SYMMETRICA-MANUAL -- Building

Building

Building

In order to build or to make a new SCHURobject we have the following routines (cf. also the corresponding routines for POLYNOMobjects) which differ in the handling of the various inputs: make does a copy and build does not copy.

NAMES: b_skn_s, m_skn_s
SYNOPSES:

INT b_skn_s(OP self,koeff,next,result)

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.
RETURN: OK or ERROR

Sometimes you have a PARTITION and you want to build a SCHURobject out of it. This you can do as follows:

NAMES: b_pa_s, m_pa_s
SYNOPSES:

INT b_pa_s(OP part,result)

INT m_pa_s(OP part,result)
DESCRIPTION: you call b_skn_s (or m_skn_s) with self==part koeff is one and next==NULL.
RETURN: OK or ERROR

Sometimes you even have a VECTORobject of INTEGERobject, which you want to sort to a PARTITIONobject, and then you want to build a SCHURobject out of it

NAMES: b_v_s, m_v_s
SYNOPSES:

INT b_v_s(OP vec, result)

INT m_v_s(OP vec, result)
DESCRIPTION: the difference is again in copying or not copying the vec, which must be a VECTOR of INTEGER. The INTEGERs are not allowed to be negative. Look at m_v_pa().
RETURN: OK or ERROR

Sometimes you may want to know the common constituents (including their multiplicities) of two SCHURobjects (which are linear combinations of Schur polynomials), they can be evaluated using the following routine:

NAME: schnitt_schur
SYNOPSIS: INT schnitt_schur(OP a,b,c)
DESCRIPTION: you enter two SCHURobjects, and the result is the common part.

The outer product of two Schur polynomials (= the linear combination, the coefficients of which obey the Littlewood-Richardson rule) is evaluated as follows:

NAME: outerproduct_schur
SYNOPSIS:

INT outerproduct_schur(OP parta,partb,result)
DESCRIPTION: you enter two PARTITIONobjects, and the result is a SCHURobject, which is the expansion of the product of the two schurfunctions, labeled by the two PARTITIONobjects parta and partb.

The symmetric part of a Schubert polynomial in terms of Schur polynomials:

NAME: newtrans
SYNOPSIS: INT newtrans(OP perm, schur)
DESCRIPTION: computes the decomposition of a schubertpolynomial labeled by the permutation perm, as a sum of Schur polynomials.

Expansion of Schur polynomials with prescribed alphabet:

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:

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

Expansion of a complete symmetric function:

NAME: compute_complete_with_alphabet
SYNOPSIS:

INT compute_complete_with_alphabet(OP num,length,res)
DESCRIPTION: computes the expansion of a complete symmetric function labeled by an INTEGER num as a POLYNOM res. The INTEGER length specifies the length of the alphabet.

Expansion of monomial symmetric functions:

NAME: compute_monomial_with_alphabet
SYNOPSIS:

INT compute_monomial_with_alphabet(OP num,length,res)
DESCRIPTION: computes the expansion of a monomial symmetric function labeled by a PARTITION num as a POLYNOM res. The INTEGER length specifies the length of the alphabet.

Example:

...
scan(PARTITION,a); scan(INTEGER,b);
compute_monomial_with_alphabet(a,b,c); println(c);
...

Expansion of power sum symmetric functions:

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 as a POLYNOM erg. The INTEGER laenge specifies the length of the alphabet.

Another expansion of Schur polynomials:

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 the INTEGER length.

Expansion of skew Schur polynomials:

NAME: compute_skewschur_with_alphabet_det
SYNOPSIS:

INT compute_skewschur_with_alphabet _det(OP sp,l,res)
DESCRIPTION: computes the expansion of a skewschurfunction labeled by the SKEWPARTITION sp, using the Jacobi Trudi Identity, the result is the POLYNOM res, the length of the alphabet is given by the INTEGER l.

Example:

...
scan(SKEWPARTITION,a);
scan(INTEGER,b);
compute_skewschur_with_alphabet_det(a,b,d);println(d);
...

The evaluation of the Hall-Littlewood polynomials:

NAME: hall_littlewood
SYNOPSIS: INT hall_littlewood(OP part, OP res)
DESCRIPTION: computes the-so called Hall-Littlewood polynomials, i.e. a SCHURobject, 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)

harald.fripertinger "at" uni-graz.at, May 26, 2011

Building