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Maple functions for computing
Macdonald polynomials



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SF
Main functions:
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

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ACE
Main functions:
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

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Macdonald.html
Main functions:
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.

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JackoMaco.html
Main functions:
Functions for creating symmetric and non-symmetric Jack and Macdonald symmetric functions
Cherednick's double affine Hecke algebra functions

Provided by A. Lascoux

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Determ.html
Main functions:
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

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ribbon.html

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.

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 Vertex.html
Main functions:
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.

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 nabla.html
Main 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)

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 qtCat.html
Main functions:
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
 

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q,t Kostka tables

Main functions:
These files contain tables of q,t-Kostka polynomials up to degree 10, created with the SF package.

John Stembridge

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Plethysm

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.

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If you have suggestions or other Maple functions/packages/efficiency improvements e-mail me at .
This page last updated on Jan 1, 2003