Coverage for local/lib/python2.7/site-packages/sage/combinat/sf/multiplicative.py : 45%

""" 

Multiplicative symmetric functions 

A realization `h` of the ring of symmetric functions is multiplicative if for 

a partition `\lambda = (\lambda_1,\lambda_2,\ldots)` we have 

`h_\lambda = h_{\lambda_1} h_{\lambda_2} \cdots`. 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>, 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code 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. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from . import classical 

import sage.combinat.partition 

class SymmetricFunctionAlgebra_multiplicative(classical.SymmetricFunctionAlgebra_classical): 

""" 

The class of multiplicative bases of the ring of symmetric functions. 

A realization `q` of the ring of symmetric functions is multiplicative if 

for a partition `\lambda = (\lambda_1,\lambda_2,\ldots)` we have 

`q_\lambda = q_{\lambda_1} q_{\lambda_2} \cdots` (with `q_0` meaning `1`). 

Examples of multiplicative realizations are the elementary symmetric basis, 

the complete homogeneous basis, the powersum basis (if the base ring is a 

`\QQ`-algebra), and the Witt basis (but not the Schur basis or the 

monomial basis). 

""" 

def _multiply_basis(self, left, right): 

""" 

Return the product of ``left`` and ``right``. 

INPUT: 

- ``left``, ``right`` -- partitions 

OUTPUT: 

- an element of ``self`` 

TESTS:: 

sage: e = SymmetricFunctions(QQ).e() 

sage: e([2,1])^2 # indirect doctest 

e[2, 2, 1, 1] 

:: 

sage: h = SymmetricFunctions(QQ).h() 

sage: h([2,1])^2 

h[2, 2, 1, 1] 

:: 

sage: p = SymmetricFunctions(QQ).p() 

sage: p([2,1])^2 

p[2, 2, 1, 1] 

:: 

sage: QQx.<x> = QQ[] 

sage: p = SymmetricFunctions(QQx).p() 

sage: (x*p([2]))^2 

x^2*p[2, 2] 

sage: TestSuite(p).run() # to silence sage -coverage 

""" 

m = list(left)+list(right) 

m.sort(reverse=True) 

return sage.combinat.partition.Partition(m) 

def coproduct_on_basis(self, mu): 

r""" 

Return the coproduct on a basis element for multiplicative bases. 

INPUT: 

- ``mu`` -- a partition 

OUTPUT: 

- the image of ``self[mu]`` under comultiplication; this is an 

element of the tensor square of ``self`` 

EXAMPLES:: 

sage: Sym = SymmetricFunctions(QQ) 

sage: p = Sym.powersum() 

sage: p.coproduct_on_basis([2,1]) 

p[] # p[2, 1] + p[1] # p[2] + p[2] # p[1] + p[2, 1] # p[] 

sage: e = Sym.elementary() 

sage: e.coproduct_on_basis([3,1]) 

e[] # e[3, 1] + e[1] # e[2, 1] + e[1] # e[3] + e[1, 1] # e[2] + e[2] # e[1, 1] + e[2, 1] # e[1] + e[3] # e[1] + e[3, 1] # e[] 

sage: h = Sym.homogeneous() 

sage: h.coproduct_on_basis([3,1]) 

h[] # h[3, 1] + h[1] # h[2, 1] + h[1] # h[3] + h[1, 1] # h[2] + h[2] # h[1, 1] + h[2, 1] # h[1] + h[3] # h[1] + h[3, 1] # h[] 

""" 

T = self.tensor_square() 

return T.prod(self.coproduct_on_generators(p) for p in mu)