repthy
Class GpCharacter

Object
  ClassFunction
      GpCharacter

public class GpCharacter

extends ClassFunction

The character of a complex representation of the Group G, or an element of the Z-lattice generated by such characters (a virtual character).

In practice, the difference between a GpCharacter and a ClassFunction is that the inner product of two GpCharacters belongs to Z. We therefore assume that rounding the product off to the nearest int or long gives the correct value. There is no error-checking in this regard.

Operations that combine two characters will throw an IllegalArgumentException if the underlying groups aren't equal.

Author:

Mark McConnell

Field Summary

static int	COMPLEX Return value for getRealType().
static Comparator	NORM Assumes the arguments are GpCharacters on the same group, and sorts them in ascending order of normSqInt().
static int	QUATERNIONIC Return value for getRealType().
static int	REAL Return value for getRealType().
static int	REGULAR Used with the (Group, int) constructor to construct the character of the regular representation.
static int	SIGN When the group is a PermGp, is used with the (Group, int) constructor to construct the character that's 1 or -1 according to the sign of the permutation.
static int	STANDARD When the group is a PermGp, is used with the (Group, int) constructor to construct the character of the standard representation (the underlying permutation representation minus the identity).
static int	TRIVIAL Used with the (Group, int) constructor to construct the trivial character.
static int	ZERO Used with the (Group, int) constructor to construct the zero character.

Constructor Summary

GpCharacter(Group G, Complex[] val, int cycloField)
Constructs the character on G whose value on the i-th conjugacy class (as returned by Group.getConjClass(int)) is val[i].

GpCharacter(Group G, Complex[] val, int cycloField, String name)
Same as the three-argument constructor, but allows the user to specify the name.

GpCharacter(Group G, int type)
Constructs the specified character on G.

Method Summary

GpCharacter	dual() The character of the dual of the representation for this.
boolean	equals(Object o)
int	getCycloField() Returns an m ≥ 1 such that all values of this character lie in the cyclotomic field Q(ζm) of m-th roots of unity.
int	getDegree() Returns the degree of the representation associated to this character.
Subgroup	getKernel() Returns (G, H), where H is the set of elements of G that map to the identity under the representation associated to this character.
String	getName()
int	getRealType() Returns REAL if the (complex) representation associated to this character is a real representation tensored with C, QUATERNIONIC if it is quaternionic (has a G-invariant conjugate-linear automorphism J with J^2 = -I), and COMPLEX otherwise.
int	hashCode()
boolean	isIrreducible()
boolean	isZero()
GpCharacter	minus(GpCharacter chi) Returns this - chi.
int	normSqInt() Rounds ClassFunction.normSq() to the nearest int.
long	normSqLong() Rounds ClassFunction.normSq() to the nearest long.
void	setName(String name)
GpCharacter	sum(GpCharacter chi) The character of the direct sum of the representations for this and for chi.
GpCharacter	sumTimes(int n) The character of the representation for this direct-summed with itself n times.
GpCharacter	sym2() The character of the second symmetric power of the representation for this.
GpCharacter	tensor(GpCharacter chi) The character of the tensor product of the representations for this and for chi.
String	toString() Returns w - k*v with k chosen to make the difference as short as possible.
GpCharacter	wedge2() The character of the second exterior power of the representation for this.

Methods inherited from class ClassFunction

apply, conjugate, getGroup, getValue, innerProduct, minus, normSq, sum, times, times

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail

TRIVIAL

public static final int TRIVIAL

Used with the (Group, int) constructor to construct the trivial character.

See Also:

Constant Field Values

REGULAR

public static final int REGULAR

Used with the (Group, int) constructor to construct the character of the regular representation.

See Also:

Constant Field Values

SIGN

public static final int SIGN

When the group is a PermGp, is used with the (Group, int) constructor to construct the character that's 1 or -1 according to the sign of the permutation.

See Also:

Constant Field Values

STANDARD

public static final int STANDARD

When the group is a PermGp, is used with the (Group, int) constructor to construct the character of the standard representation (the underlying permutation representation minus the identity).

See Also:

Constant Field Values

ZERO

public static final int ZERO

Used with the (Group, int) constructor to construct the zero character.

See Also:

Constant Field Values

REAL

public static final int REAL

Return value for getRealType().

See Also:

Constant Field Values

COMPLEX

public static final int COMPLEX

Return value for getRealType().

See Also:

Constant Field Values

QUATERNIONIC

public static final int QUATERNIONIC

Return value for getRealType().

See Also:

Constant Field Values

NORM

public static final Comparator NORM

Assumes the arguments are GpCharacters on the same group, and sorts them in ascending order of normSqInt().

Constructor Detail

GpCharacter

public GpCharacter(Group G,
                   Complex[] val,
                   int cycloField)

Constructs the character on G whose value on the i-th conjugacy class (as returned by Group.getConjClass(int)) is val[i]. The argument val is cloned.

Parameters:

cycloField - An m ≥ 1 such that all values of this character lie in the cyclotomic field Q(ζ_m) of m-th roots of unity. They may actually lie in a subfield of this field.

Throws:

IllegalArgumentException - if val.length isn't equal to the number of conjugacy classes.

GpCharacter

public GpCharacter(Group G,
                   Complex[] val,
                   int cycloField,
                   String name)

Same as the three-argument constructor, but allows the user to specify the name.

Parameters:

name - A short name for a row in a character table.

GpCharacter

public GpCharacter(Group G,
                   int type)

Constructs the specified character on G.

Parameters:

type - One of the int constants in this class: TRIVIAL, etc.

Throws:

IllegalArgumentException - If SIGN or STANDARD is used when G is not a PermGp.

Method Detail

getName

public final String getName()

setName

public final void setName(String name)

Parameters:

name - A short name for a row in a character table.

getCycloField

public final int getCycloField()

Returns an m ≥ 1 such that all values of this character lie in the cyclotomic field Q(ζ_m) of m-th roots of unity. They may actually lie in a subfield of this field. Odd m are replaced with 2m, because Q(-1) = Q, Q(ζ₆) = Q(ζ₃), etc.

normSqInt

public int normSqInt()

Rounds ClassFunction.normSq() to the nearest int.

normSqLong

public long normSqLong()

Rounds ClassFunction.normSq() to the nearest long.

isIrreducible

public boolean isIrreducible()

isZero

public boolean isZero()

equals

public boolean equals(Object o)

hashCode

public int hashCode()

getDegree

public int getDegree()

Returns the degree of the representation associated to this character.

getKernel

public Subgroup getKernel()

Returns (G, H), where H is the set of elements of G that map to the identity under the representation associated to this character.

getRealType

public int getRealType()

Returns

• REAL if the (complex) representation associated to this character is a real representation tensored with C,
• QUATERNIONIC if it is quaternionic (has a G-invariant conjugate-linear automorphism J with J^2 = -I), and
• COMPLEX otherwise.

sum

public GpCharacter sum(GpCharacter chi)

The character of the direct sum of the representations for this and for chi. Compare ClassFunction.sum(repthy.ClassFunction).

minus

public GpCharacter minus(GpCharacter chi)

Returns this - chi. Compare ClassFunction.minus(repthy.ClassFunction).

sumTimes

public GpCharacter sumTimes(int n)

The character of the representation for this direct-summed with itself n times. Compare ClassFunction.times(int).

tensor

public GpCharacter tensor(GpCharacter chi)

The character of the tensor product of the representations for this and for chi. Compare ClassFunction.times(ClassFunction).

dual

public GpCharacter dual()

The character of the dual of the representation for this. Compare ClassFunction.conjugate().

wedge2

public GpCharacter wedge2()

The character of the second exterior power of the representation for this.

sym2

public GpCharacter sym2()

The character of the second symmetric power of the representation for this.

toString

public String toString()

Returns w - k*v with k chosen to make the difference as short as possible. Returns null if it can't return a vector shorter than w.

Overrides:

toString in class ClassFunction