Representing symmetric polynomials

[Fulvio Gesmundo]

Hi,

Is there a way to express a symmetric polynomial in terms of elementary symmetric functions (or power sums, or Schur functions or etc)?

The SchurRings package has commands to pass from one representation to the other (toE, toP, toH) but I could not find a way to start from an actual polynomial in, say, x_1, ... , x_5 which happens to be symmetric and obtain its representation in terms of, for instance, elementary symmetric functions.

Thank you

Best

Fulvio

[Paul Zinn-Justin]

not sure if it's already coded somewhere, but

here's the code I use in my upcoming package "CotangentSchubert":

elem := (i,vrs) -> sum(subsets(vrs,i), product);
expandElem := (P,vrs,els) -> (
    if P == 0 then return 0;
    c := coefficients(P,Variables=>vrs);
    M := c#0_(0,0); C := c#1_(0,0);
    e := append((first exponents M)_(apply(vrs,index)),0);
    ee := apply(#vrs, i -> e#i - e#(i+1));
    if any(ee, i->i<0) then error "nonsymmetric polynomial";
    Q := P - C * product(#vrs, i -> (elem(i+1,vrs))^(ee#i));
    sub(C,ring first els) * product(#vrs, i -> (els#i)^(ee#i)) + expandElem(Q,vrs,els)
    )

ex of use:

i3 : R=QQ[x_1..x_5]

o3 = R

o3 : PolynomialRing

i4 : S=QQ[e_1..e_5]

o4 = S

o4 : PolynomialRing

i5 : expandElem(sum(1..5,i->x_i^3),gens R,gens S)

      3
o5 = e  - 3e e  + 3e
      1     1 2     3

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[d.markushevich]

You can use the command "elementarySymmetric" from the package "SymmetricPolynomials".
(Not appropriate when the number of variables is big.)

An example of use:

R=QQ[x_0..x_5];
loadPackage "SymmetricPolynomials";
-- Sum of powers of variables:
SumP = n -> sum(#(gens R),i->R_i^n);
for i from 1 to 6 do (print "\n"; print elementarySymmetric (SumP i));

Output:

i4 : for i from 1 to 6 do (print "\n"; print elementarySymmetric (SumP i));

e
 1


 2
e  - 2e
 1     2


 3

e  - 3e e  + 3e
 1     1 2     3

 4     2               2
e  - 4e e  + 4e e  + 2e  - 4e
 1     1 2     1 3     2     4


 5     3       2         2
e  - 5e e  + 5e e  + 5e e  - 5e e  - 5e e  + 5e
 1     1 2     1 3     1 2     1 4     2 3     5


 6     4       3       2 2     2                          3             2
e  - 6e e  + 6e e  + 9e e  - 6e e  - 12e e e  + 6e e  - 2e  + 6e e  + 3e  - 6e
 1     1 2     1 3     1 2     1 4      1 2 3     1 5     2     2 4     3     6

[Luke Oeding]

I stumbled across this in my inbox. Here's a sample of code I've used to get the change of basis matrices between different bases of the ring of symmetric functions:

--

s = 5
R = QQ[x_0..x_s]
L  = apply(partitions(s), k-> toList k)
E = id_(ZZ^(s));
repeats = P -> entries sum(P,p-> E_(p-1))
Mon1 = P-> product(#P, j-> x_j^(P_j))
Mon = L -> 1/(product(repeats L, l-> l!))* sum(subsets(s,#L), p -> sum(permutations p, q-> product(#L, i-> x_(q_i)^(L_i))))
Mon {2,2}
Pow = L -> product(L, l-> sum(s, i->x_i^l))
Pow {3,2}

elem = r-> sum(subsets(s,r), L-> product(L, l-> x_l))
elem1 = r-> product(0..(r-1), l-> x_l)

Elem = L -> product(L, l-> elem l)
Elem {1,1}
Elem1 = L -> product(L, l-> elem1 l)
Elem1 {1,1}
Elem {3,2}

elem 3


homog1 = r-> sum(partitions r, P-> product(#P, l-> x_l^(P_l)))
homog1 5
homog = r -> sum(partitions r, P -> Mon toList P  )
Homog = L -> product(L, l-> homog l)
Homog1 = L -> product(L, l-> homog1 l)
homog 3


MM = matrix apply(L, l-> apply(L, k-> contract(Mon1 k, Mon l)))
contract(x_0+x_1, x_0^3 + x_1^2*x_0)

contract( Mon1 {5}, Elem {5})

ElM = matrix  apply(L, l-> apply(L, k-> contract(Mon1 k, Elem l)))

PM = matrix apply(L, l-> apply(L, k-> contract(Mon1 k, Pow l)))

HM = matrix apply(L, l-> apply(L, k-> contract(Mon1 k, Homog l)))

EE = matrix apply(L, l-> apply(L, k-> contract(Elem1 k, Elem l)))
ME = matrix apply(L, l-> apply(L, k-> contract(Elem1 k, Mon l)))
PE = matrix apply(L, l-> apply(L, k-> contract(Elem1 k, Pow l)))
HE = matrix apply(L, l-> apply(L, k-> contract(Elem1 k, Homog l)))


HoH = matrix apply(L, l-> apply(L, k-> contract(Homog1 k, Homog l)))
EH = matrix apply(L, l-> apply(L, k-> contract(Homog1 k, Elem l)))
MH = matrix apply(L, l-> apply(L, k-> contract(Homog1 k, Mon l)))
PH = matrix apply(L, l-> apply(L, k-> contract(Homog1 k, Pow l)))

[Fulvio Gesmundo]

Thank you for the answers!

Best

Fulvio