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newPackage(
"SymmetricPolynomials",
Version => "1.0",
Date => "May 20 2009",
Authors => {{Name => "Alexandra Seceleanu", HomePage => "http://www.math.uiuc.edu/~asecele2/"}},
Headline => "symmetric polynomials",
Keywords => {"Combinatorics", "Representation Theory"},
DebuggingMode => false
)
export {"buildSymmetricGB","elementarySymmetric"}
mons = (X,i)-> (
n := #X;
a := unique flatten apply( apply( partitions (i), p-> toList p| for i from #p to n-1 list 0),q-> permutations q);
return apply(a, r-> product for i to n-1 list X_i^(r_i))
)
symring= R->(
X:= flatten entries vars R;
n := #X;
w := (for i to n-1 list (1))|toList(1..n);
e := getSymbol "e";
S := (coefficientRing R)[X,e_1..e_n,MonomialOrder=>{Weights => w,Lex}];
return S
)
protect elementarySymmetricRing
elementarySymmetricPolynomialRing =(cacheValue symbol elementarySymmetricRing)symring
buildSymmetricGB =method();
buildSymmetricGB (PolynomialRing) := R -> (
n := # flatten entries vars R;
S := elementarySymmetricPolynomialRing R;
xvars := select(n,gens S,i->true);
svars :=select(2*n,gens S,i->true)-set xvars;
x := local x;
A :=coefficientRing(R)[svars,xvars][x];
svars =apply(svars,i->(map(A,S))(i));
xvars =apply(xvars,i->(map(A,S))(i));
g :=x^n+sum for i to n-1 list svars_i*x^(n-i-1);
l := {};
for i to n-1 do (
f :=sub(g,x=>-xvars_(n-i-1));
l = append(l,f);
g=g//(x+xvars_(n-i-1));
);
F := map(S,A);
l= apply(l,i-> F i);
use R;
return l
)
elementarySymmetric = method();
elementarySymmetric (RingElement):= f -> (
R := ring f;
n := # flatten entries vars R;
if n<2 then return f;
l := buildSymmetricGB(R);
I := ideal l;
S := ring I;
forceGB matrix {l};
F := map(S,R);
answer := F(f)%I;
xvars := select(n,gens S,i->true);
svars := gens S -set xvars;
if isSubset(support(answer),svars) then (use R; return answer)
else (use R; print "your input is not a symmetric polynomial")
)
elementarySymmetric (PolynomialRing):= R->(
forceGB matrix{buildSymmetricGB R};
return map(elementarySymmetricPolynomialRing R,R)
)
beginDocumentation()
document {
Key =>SymmetricPolynomials,
Headline => "the algebra of symmetric polynomials",
PARA{"This package uses an explicit description of the Groebner basis of the ideal of obvious relations in this algebra based on:"},
PARA{"Grayson, Stillmann - Computations in the intersection theory of flag varieties, preprint, 2009"},
PARA{"Sturmfels - Algorithms in Invariant Theory, Springer Verlag, Vienna, 1993"}
}
document {
Key =>elementarySymmetric,
Headline => "expression in terms of elementary symmetric polynomials",
Inputs => {"f, a symmetric", TO RingElement},
Outputs =>{"the expression of f in terms of the elementary symmetric functions e_i"},
Usage => "elementalSymm f",
Caveat => {"if the input is not symmetric the function will announce this"},
EXAMPLE lines ///
n=5;
R=QQ[x_1..x_n];
f=(product gens R)*(sum gens R);
elementarySymmetric f
///,
PARA{"This function should work up to a size of 15 variables in the base ring."},
PARA{"This function is part of the package SymmetricPolynomials."}
}
document {
Key =>(elementarySymmetric,RingElement),
Headline => "expression in terms of elementary symmetric polynomials",
Inputs => {"f"=> {"a symmetric", TO RingElement}},
Outputs =>{"the expression of f in terms of the elementary symmetric functions e_i"},
Usage => "elementalSymm f",
Caveat => {"if the input is not symmetric the function will announce this"},
EXAMPLE lines ///
n=5;
R=QQ[x_1..x_n];
f=(product gens R)*(sum gens R);
elementarySymmetric f
///,
PARA{"This function should work up to a size of 15 variables in the base ring"},
PARA{"This function is part of the package SymmetricPolynomials."}
}
document {
Key =>(elementarySymmetric,PolynomialRing),
Headline => "elementary symmetric polynomials algebra",
Inputs => {"R"=>{ "a", TO PolynomialRing}},
Outputs =>{"a map from R adjoin the elementary symmetric functions e_i to R"},
Usage => "elementalSymm R",
EXAMPLE lines ///
n=5;
R=QQ[x_1..x_n];
elementarySymmetric R
///,
PARA{"This function should work up to a size of 15 variables in the base ring"},
PARA{"This function is part of the package SymmetricPolynomials."}
}
document {
Key =>buildSymmetricGB,
Headline => "Groebner basis of elementary symmetric polynomials algebra",
Inputs => {"R"=>{ "a", TO Ring}},
Outputs =>{"the Groebner basis of the elementary symmetric algebra"},
Usage => "buildSymmetricGB R",
EXAMPLE lines ///
n=5;
R=QQ[x_1..x_n];
buildSymmetricGB R
///,
PARA{"This function should work up to a size of 15 variables in the base ring"},
PARA{"This function is part of the package SymmetricPolynomials."}
}
document {
Key =>(buildSymmetricGB,PolynomialRing),
Headline => "Groebner basis of elementary symmetric polynomials algebra",
Inputs => {"R"=>{ "a", TO PolynomialRing}},
Outputs =>{"the Groebner basis of the elementary symmetric algebra"},
Usage => "buildSymmetricGB R",
EXAMPLE lines ///
n=5;
R=QQ[x_1..x_n];
buildSymmetricGB R
///,
PARA{"This function should work up to a size of 15 variables in the base ring"},
PARA{"This function is part of the package SymmetricPolynomials."}
}
end
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