NAME
YDLCreator - Young Diagram Lattice Calculator
SYNOPSIS
use YDL::Compute;
use YDL::Draw;
my $YDLcomp = YDL::Compute->new();
my $YDLdraw = YDL::Draw->new();
# ----- Compute YDL ----- #
my $compute = $YDLcomp->YDLCompute(states => $N);
# ----- Get YDL lattice ----- #
$l = $compute->GetLattice();
# ----- Parse YDL lattice ----- #
my $table = ParseLattice($l);
sub ParseLattice {
my ($array) = @_;
my %latt;
foreach my $part (@{$array}){
my @x = split("\t", $part);
push(@{$latt{$x[1]}}, $x[0]);
}
return \%latt;
}
# ----- Draw YDL ----- #
# $conf = [N, 0,0,..,0(N-1)]
$YDLdraw->YDLDraw(start => $conf, lattice => $table);
DESCRIPTION
In mathematics, Young's lattice is a partially ordered set of Young
diagrams that is formed by connecting all integer partitions of a given
diagram. It was named after Alfred Young, who developed the
representation theory of the symmetric groups. In Young's theory, the
objects (now called Young diagrams) can be partially ordered by
inclusion resulting in a so called partially ordered set termed Young's
lattice.
Moreover, given M^l it is possible to extract a complete list of
irreducible subrepresentations (S^l) form a complete list as l varies
over all possible partitions. Here one such set is computed and its
representation illustrated using a Hesse diagram:
A
|
B
/ \
C D
\ /
E
|
F
|
G
Where A, B, ...G are Young diagrams depicting a given partition.
AUTHOR
Robert Bakaric <rbakaric@exaltum.eu>
LICENSE
# Copyright 2016 Robert Bakaric <rbakaric@exaltum.eu>
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, write to the Free Software
# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
# MA 02110-1301, USA.
#
#
ACKNOWLEDGEMENT
1. William Fulton (1997) Young Tableaux: With Applications to
Representation Theory and Geometry. Cambridge University Press.