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COMMENT:
SCHUR
-----
In general an object of the type SCHUR is a special LIST object.
The self components of the LIST are MONOM objects. And the
self part of the MONOM object are PARTITION objects.
(Look at the documentation on PARTITION and LIST)
If we want to build a SCHUR object, we need three parts
the PARTITION object, called self, the next part, which is the
next entry in the list, which is NULL at the end, or a
SCHUR object, with a bigger PARTITION, and the coefficent,
which is of arbitray type.
To access and change the parts of a SCHUR object we have
the following routines
routine macro description type if posible
----------------------------------------------------------------
s_s_s S_S_S select_schur_self PARTITION
s_s_n S_S_N select_schur_next SCHUR or NULL
s_s_k S_S_K select_schur_koeff arbitrary
s_s_si S_S_SI select_schur_self_ith INTEGER
s_s_sl S_S_SL select_schur_self_length INTEGER
s_s_sii S_S_SII select_schur_self_ith_int INT
s_s_sli S_S_SLI select_schur_self_length_int INT
There are also other object types for the different bases of
algebra of symmetric functions. They are also a list of
monoms with partition as the self part. the names are
POW_SYM symmetric powersum
HOM_SYM complete homogenous symmetric functions
MONOMIAL momomial symmetric functions
ELM_SYM elementary symmetric functions
to access parts of it you use the select-functions (macros) as
for the schur functions.
To build a new SCHUR object we have the routine build_..
and the routine make_.., which differ in the handling of
the inputs, make does a copy and build does no copy.
NAME:
b_skn_s
SYNOPSIS:
INT b_skn_s(OP self,koeff,next,result)
DESCRIPTION:
first the result is freed to an empty object,
if it is not yet an empty object. Then the SCHUR
object is build out of the components, given as arguments.
in m_skn_s the components are copied, so you may use
them in the further programm, in b_skn_s they become
part of result, so using them may destroy the SCHUR
object.
NAME:
m_skn_s
SYNOPSIS:
INT m_skn_s(OP self,koeff,next,result)
DESCRIPTION:
first the result is freed to an empty object,
if it is not yet an empty object. Then the SCHUR
object is build out of the components, given as arguments.
in m_skn_s the components are copied, so you may use
them in the further programm, in b_skn_s they become
part of result, so using them may destroy the SCHUR
object.
COMMENT:
sometimes you have a PARTITION and you want to build a
SCHUR object out of it
NAME:
b_pa_s
SYNOPSIS:
INT b_pa_s(OP part,result)
DESCRIPTION:
you call b_skn_s with self==part
koeff is one and next==NULL.
NAME:
m_pa_s
SYNOPSIS:
INT m_pa_s(OP part,result)
DESCRIPTION:
you call m_skn_s with self==part
koeff is one and next==NULL.
COMMENT:
and sometimes you even have a VECTOR object of INTEGER object,
which you want to sort to a PARTITION object, and then you
want to build a SCHUR object out of it
NAME:
m_v_s
SYNOPSIS:
INT m_v_s(OP vec, result)
DESCRIPTION:
you enter a VECTOR object vec, which must be an VECTOR of INTEGER.
The INTEGERs are not allowd to be negative. Look at m_v_pa(). These
are sorted and build a PARTITION which labeles the resulting SCHUR object
result. In b_v_s the VECTOR object will be part of the resulting OBJECT
in m_v_s a copy of the input will be part of the result.
NAME:
b_v_s
SYNOPSIS:
INT b_v_s(OP vec, result)
DESCRIPTION:
you enter a VECTOR object vec, which must be an VECTOR of INTEGER.
The INTEGERs are not allowd to be negative. Look at m_v_pa(). These
are sorted and build a PARTITION which labeles the resulting SCHUR object
result. In b_v_s the VECTOR object will be part of the resulting OBJECT
in m_v_s a copy of the input will be part of the result.
COMMENT:
sometimes you want to build a Schur function representing an
scalar, i.e. the Schur function labeled by the zero partition,
for this you have
NAME:
m_scalar_schur
SYNOPSIS:
INT m_scalar_schur(OP scal, schur)
DESCRIPTION:
this results in a SCHUR object schur, which
consits of one monomial whose self part is a partition
of length one with entry zero and a coeff part, which
is the object scal.
NAME:
dimension_schur
SYNOPSIS:
INT dimension_schur(OP a,b)
DESCRIPTION:
you enter a SCHUR object a, and the result is the
dimension of the corresponding representation of the
symmetric group sn.
NAME:
schnitt_schur
SYNOPSIS:
INT schnitt_schur(OP a,b,c)
DESCRIPTION:
you enter two SCHUR objects, and the result is the
common part.
NAME:
outerproduct_schur
SYNOPSIS:
INT outerproduct_schur(OP parta,partb,result)
DESCRIPTION:
you enter two PARTITION objects, and the result is
a SCHUR object, which is the expansion of the product
of the two schurfunctions, labbeled by
the two PARTITION objects parta and partb.
Of course this can also be interpreted as the decomposition of the
outer tensor product of two irreducibe representations of the
symmetric group.
NAME:
part_part_skewschur
SYNOPSIS:
INT part_part_skewschur(OP big,small,result)
DESCRIPTION:
you enter two PARTITION objects big and small, where big is
a partition which contains small, and result becomes a SCHUR
object, which represents the decomposition of the corresponding
skew partition.
NAME:
newtrans
SYNOPSIS:
INT newtrans(OP perm, schur)
DESCRIPTION:
computes the decomposition of a schubertpolynomial labeled by
the permutation perm, as a sum of Schurfunction.
NAME:
compute_schur_with_alphabet
SYNOPSIS:
INT compute_schur_with_alphabet(OP part,laenge,erg)
DESCRIPTION:
computes the expansion of a schurfunction labeled by a
partition PART as a POLYNOM erg. The INTEGER laenge specifies the
length of the alphabet.
EXAMPLE:
#include "def.h"
#include "macro.h"
BEGIN
scan(PARTITION,a);
scan(INTEGER,b);
compute_schur_with_alphabet(a,b,c);println(c);
END
NAME:
compute_homsym_with_alphabet
SYNOPSIS:
INT compute_homsym_with_alphabet(OP number,laenge,erg)
DESCRIPTION:
computes the expansion of a homogenous(=complete) symmetric
function labeled by a INTEGER number as a POLYNOM erg.
The object number may also be a PARTITION or a HOM_SYM object.
The INTEGER laenge specifies the length of the alphabet.
Both routines are the same.
EXAMPLE:
we check whether the multiplication of POLYNOM objects
and HOM_SYM objects gives the same.
#include "def.h"
#include "macro.h"
main()
{
OP a,b,c,d,e;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
d=callocobject(); e=callocobject();
scan(HOM_SYM,a); println(a);
scan(INTEGER,c); compute_homsym_with_alphabet(a,c,d);
println(d);
mult(a,a,b);
mult(d,d,e); println(b);
compute_homsym_with_alphabet(b,c,d);
println(d);
if (NEQ(d,e))
println(e);
freeall(a); freeall(d); freeall(e); freeall(b); freeall(c);
ende();
}
NAME:
compute_elmsym_with_alphabet
SYNOPSIS:
INT compute_elmsym_with_alphabet(OP label,laenge,erg)
DESCRIPTION:
computes the expansion of a elementary symmetric
function labeled by a INTEGER label or by a PARTITION label
or a ELM_SYM label as a POLYNOM erg.
The INTEGER laenge specifies the length of the alphabet.
EXAMPLE:
compares the multiplication of ELM_SYM and POLYNOM
#include "def.h"
#include "macro.h"
main()
{
OP a,b,c,d,e;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
d=callocobject(); e=callocobject();
scan(ELM_SYM,a);
println(a);
scan(INTEGER,c);
compute_elmsym_with_alphabet(a,c,d); println(d);
mult(a,a,b); mult(d,d,e); println(b);
compute_elmsym_with_alphabet(b,c,d); println(d);
if (NEQ(d,e))
println(e);
freeall(a); freeall(d); freeall(e); freeall(b); freeall(c);
ende();
}
NAME:
compute_monomial_with_alphabet
SYNOPSIS:
INT compute_monomial_with_alphabet(OP number,laenge,erg)
DESCRIPTION:
computes the expansion of a monomial symmetric
function labeled by a PARTITION number as a POLYNOM erg.
The INTEGER laenge specifies the length of the alphabet.
EXAMPLE:
#include "def.h"
#include "macro.h"
main()
{
OP a,b,c;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
scan(PARTITION,a); scan(INTEGER,b);
compute_monomial_with_alphabet(a,b,c); println(c);
freeall(a); freeall(b); freeall(c);
ende();
}
NAME:
compute_power_with_alphabet
SYNOPSIS:
INT compute_power_with_alphabet(OP label,laenge,erg)
DESCRIPTION:
computes the expansion of a power symmetric
function labeled by a INTEGER label or by a PARTITION label
or a POW_SYM label as a POLYNOM erg.
The INTEGER laenge specifies the length of the alphabet.
another name for this routine is compute_powsym_with_alphabet().
EXAMPLE:
compares the multiplication of POW_SYM and POLYNOM
#include "def.h"
#include "macro.h"
main()
{
OP a,b,c,d,e;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
d=callocobject(); e=callocobject();
scan(POW_SYM,a);
println(a);
scan(INTEGER,c);
compute_power_with_alphabet(a,c,d); println(d);
mult(a,a,b); mult(d,d,e); println(b);
compute_power_with_alphabet(b,c,d); println(d);
if (NEQ(d,e))
println(e);
freeall(a); freeall(d); freeall(e); freeall(b); freeall(c);
ende();
}
NAME:
compute_powsym_with_alphabet
SYNOPSIS:
INT compute_powsym_with_alphabet(OP label,laenge,erg)
DESCRIPTION:
computes the expansion of a power symmetric
function labeled by a INTEGER label or by a PARTITION label
or a POW_SYM label as a POLYNOM erg.
The INTEGER laenge specifies the length of the alphabet.
another name for this routine is compute_power_with_alphabet().
EXAMPLE:
compares the multiplication of POW_SYM and POLYNOM
#include "def.h"
#include "macro.h"
main()
{
OP a,b,c,d,e;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
d=callocobject(); e=callocobject();
scan(POW_SYM,a);
println(a);
scan(INTEGER,c);
compute_powsym_with_alphabet(a,c,d); println(d);
mult(a,a,b); mult(d,d,e); println(b);
compute_powsym_with_alphabet(b,c,d); println(d);
if (NEQ(d,e))
println(e);
freeall(a); freeall(d); freeall(e); freeall(b); freeall(c);
ende();
}
NAME:
compute_schur_with_alphabet_det
SYNOPSIS:
INT compute_schur_with_alphabet_det(OP part,length,erg)
DESCRIPTION:
computes the expansion of a schurfunction labeled by the
PARTITION part, using the Jacobi Trudi Identity, the result is the
POLYNOM erg, the length of the alphabet is given by INTEGER length.
NAME:
compute_skewschur_with_alphabet_det
SYNOPSIS:
INT compute_skewschur_with_alphabet_det(OP skewpart,length,erg)
DESCRIPTION:
computes the expansion of a skewschurfunction labeled by the
SKEWPARTITION skewpart, using the Jacobi Trudi Identity,
the result is the
POLYNOM erg, the length of the alphabet is given by INTEGER length.
EXAMPLE:
#include "def.h"
#include "macro.h"
main()
{
OP a = callocobject(), b = callocobject(), c = callocobject();
OP d = callocobject();
anfang();
scan(SKEWPARTITION,a);
scan(INTEGER,b);
compute_skewschur_with_alphabet_det(a,b,d);println(d);
freeall(a); freeall(d); freeall(b); freeall(c);
ende();
}
COMMENT:
Now there are several routines to change basis in the space of symmetric
functions:
NAME:
t_SCHUR_MONOMIAL
SYNOPSIS:
INT t_SCHUR_MONOMIAL(OP s,m)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_MONOMIAL_SCHUR
SYNOPSIS:
INT t_MONOMIAL_SCHUR(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_MONOMIAL_HOMSYM
SYNOPSIS:
INT t_MONOMIAL_HOMSYM(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_SCHUR_HOMSYM
SYNOPSIS:
INT t_SCHUR_HOMSYM(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_ELMSYM_SCHUR
SYNOPSIS:
INT t_ELMSYM_SCHUR(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_ELMSYM_MONOMIAL
SYNOPSIS:
INT t_ELMSYM_MONOMIAL(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_ELMSYM_HOMSYM
SYNOPSIS:
INT t_ELMSYM_HOMSYM(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_HOMSYM_SCHUR
SYNOPSIS:
INT t_HOMSYM_SCHUR(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
t_HOMSYM_MONOMIAL
SYNOPSIS:
INT t_HOMSYM_MONOMIAL(OP m,s)
DESCRIPTION:
you input an object of the first kind, i.e. in the case
of t_SCHUR_MONOMIAL an object of type SCHUR, and the output is the
same symmetric function written in the second basis, i.e. in the
given case an object of the type MONOMIAL.
BUG:
up to now the two objects must be different
NAME:
hall_littlewood
SYNOPSIS:
INT hall_littlewood(OP part, OP res)
DESCRIPTION:
computes the so called Hall Littlewood Polynomials, i.e.
a SCHUR object, whose coefficient are polynomials in one
variable. The method, which is used for the computation is described
in the paper: A.O. Morris The Characters of the group GL(n,q)
Math Zeitschr 81, 112-123 (1963)
NAME:
compute_zonal_with_alphabet
SYNOPSIS:
INT compute_zonal_with_alphabet(part,l,res) OP part,l,res;
DESCRIPTION:
computes the zonal polynomial, a POLYNOM object, labeled by the
PARTITION object part, for the alphabet in l variables.
COMMENT:
General Routines
---------------
add
add_apply
addinvers
addinvers_apply
comp
freeall
freeself
fprint
fprintln
mult
mult_apply
objectread
objectwrite
print
println
scalarproduct
sub
tex
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