brauer table at position p , and calls the operation BrauerTableOp for computing values that are not yet known. So if one wants to install a new method for computing Brauer tables then it is sufficient to install it for BrauerTableOp. For an ordinary character table tbl and a prime integer p , CharacterTableRegular returns the "table head" of the p -modular Brauer character table of tbl. This is the restriction of tbl to its p -regular classes, like the return value of BrauerTable In general, these characters are hard to compute, and BrauerTable The returned table head can be used to create p -modular Brauer characters, by restricting ordinary characters, for example when one is interested in approximations of the unknown irreducible Brauer characters.
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SupportedCharacterTableInfo is a list that contains at position 3i-2 an attribute getter function, at position 3i-1 the name of this attribute, and at position 3i a list containing a subset of [ "character", "class", "mutable" ] , depending on whether the attribute value relies on the ordering of characters or classes, or whether the attribute value is a mutable list or record. When ordinary or Brauer character table objects are created from records, using ConvertToCharacterTable Let record be a record.
The values of those components of record whose names occur in SupportedCharacterTableInfo All other components of the record simply become components of the character table object. If inconsistencies in record are detected, fail is returned. UnderlyingCharacteristic We want to distinguish ordinary and Brauer tables because a Brauer table may delegate tasks to the ordinary table of the same group, for example the computation of power maps.
We want to distinguish these tables from partially known ordinary tables that cannot be asked for all power maps or all irreducible characters. This means that not all irreducible characters, not all power maps are known, and perhaps even the number of classes and the centralizer orders are known.
Such tables occur when the character table of a group G is constructed using character tables of related groups and information about G but for example without explicitly computing the conjugacy classes of G. An object in IsCharacterTableInProgress is first of all mutable , so nothing is stored automatically on such a table, since otherwise one has no control of side-effects when a hypothesis is changed.
Incomplete tables in this sense are currently not supported and will be described in a chapter of their own when they become available. Note that the term "incomplete table" shall express that GAP cannot compute certain values such as irreducible characters or power maps. A table with access to its group is therefore always complete, also if its irreducible characters are not yet stored. The class of the identity element is expected to be the first one; thus the degree of a character is the character value at position 1.
The trivial character of a character table need not be the first in the list of irreducibles. Most functions that take a character table as an argument and work with characters expect these characters as an argument, too. For some functions, the list of irreducible characters serves as the default, i. For a stored class fusion, the image table is denoted by its Identifier Character tables constructed from groups know these values upon construction, and for character tables constructed without groups, these values are usually not known and cannot be computed from the table.
Tasks may be delegated from a group to its character table or vice versa only if these three attribute values are stored in the character table. For an ordinary character table ordtbl of a finite group, the group can be stored as value of UnderlyingGroup. For a character table tbl with known underlying group G , the ConjugacyClasses value of tbl is a list of conjugacy classes of G. All those lists stored in the table that are related to the ordering of conjugacy classes such as sizes of centralizers and conjugacy classes, orders of representatives, power maps, and all class functions refer to the ordering of this list.
One reason for this is that otherwise we would not be allowed to use a library table as the character table of a group for which the conjugacy classes are stored already. Another, less important reason is that we can use the same group as underlying group of character tables that differ only w. If tbl was constructed from G then the conjugacy classes have been stored at the same time when G was stored. If G and tbl have been connected later than in the construction of tbl , the recommended way to do this is via CharacterTableWithStoredGroup So there is no method for ConjugacyClasses that computes the value for tbl if it is not yet stored.
For an ordinary character table tbl with known underlying group G , IdentificationOfConjugacyClasses returns a list of positive integers that contains at position i the position of the i -th conjugacy class of tbl in the ConjugacyClasses Let G be a group and tbl a character table of a group isomorphic to G , such that G does not store its OrdinaryCharacterTable Otherwise, i.
If a record is present as the third argument info , its meaning is the same as the optional argument arec for CompatibleConjugacyClasses If a list is entered as third argument info it is used as value of IdentificationOfConjugacyClasses If the arguments G and ccl are present then ccl must be a list of the conjugacy classes of the group G , and tbl the ordinary character table of G. If tbl is the first argument then it must be an ordinary character table, and CompatibleConjugacyClasses checks whether the columns of tbl can be identified with the conjugacy classes of a group isomorphic to that for which tbl is the character table; the return value is a list of all those sets of class positions for which the columns of tbl cannot be distinguished with the invariants used, up to automorphisms of tbl.
So the identification is unique if and only if the returned list is empty.
The usual approach is that one first calls CompatibleConjugacyClasses in the second form for checking quickly whether the first form will be successful, and only if this is the case the more time consuming calculations with both group and character table are done. If the optional argument arec is present then it must be a record whose components describe additional information for the class identification. The following components are supported.
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These are first those that have the same meaning for both the group and its character table, and whose values can be read off or computed, respectively, from the character table, such as Size Second, there are attributes whose meaning for character tables is different from the meaning for groups, such as ConjugacyClasses In the first form, CharacterDegrees returns a collected list of the degrees of the absolutely irreducible characters of the group G ; the optional second argument p must be either zero or a prime integer denoting the characteristic, the default value is zero.
In the second form, tbl must be an ordinary or Brauer character table, and CharacterDegrees returns a collected list of the degrees of the absolutely irreducible characters of tbl. The default method for the call with only argument a group is to call the operation with second argument 0. Called with a group G , Irr returns the irreducible characters of the ordinary character table of G. Called with a group G and a prime integer p , Irr returns the irreducible characters of the p -modular Brauer table of G.
Called with an ordinary or Brauer character table tbl , Irr returns the list of all complex absolutely irreducible characters of tbl. For a character table tbl with underlying group, Irr may delegate to the group. For a group G , Irr may delegate to its character table only if the irreducibles are already stored there.
Note that the ordering of columns in the Irr matrix of the group G refers to the ordering of conjugacy classes in the CharacterTable As an extreme example, for a character table obtained from sorting the classes of the CharacterTable The ordering of the entries in the attribute Irr of a group need not coincide with the ordering of its IrreducibleRepresentations LinearCharacters returns the linear i.
In the second form, LinearCharacters returns the p -modular linear characters of the group G. For a character table tbl with underlying group, LinearCharacters may delegate to the group. For a group G , LinearCharacters may delegate to its character table only if the irreducibles are already stored there. OrdinaryCharacterTable returns the ordinary character table of the group G or the Brauer character table modtbl , respectively. Since Brauer character tables are constructed from ordinary tables, the attribute value for modtbl is already stored cf. These operations for groups are applicable to character tables and mean the same for a character table as for its underlying group; see Chapter 39 for the definitions.
The operations are mainly useful for selecting character tables with certain properties, also for character tables without access to a group. The following three attributes for character tables — OrdersClassRepresentatives This is because the values depend on the ordering of conjugacy classes stored as the value of ConjugacyClasses Note that for character tables, the consistency of attribute values must be guaranteed, whereas for groups, there is no need to impose such a consistency rule.
For an ordinary character table tbl , the result is 0 , for a p -modular Brauer table tbl , it is p. The underlying characteristic of a class function psi is equal to that of its underlying character table. The underlying characteristic must be stored when the table is constructed, there is no method to compute it. We cannot use the attribute Characteristic ClassNames and CharacterNames return lists of strings, one for each conjugacy class or irreducible character, respectively, of the character table tbl.
These names are used when tbl is displayed. The default method for ClassNames computes class names consisting of the order of an element in the class and at least one distinguishing letter. The default method for CharacterNames returns the list [ "X. The position of the class with name name in tbl can be accessed as tbl.
The values of these attributes are lists containing a parameter for each conjugacy class or irreducible character, respectively, of the character table tbl. It depends on tbl what these parameters are, so there is no default to compute class and character parameters.
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For example, the classes of symmetric groups can be parametrized by partitions, corresponding to the cycle structures of permutations. Character tables constructed from generic character tables see the manual of the GAP Character Table Library usually have class and character parameters stored. It is used mainly for class fusions into tbl that are stored on other character tables. For character tables without group, the identifier is also used to print the table; this is the case for library tables, but also for tables that are constructed as direct products, factors etc.
The default method for ordinary tables constructs strings of the form "CT n " , where n is a positive integer. The default method for Brauer tables returns the concatenation of the identifier of the ordinary table, the string "mod" , and the string of the underlying characteristic. There is no default method to create an info text. Usual parts of the information are the origin of the table, tests it has passed 1. For a character table tbl , InverseClasses returns the list mapping each conjugacy class to its inverse class. For a character table tbl , RealClasses returns the strictly sorted list of positions of classes in tbl that consist of real elements.
That is, exactly the classes at positions given by the list returned by ClassOrbit contain generators of the cyclic group generated by an element in the cc -th class. For a character table tbl , ClassRoots returns a list containing at position i the list of positions of the classes of all nontrivial p -th roots, where p runs over the prime divisors of the Size The following attributes for a character table tbl correspond to attributes for the group G of tbl.
But instead of a normal subgroup or a list of normal subgroups of G , they return a strictly sorted list of positive integers or a list of such lists which are the positions —relative to the ConjugacyClasses The entries of the result lists are sorted according to increasing length. So this total order respects the partial order of normal subgroups given by inclusion.
Let tbl be the ordinary character table of the group G , say. Called with second argument a list nclasses of class positions of a normal subgroup N of G , ClassPositionsOfDirectProductDecompositions returns the list of pairs describing the decomposition of N as a direct product of two normal subgroups of G.
Originally written in , this book remains a classic source on representations and characters of finite and compact groups. The book starts with necessary information about matrices, algebras, and groups. Then the author proceeds to representations of finite groups. Of particular interest in this part of the book are several chapters devoted to representations and characters of symmetric groups and the closely related theory of symmetric polynomials. The concluding chapters present the representation theory of classical Lie groups, including a detailed description of representations of the unitary and orthogonal groups.
The book, which can be read with minimal prerequisites an undergraduate algebra course , allows the reader to get a good understanding of beautiful classical results about group representations. By the way, on p. It should be stressed, however, that the reader would be ill-advised to come to the book under review without the background referred to above, given that Littlewood not only wastes no time, but arranges his presentation in what is today an alien style: the organization of the book is about as far away from, say, a Landau-type telegraph-style as can be imagined.