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 coefficient,
	which is of arbitrary type.

	To access and change the parts of a SCHUR object we have
	the following routines

	routine     macro       description             type if possible
	----------------------------------------------------------------
	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 homogeneous 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:
	s_s_s
	s_s_n
	s_s_k
	s_s_si
	s_s_sl
	s_s_sii
	s_s_sli
SYNOPSIS:
	OP s_s_s(OP a)
	OP s_s_n(OP a)
	OP s_s_k(OP a)
	OP s_s_si(OP a, INT i)
	OP s_s_sl(OP a)
	INT s_s_sii(OP a, INT i)
	INT s_s_sli(OP a)
MACRO:
	S_S_S
	S_S_N
	S_S_K
	S_S_SI
	S_S_SL
	S_S_SII
	S_S_SLI
DESCRIPTION:
	see routine chart above

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 program, 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 program, 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
	consists 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, labeled 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 irreducible 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 <symmetrica.h>

	SYM_BEGIN
	scan(PARTITION,a);
	scan(INTEGER,b);
	compute_schur_with_alphabet(a,b,c);println(c);
	SYM_END


NAME:
	compute_homsym_with_alphabet
SYNOPSIS:
	INT compute_homsym_with_alphabet(OP number,laenge,erg)
DESCRIPTION:
	computes the expansion of a homogeneous(=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 <symmetrica.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 <symmetrica.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 <symmetrica.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 <symmetrica.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 <symmetrica.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 <symmetrica.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(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
