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This is an
integrated package for studying symmetric functions
Includes limited
functions for handling partitions
SF updated and should work
on all versions of Maple.
John Stembridge
This is a
complete sytem for studying algebraic combinatorics
Symmetric
Functions on different Alphabets, Free Module over symmetric polynomials,
NonCommutative Symmetric Functions
Characters
of Linear Groups, Schubert Polynomials, SYMmetric Functions, TABleaux,
FREE algebra, hyperoctahedral Algebras
Symmetric
Group Algebra, NilCoxeter Algebra, IDCoxeter Algebra, HEcKe Algebra, hyperoctahedral
groups, Symmetric Groups
Classical
Groups, PARTitions, COMPositions
Sébastien Veigneau, Alain Lascoux and Jean-Yves Thibon
defintions
of Macdonald polynomials P_\mu[X;q,t], J_\mu[X;q,t], H_\mu[X;q,t]
Vertex operators
for J & H basis to add row and column of size 2
Vertex operators
for J & H that add a row of size 3 to a polynomial indexed by a two
column shape
Functions
to compute J & H basis that are FAST when the case is covered by the
vertex operator formulas
Hall-Littlewood
SFs and vertex operator
Vertex operator formulas are from:
M. A. Zabrocki, A Macdonald Vertex Operator and
Standard Tableaux Statistics for the Two-Column $(q,t)$-Kostka Coefficients,
Electron. J.
Combinat. 5,
R45 (1998), 46pp.
M. A. Zabrocki, Positivity for special cases of $(q,t)$-Kostka coefficients and standard tableaux statistics.
Functions
for creating symmetric and non-symmetric Jack and Macdonald symmetric functions
Cherednick's
double affine Hecke algebra functions
Provided by A. Lascoux
Determinental
formulas for the Macdonald polynomials H_\mu[X;q,t] expanded in terms of
s_\la[X/(1-q)] basis
Functions
to expand these determinants
Formulas are from:
L. Lapointe, A. Lascoux, J. Morse, Determinantal expressions of Macdonald polynomials
Main functions:
operator that
adds a ribbon to the left of a Schur function
operators
that add a column on the homogeneous & Hall-Littlewood functions
q-deformations
of the ribbon operators
operators
that add a column on the Macdonald functions
Formulas are from:
Ribbon Operators
and Hall-Littlewood Symmetric Functions, Adv. in Math., 156 (2000),
pp. 33-43.
Vertex operators
of the form \sum_i f_i g_i^\perp that add a column to elementary and homogeneous
symmetric functions
Symmetric
function operators to add a row or a sequence of columns to the monomial
and forgotten symmetric functions
Symmetric
function operators to add a row or a sequence of columns to the Schur symmetric
functions.
Symmetric
function operators to add a sequence of columns to the power symmetric
functions.
For monomial,
forgotten and Schur, the column operators has the property that if length(\la)
> the length of the column being added then the result is 0.
Formulas are from:
M. A. Zabrocki, Vertex operators for standard bases of the symmetric functions.
A function
to compute the operator nabla on arbitrary
homogeneous symmetric functions
Many
q,t
analogs of sequences assocated with trees, Dyck paths and parking functions
come from this operator, and included are the ones corresponding to n^n,
(n+1)^(n-1),
and the Catalan numbers
Formulas are from:
Explicit Plethystic Formulas for the Macdonald q,t-Kostka Coefficients (A. Garsia, M. Haiman and G. Tesler)
Identities and positivity conjectures for some remarkable operators in the theory of symmetric function (F. Bergeron, A. Garsia, M. Haiman, and G. Tesler)
Functions
to compute the qt-Catalan numbers
directly from the definition.
It is possible to compute this way for much larger n than using the operator
nabla.
Formulas are from:
A REMARKABLE q,t-CATALAN SEQUENCE
AND q-LAGRANGE INVERSION, Adriano
Garsia, Mark Haiman
Main functions:
These files
contain tables of q,t-Kostka polynomials up to degree 10, created with
the SF package.
John Stembridge
Main functions:
A Maple program
to compute the relevant formulas given in the paper listed below.
A table of
the functions k_gamma[X;q,t] in the paper, sufficient to compute K_{lambda,mu}(q,t)
up to partitions of size 12, and certain infinite families beyond this.
A maple script
to produce and save k_gamma's and intermediate computations in "alltables".
This directory contains Maple programs for computing q,t-Kostka polynomials
by means of the formulas given in the paper
A. M. Garsia and G. Tesler, Plethystic Formulas for Macdonald $q,t$-Kostka
Coefficients
Advances in Mathematics, volume 123, number 2, November 10, 1996, pp.
143-222.
.