|
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Götz Pfeiffer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declaration of those functions that are needed to
## compute and test possible permutation characters.
##
#T TODO:
#T - small improvement:
#T if a prescribed value is equal to the degree then restrict the
#T constituents to those having this class in the kernel
#T - use roots in `PermCandidates' (cf. `PermCandidatesFaithful'),
#T in order to guarantee property (d) already in the construction!
#T - check and document `PermCandidatesFaithful'
#T - `IsPermChar( <tbl>, <pc> )'
#T (check whether <pc> can be a permutation character of <tbl>;
#T use also the kernel of <pc>, i.e., check whether the kernel factor
#T of <pc> can be a permutation character of the factor of <tbl> by the
#T kernel; one example where this helps is the sum of characters of S3
#T in O8+(2).3.2)
#T - `Constituent' und `Maxdeg' - Optionen in `PermComb'
#############################################################################
##
## <#GAPDoc Label="[1]{ctblpope}">
## <Index Subkey="permutation">characters</Index>
## <Index Subkey="for permutation characters">candidates</Index>
## <Index>possible permutation characters</Index>
## <Index Subkey="possible">permutation characters</Index>
## For groups <M>H</M> and <M>G</M> with <M>H \leq G</M>,
## the induced character <M>(1_G)^H</M> is called the
## <E>permutation character</E> of the operation of <M>G</M>
## on the right cosets of <M>H</M>.
## If only the character table of <M>G</M> is available and not the group
## <M>G</M> itself,
## one can try to get information about possible subgroups of <M>G</M>
## by inspection of those <M>G</M>-class functions that might be
## permutation characters,
## using that such a class function <M>\pi</M> must have at least the
## following properties.
## (For details, see <Cite Key="Isa76" Where="Theorem 5.18."/>),
##
## <List>
## <Mark>(a)</Mark>
## <Item>
## <M>\pi</M> is a character of <M>G</M>,
## </Item>
## <Mark>(b)</Mark>
## <Item>
## <M>\pi(g)</M> is a nonnegative integer for all <M>g \in G</M>,
## </Item>
## <Mark>(c)</Mark>
## <Item>
## <M>\pi(1)</M> divides <M>|G|</M>,
## </Item>
## <Mark>(d)</Mark>
## <Item>
## <M>\pi(g^n) \geq \pi(g)</M> for <M>g \in G</M> and integers <M>n</M>,
## </Item>
## <Mark>(e)</Mark>
## <Item>
## <M>[\pi, 1_G] = 1</M>,
## </Item>
## <Mark>(f)</Mark>
## <Item>
## the multiplicity of any rational irreducible <M>G</M>-character
## <M>\psi</M> as a constituent of <M>\pi</M> is at most
## <M>\psi(1)/[\psi, \psi]</M>,
## </Item>
## <Mark>(g)</Mark>
## <Item>
## <M>\pi(g) = 0</M> if the order of <M>g</M> does not divide
## <M>|G|/\pi(1)</M>,
## </Item>
## <Mark>(h)</Mark>
## <Item>
## <M>\pi(1) |N_G(g)|</M> divides <M>\pi(g) |G|</M>
## for all <M>g \in G</M>,
## </Item>
## <Mark>(i)</Mark>
## <Item>
## <M>\pi(g) \leq (|G| - \pi(1)) / (|g^G| |Gal_G(g)|)</M>
## for all nonidentity <M>g \in G</M>,
## where <M>|Gal_G(g)|</M> denotes the number of conjugacy classes
## of <M>G</M> that contain generators of the group
## <M>\langle g \rangle</M>,
## </Item>
## <Mark>(j)</Mark>
## <Item>
## if <M>p</M> is a prime that divides <M>|G|/\pi(1)</M> only once then
## <M>s/(p-1)</M> divides <M>|G|/\pi(1)</M> and is congruent to <M>1</M>
## modulo <M>p</M>,
## where <M>s</M> is the number of elements of order <M>p</M> in the
## (hypothetical) subgroup <M>H</M> for which <M>\pi = (1_H)^G</M>
## holds.
## (Note that <M>s/(p-1)</M> equals the number of Sylow <M>p</M>
## subgroups in <M>H</M>.)
## </Item>
## </List>
##
## Any <M>G</M>-class function with these properties is called a
## <E>possible permutation character</E> in &GAP;.
## <P/>
## (Condition (d) is checked only for those power maps that are stored in
## the character table of <M>G</M>;
## clearly (d) holds for all integers if it holds for all prime divisors of
## the group order <M>|G|</M>.)
## <P/>
## &GAP; provides some algorithms to compute
## possible permutation characters (see <Ref Func="PermChars"/>),
## and also provides functions to check a few more criteria whether a
## given character can be a transitive permutation character
## (see <Ref Func="TestPerm1"/>).
## <P/>
## Some information about the subgroup <M>U</M> can be computed from the
## permutation character <M>(1_U)^G</M> using <Ref Func="PermCharInfo"/>.
## <#/GAPDoc>
##
#############################################################################
##
#F PermCharInfo( <tbl>, <permchars>[, <format> ] )
##
## <#GAPDoc Label="PermCharInfo">
## <Index Subkey="for permutation characters">LaTeX</Index>
## <ManSection>
## <Func Name="PermCharInfo" Arg='tbl, permchars[, format ]'/>
##
## <Description>
## Let <A>tbl</A> be the ordinary character table of the group <M>G</M>,
## and <A>permchars</A> either the permutation character <M>(1_U)^G</M>,
## for a subgroup <M>U</M> of <M>G</M>, or a list of such permutation
## characters.
## <Ref Func="PermCharInfo"/> returns a record with the following components.
## <List>
## <Mark><C>contained</C>:</Mark>
## <Item>
## a list containing, for each character <M>\psi = (1_U)^G</M> in
## <A>permchars</A>, a list containing at position <M>i</M> the number
## <M>\psi[i] |U| /</M> <C>SizesCentralizers( </C><A>tbl</A><C> )</C><M>[i]</M>,
## which equals the number of those elements of <M>U</M>
## that are contained in class <M>i</M> of <A>tbl</A>,
## </Item>
## <Mark><C>bound</C>:</Mark>
## <Item>
## a list containing,
## for each character <M>\psi = (1_U)^G</M> in <A>permchars</A>,
## a list containing at position <M>i</M> the number
## <M>|U| / \gcd( |U|,</M> <C>SizesCentralizers( <A>tbl</A> )</C><M>[i] )</M>,
## which divides the class length in <M>U</M> of an element in class <M>i</M>
## of <A>tbl</A>,
## </Item>
## <Mark><C>display</C>:</Mark>
## <Item>
## a record that can be used as second argument of <Ref Oper="Display"/>
## to display each permutation character in <A>permchars</A> and the
## corresponding components <C>contained</C> and <C>bound</C>,
## for those classes where at least one character of <A>permchars</A> is
## nonzero,
## </Item>
## <Mark><C>ATLAS</C>:</Mark>
## <Item>
## a list of strings describing the decomposition of the permutation
## characters in <A>permchars</A> into the irreducible characters of
## <A>tbl</A>, given in an &ATLAS;-like notation.
## This means that the irreducible constituents are indicated by their
## degrees followed by lower case letters <C>a</C>, <C>b</C>, <C>c</C>,
## <M>\ldots</M>,
## which indicate the successive irreducible characters of <A>tbl</A>
## of that degree,
## in the order in which they appear in <C>Irr( </C><A>tbl</A><C> )</C>.
## A sequence of small letters (not necessarily distinct) after a single
## number indicates a sum of irreducible constituents all of the same
## degree, an exponent <A>n</A> for the letter <A>lett</A> means that
## <A>lett</A> is repeated <A>n</A> times.
## The default notation for exponentiation is
## <C><A>lett</A>^{<A>n</A>}</C>,
## this is also chosen if the optional third argument <A>format</A> is
## the string <C>"LaTeX"</C>;
## if the third argument is the string <C>"HTML"</C> then exponentiation
## is denoted by <C><A>lett</A><sup><A>n</A></sup></C>.
## </Item>
## </List>
## <P/>
## <Example><![CDATA[
## gap> t:= CharacterTable( "A6" );;
## gap> psi:= Sum( Irr( t ){ [ 1, 3, 6 ] } );
## Character( CharacterTable( "A6" ), [ 15, 3, 0, 3, 1, 0, 0 ] )
## gap> info:= PermCharInfo( t, psi );
## rec( ATLAS := [ "1a+5b+9a" ], bound := [ [ 1, 3, 8, 8, 6, 24, 24 ] ],
## contained := [ [ 1, 9, 0, 8, 6, 0, 0 ] ],
## display :=
## rec(
## chars := [ [ 15, 3, 0, 3, 1, 0, 0 ], [ 1, 9, 0, 8, 6, 0, 0 ],
## [ 1, 3, 8, 8, 6, 24, 24 ] ], classes := [ 1, 2, 4, 5 ],
## letter := "I" ) )
## gap> Display( t, info.display );
## A6
##
## 2 3 3 . 2
## 3 2 . 2 .
## 5 1 . . .
##
## 1a 2a 3b 4a
## 2P 1a 1a 3b 2a
## 3P 1a 2a 1a 4a
## 5P 1a 2a 3b 4a
##
## I.1 15 3 3 1
## I.2 1 9 8 6
## I.3 1 3 8 6
## gap> j1:= CharacterTable( "J1" );;
## gap> psi:= TrivialCharacter( CharacterTable( "7:6" ) )^j1;
## Character( CharacterTable( "J1" ), [ 4180, 20, 10, 0, 0, 2, 1, 0, 0,
## 0, 0, 0, 0, 0, 0 ] )
## gap> PermCharInfo( j1, psi ).ATLAS;
## [ "1a+56aabb+76aaab+77aabbcc+120aaabbbccc+133a^{4}bbcc+209a^{5}" ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PermCharInfo" );
#############################################################################
##
#F PermCharInfoRelative( <tbl>, <tbl2>, <permchars> )
##
## <#GAPDoc Label="PermCharInfoRelative">
## <ManSection>
## <Func Name="PermCharInfoRelative" Arg='tbl, tbl2, permchars'/>
##
## <Description>
## Let <A>tbl</A> and <A>tbl2</A> be the ordinary character tables of two
## groups <M>H</M> and <M>G</M>, respectively,
## where <M>H</M> is of index two in <M>G</M>,
## and <A>permchars</A> either the permutation character <M>(1_U)^G</M>,
## for a subgroup <M>U</M> of <M>G</M>,
## or a list of such permutation characters.
## <Ref Func="PermCharInfoRelative"/> returns a record with the same
## components as <Ref Func="PermCharInfo"/>, the only exception is that the
## entries of the <C>ATLAS</C> component are names relative to <A>tbl</A>.
## <P/>
## More precisely, the <M>i</M>-th entry of the <C>ATLAS</C> component is a
## string describing the decomposition of the <M>i</M>-th entry in
## <A>permchars</A>.
## The degrees and distinguishing letters of the constituents refer to
## the irreducibles of <A>tbl</A>, as follows.
## The two irreducible characters of <A>tbl2</A> of degree <M>N</M>
## that extend the irreducible character <M>N</M> <C>a</C> of <A>tbl</A>
## are denoted by <M>N</M> <C>a</C><M>^+</M> and <M>N </M><C>a</C><M>^-</M>.
## The irreducible character of <A>tbl2</A> of degree <M>2N</M> whose
## restriction to <A>tbl</A> is the sum of the irreducible characters
## <M>N</M> <C>a</C> and <M>N</M> <C>b</C> is denoted as <M>N</M> <C>ab</C>.
## Multiplicities larger than <M>1</M> of constituents are denoted by
## exponents.
## <P/>
## (This format is useful mainly for multiplicity free permutation
## characters.)
## <P/>
## <Example><![CDATA[
## gap> t:= CharacterTable( "A5" );;
## gap> t2:= CharacterTable( "A5.2" );;
## gap> List( Irr( t2 ), x -> x[1] );
## [ 1, 1, 6, 4, 4, 5, 5 ]
## gap> List( Irr( t ), x -> x[1] );
## [ 1, 3, 3, 4, 5 ]
## gap> permchars:= List( [ [1], [1,2], [1,7], [1,3,4,4,6,6,7] ],
## > l -> Sum( Irr( t2 ){ l } ) );
## [ Character( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
## Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ),
## Character( CharacterTable( "A5.2" ), [ 6, 2, 0, 1, 0, 2, 0 ] ),
## Character( CharacterTable( "A5.2" ), [ 30, 2, 0, 0, 6, 0, 0 ] ) ]
## gap> info:= PermCharInfoRelative( t, t2, permchars );;
## gap> info.ATLAS;
## [ "1a^+", "1a^{\\pm}", "1a^++5a^-",
## "1a^++3ab+4(a^+)^{2}+5a^+a^{\\pm}" ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PermCharInfoRelative" );
#############################################################################
##
#F TestPerm1( <tbl>, <char> ) . . . . . . . . . . . . . . . . test permchar
#F TestPerm2( <tbl>, <char> ) . . . . . . . . . . . . . . . . test permchar
#F TestPerm3( <tbl>, <chars> ) . . . . . . . . . . . . . . . test permchars
#F TestPerm4( <tbl>, <chars> ) . . . . . . . . . . . . . . . test permchars
#F TestPerm5( <tbl>, <chars>, <modtbl> ) . . . . . . . . . . test permchars
##
## <#GAPDoc Label="TestPerm1">
## <ManSection>
## <Heading>TestPerm1, ..., TestPerm5</Heading>
## <Func Name="TestPerm1" Arg='tbl, char'/>
## <Func Name="TestPerm2" Arg='tbl, char'/>
## <Func Name="TestPerm3" Arg='tbl, chars'/>
## <Func Name="TestPerm4" Arg='tbl, chars'/>
## <Func Name="TestPerm5" Arg='tbl, chars, modtbl'/>
##
## <Description>
## The first three of these functions implement tests of the properties of
## possible permutation characters listed in
## Section <Ref Sect="Possible Permutation Characters"/>,
## The other two implement test of additional properties.
## Let <A>tbl</A> be the ordinary character table of a group <M>G</M>,
## <A>char</A> a rational character of <A>tbl</A>,
## and <A>chars</A> a list of rational characters of <A>tbl</A>.
## For applying <Ref Func="TestPerm5"/>, the knowledge of a <M>p</M>-modular
## Brauer table <A>modtbl</A> of <M>G</M> is required.
## <Ref Func="TestPerm4"/> and <Ref Func="TestPerm5"/> expect the characters
## in <A>chars</A> to satisfy the conditions checked by
## <Ref Func="TestPerm1"/> and <Ref Func="TestPerm2"/> (see below).
## <P/>
## The return values of the functions were chosen parallel to the tests
## listed in <Cite Key="NPP84"/>.
## <P/>
## <Ref Func="TestPerm1"/> return <C>1</C> or <C>2</C> if <A>char</A> fails
## because of (T1) or (T2), respectively;
## this corresponds to the criteria (b) and (d).
## Note that only those power maps are considered that are stored on
## <A>tbl</A>.
## If <A>char</A> satisfies the conditions, <C>0</C> is returned.
## <P/>
## <Ref Func="TestPerm2"/> returns <C>1</C> if <A>char</A> fails because of
## the criterion (c),
## it returns <C>3</C>, <C>4</C>, or <C>5</C> if <A>char</A> fails because
## of (T3), (T4), or (T5), respectively;
## these tests correspond to (g), a weaker form of (h), and (j).
## If <A>char</A> satisfies the conditions, <C>0</C> is returned.
## <P/>
## <Ref Func="TestPerm3"/> returns the list of all those class functions in
## the list <A>chars</A> that satisfy criterion (h);
## this is a stronger version of (T6).
## <P/>
## <Ref Func="TestPerm4"/> returns the list of all those class functions in
## the list <A>chars</A> that satisfy (T8) and (T9) for each prime divisor
## <M>p</M> of the order of <M>G</M>;
## these tests use modular representation theory but do not require the
## knowledge of decomposition matrices
## (cf. <Ref Func="TestPerm5"/> below).
## <P/>
## (T8) implements the test of the fact that in the case that <M>p</M>
## divides <M>|G|</M> and the degree of a transitive permutation character
## <M>\pi</M> exactly once,
## the projective cover of the trivial character is a summand of <M>\pi</M>.
## (This test is omitted if the projective cover cannot be identified.)
## <P/>
## Given a permutation character <M>\pi</M> of a group <M>G</M> and a prime
## integer <M>p</M>,
## the restriction <M>\pi_B</M> to a <M>p</M>-block <M>B</M> of <M>G</M> has
## the following property, which is checked by (T9).
## For each <M>g \in G</M> such that <M>g^n</M> is a <M>p</M>-element of
## <M>G</M>, <M>\pi_B(g^n)</M> is a nonnegative integer that satisfies
## <M>|\pi_B(g)| \leq \pi_B(g^n) \leq \pi(g^n)</M>.
## (This is <Cite Key="Sco73" Where="Corollary A on p. 113"/>.)
## <P/>
## <Ref Func="TestPerm5"/> requires the <M>p</M>-modular Brauer table
## <A>modtbl</A> of <M>G</M>, for some prime <M>p</M> dividing the order of
## <M>G</M>,
## and checks whether those characters in the list <A>chars</A> whose degree
## is divisible by the <M>p</M>-part of the order of <M>G</M> can be
## decomposed into projective indecomposable characters;
## <Ref Func="TestPerm5"/> returns the sublist of all those characters in
## <A>chars</A> that either satisfy this condition or to which the test does
## not apply.
## <P/>
## <!-- Say a word about (T7)?-->
## <!-- This is the check whether the cycle structure of elements is well-defined;-->
## <!-- the check is superfluous (at least) for elements of prime power order-->
## <!-- or order equal to the product of two primes (see <Cite Key="NPP84"/>);-->
## <!-- note that by construction, the numbers of <Q>cycles</Q> are always integral,-->
## <!-- the only thing to test is whether they are nonnegative.-->
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "A5" );;
## gap> rat:= RationalizedMat( Irr( tbl ) );
## [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ),
## Character( CharacterTable( "A5" ), [ 6, -2, 0, 1, 1 ] ),
## Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ),
## Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ]
## gap> tup:= Filtered( Tuples( [ 0, 1 ], 4 ), x -> not IsZero( x ) );
## [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 0, 0 ],
## [ 0, 1, 0, 1 ], [ 0, 1, 1, 0 ], [ 0, 1, 1, 1 ], [ 1, 0, 0, 0 ],
## [ 1, 0, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 0, 0 ],
## [ 1, 1, 0, 1 ], [ 1, 1, 1, 0 ], [ 1, 1, 1, 1 ] ]
## gap> lincomb:= List( tup, coeff -> coeff * rat );;
## gap> List( lincomb, psi -> TestPerm1( tbl, psi ) );
## [ 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0 ]
## gap> List( lincomb, psi -> TestPerm2( tbl, psi ) );
## [ 0, 5, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1 ]
## gap> Set( TestPerm3(tbl, lincomb), x -> Position(lincomb, x) );
## [ 1, 4, 6, 7, 8, 9, 10, 11, 13 ]
## gap> tbl:= CharacterTable( "A7" );
## CharacterTable( "A7" )
## gap> perms:= PermChars( tbl, rec( degree:= 315 ) );
## [ Character( CharacterTable( "A7" ), [ 315, 3, 0, 0, 3, 0, 0, 0, 0 ] )
## , Character( CharacterTable( "A7" ),
## [ 315, 15, 0, 0, 1, 0, 0, 0, 0 ] ) ]
## gap> TestPerm4( tbl, perms );
## [ Character( CharacterTable( "A7" ), [ 315, 15, 0, 0, 1, 0, 0, 0, 0
## ] ) ]
## gap> perms:= PermChars( tbl, rec( degree:= 15 ) );
## [ Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ),
## Character( CharacterTable( "A7" ), [ 15, 3, 3, 0, 1, 0, 3, 1, 1 ] )
## ]
## gap> TestPerm5( tbl, perms, tbl mod 5 );
## [ Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] )
## ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "TestPerm1" );
DeclareGlobalFunction( "TestPerm2" );
DeclareGlobalFunction( "TestPerm3" );
DeclareGlobalFunction( "TestPerm4" );
DeclareGlobalFunction( "TestPerm5" );
#############################################################################
##
#F PermChars( <tbl> )
#F PermChars( <tbl>, <degree> )
#F PermChars( <tbl>, <arec> )
##
## <#GAPDoc Label="PermChars">
## <ManSection>
## <Func Name="PermChars" Arg='tbl[, cond]'/>
##
## <Description>
## &GAP; provides several algorithms to determine
## possible permutation characters from a given character table.
## They are described in detail in <Cite Key="BP98"/>.
## The algorithm is selected from the choice of the optional argument
## <A>cond</A>.
## The user is encouraged to try different approaches,
## especially if one choice fails to come to an end.
## <P/>
## Regardless of the algorithm used in a specific case,
## <Ref Func="PermChars"/> returns a list of <E>all</E>
## possible permutation characters with the properties described by
## <A>cond</A>.
## There is no guarantee that a character of this list is in fact
## a permutation character.
## But an empty list always means there is no permutation character
## with these properties (e.g., of a certain degree).
## <P/>
## Called with only one argument, a character table <A>tbl</A>,
## <Ref Func="PermChars"/> returns the list of all possible permutation
## characters of the group with this character table.
## This list might be rather long for big groups,
## and its computation might take much time.
## The algorithm is described in <Cite Key="BP98" Where="Section 3.2"/>;
## it depends on a preprocessing step, where the inequalities
## arising from the condition <M>\pi(g) \geq 0</M> are transformed into
## a system of inequalities that guides the search
## (see <Ref Oper="Inequalities"/>).
## So the following commands compute the list of 39 possible permutation
## characters of the Mathieu group <M>M_{11}</M>.
## <P/>
## <Example><![CDATA[
## gap> m11:= CharacterTable( "M11" );;
## gap> SetName( m11, "m11" );
## gap> perms:= PermChars( m11 );;
## gap> Length( perms );
## 39
## ]]></Example>
## <P/>
## There are two different search strategies for this algorithm.
## The default strategy simply constructs all characters with nonnegative
## values and then tests for each such character whether its degree
## is a divisor of the order of the group.
## The other strategy uses the inequalities to predict
## whether a character of a certain degree can lie
## in the currently searched part of the search tree.
## To choose this strategy, enter a record as the second argument of
## <Ref Func="PermChars"/>,
## and set its component <C>degree</C> to the range of degrees
## (which might also be a range containing all divisors of the group order)
## you want to look for;
## additionally, the record component <C>ineq</C> can take the inequalities
## computed by <Ref Oper="Inequalities"/> if they are needed more than once.
## <P/>
## If a positive integer is given as the second argument <A>cond</A>,
## <Ref Func="PermChars"/> returns the list of all
## possible permutation characters of <A>tbl</A> that have degree
## <A>cond</A>.
## For that purpose, a preprocessing step is performed where
## essentially the rational character table is inverted
## in order to determine boundary points for the simplex
## in which the possible permutation characters of the given degree
## must lie (see <Ref Func="PermBounds"/>).
## The algorithm is described at the end of
## <Cite Key="BP98" Where="Section 3.2"/>.
## Note that inverting big integer matrices needs a lot of time and space.
## So this preprocessing is restricted to groups with less than 100 classes,
## say.
## <P/>
## <Example><![CDATA[
## gap> deg220:= PermChars( m11, 220 );
## [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ),
## Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ),
## Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ) ]
## ]]></Example>
## <P/>
## If a record is given as the second argument <A>cond</A>,
## <Ref Func="PermChars"/> returns the list of all
## possible permutation characters that have the properties described by
## the components of this record.
## One such situation has been mentioned above.
## If <A>cond</A> contains a degree as value of the record component
## <C>degree</C>
## then <Ref Func="PermChars"/> will behave exactly as if this degree was
## entered as <A>cond</A>.
## <P/>
## <Example><![CDATA[
## gap> deg220 = PermChars( m11, rec( degree:= 220 ) );
## true
## ]]></Example>
## <P/>
## For the meaning of additional components of <A>cond</A> besides
## <C>degree</C>, see <Ref Func="PermComb"/>.
## <P/>
## Instead of <C>degree</C>, <A>cond</A> may have the component <C>torso</C>
## bound to a list that contains some known values of the required
## characters at the right positions;
## at least the degree <A>cond</A><C>.torso[1]</C> must be an integer.
## In this case, the algorithm described in
## <Cite Key="BP98" Where="Section 3.3"/> is chosen.
## The component <C>chars</C>, if present, holds a list of all those
## <E>rational</E> irreducible characters of <A>tbl</A> that might be
## constituents of the required characters.
## <P/>
## (<E>Note</E>: If <A>cond</A><C>.chars</C> is bound and does not contain
## <E>all</E> rational irreducible characters of <A>tbl</A>,
## &GAP; checks whether the scalar products of all class functions in the
## result list with the omitted rational irreducible characters of
## <A>tbl</A> are nonnegative;
## so there should be nontrivial reasons for excluding a character
## that is known to be not a constituent of the desired possible permutation
## characters.)
## <P/>
## <Example><![CDATA[
## gap> PermChars( m11, rec( torso:= [ 220 ] ) );
## [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ),
## Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ),
## Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ) ]
## gap> PermChars( m11, rec( torso:= [ 220,,,,, 2 ] ) );
## [ Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ) ]
## ]]></Example>
## <P/>
## An additional restriction on the possible permutation characters computed
## can be forced if <A>con</A> contains, in addition to <C>torso</C>,
## the components <C>normalsubgroup</C> and <C>nonfaithful</C>,
## with values a list of class positions of a normal subgroup <M>N</M> of
## the group <M>G</M> of <A>tbl</A> and a possible permutation character
## <M>\pi</M> of <M>G</M>, respectively, such that <M>N</M> is contained in
## the kernel of <M>\pi</M>.
## In this case, <Ref Func="PermChars"/> returns the list of those possible
## permutation characters <M>\psi</M> of <A>tbl</A> coinciding with
## <C>torso</C> wherever its values are bound
## and having the property that no irreducible constituent of
## <M>\psi - \pi</M> has <M>N</M> in its kernel.
## If the component <C>chars</C> is bound in <A>cond</A> then the above
## statements apply.
## An interpretation of the computed characters is the following.
## Suppose there exists a subgroup <M>V</M> of <M>G</M> such that
## <M>\pi = (1_V)^G</M>;
## Then <M>N \leq V</M>, and if a computed character is of the form
## <M>(1_U)^G</M>, for a subgroup <M>U</M> of <M>G</M>, then <M>V = UN</M>.
## <P/>
## <Example><![CDATA[
## gap> s4:= CharacterTable( "Symmetric", 4 );;
## gap> nsg:= ClassPositionsOfDerivedSubgroup( s4 );;
## gap> pi:= TrivialCharacter( s4 );;
## gap> PermChars( s4, rec( torso:= [ 12 ], normalsubgroup:= nsg,
## > nonfaithful:= pi ) );
## [ Character( CharacterTable( "Sym(4)" ), [ 12, 2, 0, 0, 0 ] ) ]
## gap> pi:= Sum( Filtered( Irr( s4 ),
## > chi -> IsSubset( ClassPositionsOfKernel( chi ), nsg ) ) );
## Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, 2, 0 ] )
## gap> PermChars( s4, rec( torso:= [ 12 ], normalsubgroup:= nsg,
## > nonfaithful:= pi ) );
## [ Character( CharacterTable( "Sym(4)" ), [ 12, 0, 4, 0, 0 ] ) ]
## ]]></Example>
## <P/>
## The class functions returned by <Ref Func="PermChars"/> have the
## properties tested by <Ref Func="TestPerm1"/>, <Ref Func="TestPerm2"/>,
## and <Ref Func="TestPerm3"/>.
## So they are possible permutation characters.
## See <Ref Func="TestPerm1"/> for criteria whether a
## possible permutation character can in fact be a permutation character.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PermChars" );
#############################################################################
##
#O Inequalities( <tbl>, <chars>[, <option>] )
##
## <#GAPDoc Label="Inequalities">
## <ManSection>
## <Oper Name="Inequalities" Arg='tbl, chars[, option]'/>
##
## <Description>
## Let <A>tbl</A> be the ordinary character table of a group <M>G</M>.
## The condition <M>\pi(g) \geq 0</M> for every possible permutation
## character <M>\pi</M> of <M>G</M> places restrictions on the
## multiplicities <M>a_i</M> of the irreducible constituents <M>\chi_i</M>
## of <M>\pi = \sum_{{i = 1}}^r a_i \chi_i</M>.
## For every element <M>g \in G</M>,
## we have <M>\sum_{{i = 1}}^r a_i \chi_i(g) \geq 0</M>.
## The power maps provide even stronger conditions.
## <P/>
## This system of inequalities is kind of diagonalized,
## resulting in a system of inequalities restricting <M>a_i</M>
## in terms of <M>a_j</M>, <M>j < i</M>.
## These inequalities are used to construct characters with nonnegative
## values (see <Ref Func="PermChars"/>).
## <Ref Func="PermChars"/> either calls <Ref Oper="Inequalities"/> or takes
## this information from the <C>ineq</C> component of its argument record.
## <P/>
## The number of inequalities arising in the process of diagonalization may
## grow very strongly.
## <P/>
## There are two ways to organize the projection.
## The first, which is chosen if no <A>option</A> argument is present,
## is the straight approach which takes the rational irreducible
## characters in their original order and by this guarantees the character
## with the smallest degree to be considered first.
## The other way, which is chosen if the string <C>"small"</C> is entered as
## third argument <A>option</A>, tries to keep the number of intermediate
## inequalities small by eventually changing the order of characters.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "M11" );;
## gap> PermComb( tbl, rec( degree:= 110 ) );
## [ Character( CharacterTable( "M11" ),
## [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ),
## Character( CharacterTable( "M11" ),
## [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ),
## Character( CharacterTable( "M11" ), [ 110, 14, 2, 2, 0, 2, 0, 0, 0,
## 0 ] ) ]
## gap> # Now compute only multiplicity free permutation characters.
## gap> bounds:= List( RationalizedMat( Irr( tbl ) ), x -> 1 );;
## gap> PermComb( tbl, rec( degree:= 110, maxmult:= bounds ) );
## [ Character( CharacterTable( "M11" ),
## [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Inequalities", [ IsOrdinaryTable, IsList ] );
DeclareOperation( "Inequalities", [ IsOrdinaryTable, IsList, IsObject ] );
#############################################################################
##
#F Permut( <tbl>, <arec> )
##
## <ManSection>
## <Func Name="Permut" Arg='tbl, arec'/>
##
## <Description>
## <C>Permut</C> computes possible permutation characters of the character table
## <A>tbl</A> by the algorithm that solves a system of inequalities.
## This is described in <Cite Key="BP98" Where="Section 3.2"/>.
## <P/>
## <A>arec</A> must be a record.
## Only the following components are used in the function.
## <List>
## <Mark><C>ineq</C> </Mark>
## <Item>
## the result of <Ref Func="Inequalities"/>,
## will be computed if it is not present,
## <C>degree</C> &
## the list of degrees for which the possible permutation characters
## shall be computed,
## this will lead to a speedup only if the range of degrees is
## restricted.
## </Item>
## </List>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "Permut" );
#############################################################################
##
#F PermBounds( <tbl>, <d> ) . . . . . . . . . . boundary points for simplex
##
## <#GAPDoc Label="PermBounds">
## <ManSection>
## <Func Name="PermBounds" Arg='tbl, d'/>
##
## <Description>
## Let <A>tbl</A> be the ordinary character table of the group <M>G</M>.
## All <M>G</M>-characters <M>\pi</M> satisfying <M>\pi(g) > 0</M> and
## <M>\pi(1) = <A>d</A></M>,
## for a given degree <A>d</A>, lie in a simplex described by these
## conditions.
## <Ref Func="PermBounds"/> computes the boundary points of this simplex for
## <M>d = 0</M>,
## from which the boundary points for any other <A>d</A> are easily derived.
## (Some conditions from the power maps of <A>tbl</A> are also involved.)
## For this purpose, a matrix similar to the rational character table of
## <M>G</M> has to be inverted.
## These boundary points are used by <Ref Func="PermChars"/>
## to construct all possible permutation characters
## (see <Ref Sect="Possible Permutation Characters"/>) of a given
## degree.
## <Ref Func="PermChars"/> either calls <Ref Func="PermBounds"/> or takes
## this information from the <C>bounds</C> component of its argument record.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PermBounds" );
#############################################################################
##
#F PermComb( <tbl>, <arec> ) . . . . . . . . . . . . permutation characters
##
## <#GAPDoc Label="PermComb">
## <ManSection>
## <Func Name="PermComb" Arg='tbl, arec'/>
##
## <Description>
## <Ref Func="PermComb"/> computes possible permutation characters of the
## character table <A>tbl</A> by the improved combinatorial approach
## described at the end of <Cite Key="BP98" Where="Section 3.2"/>.
## <P/>
## For computing the possible linear combinations <E>without</E> prescribing
## better bounds (i.e., when the computation of bounds shall be suppressed),
## enter
## <P/>
## <C><A>arec</A>:= rec( degree := <A>degree</A>, bounds := false )</C>,
## <P/>
## where <A>degree</A> is the character degree;
## this is useful if the multiplicities are expected to be small,
## and if this is forced by high irreducible degrees.
## <P/>
## A list of upper bounds on the multiplicities of the rational irreducibles
## characters can be explicitly prescribed as a <C>maxmult</C> component in
## <A>arec</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PermComb" );
#############################################################################
##
#F PermCandidates( <tbl>, <characters>, <torso> )
##
## <ManSection>
## <Func Name="PermCandidates" Arg='tbl, characters, torso'/>
##
## <Description>
## <C>PermCandidates</C> computes possible permutation characters of the
## character table <A>tbl</A> with the strategy using Gaussian elimination,
## which is described in <Cite Key="BP98" Where="Section 3.3"/>.
## <P/>
## The class functions in the result have the additional properties that
## only the (necessarily rational) characters <A>characters</A> occur as
## constituents, and that they are all completions of <A>torso</A>.
## (Note that scalar products with rational irreducible characters of
## <A>tbl</A> that are omitted in <A>characters</A> may be negative,
## so not all class functions in the result list are necessarily characters
## if <A>characters</A> does not contain all rational irreducible characters
## of <A>tbl</A>.)
## <P/>
## Known values of the candidates must be nonnegative integers in
## <A>torso</A>, the other positions of <A>torso</A> are unbound;
## at least the degree <C><A>torso</A>[1]</C> must be an integer.
## <!-- what about choice lists ??-->
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "PermCandidates" );
#############################################################################
##
#F PermCandidatesFaithful( <tbl>, <chars>, <norm_subgrp>, <nonfaithful>,
#F <lower>, <upper>, <torso> )
##
## <ManSection>
## <Func Name="PermCandidatesFaithful"
## Arg='tbl, chars, norm_subgrp, nonfaithful, lower, upper, torso'/>
##
## <Description>
## computes certain possible permutation characters of the character table
## <A>tbl</A> with a generalization of the strategy
## using Gaussian elimination (see <Ref Func="PermCandidates"/>).
## These characters are all with the following properties.
## <P/>
## <Enum>
## <Item>
## Only the (necessarily rational) characters <A>chars</A> occur as
## constituents,
## </Item>
## <Item>
## they are completions of <A>torso</A>, and
## </Item>
## <Item>
## have the character <A>nonfaithful</A> as maximal constituent with kernel
## <A>norm_subgrp</A>.
## </Item>
## </Enum>
## <P/>
## Known values of the candidates must be nonnegative integers in
## <A>torso</A>, the other positions of <A>torso</A> are unbound;
## at least the degree <C><A>torso</A>[1]</C> must be an integer.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "PermCandidatesFaithful" );
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