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Obtaining the immanent associated to a partition

asked 3 years ago

klaaa gravatar image

updated 3 years ago

For a partition λ let yλ be the corresponding irreducible representation of the symmetric group Sn. Let pλ=πSnyλ(π)x1π(1)...xnπ(n) be the immanent corresponding to λ. (For the sign representation we will just get the determinant for example). This is a polynomial in the n2 variables xi,j over Z.

My question is how can I obtain the immanent given a parition λ using Sage?

My first problem is already that we need the polynomial ring in the n2 variables xi,j and I am not sure how to define this in Sage depending on n.

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answered 3 years ago

rburing gravatar image

updated 3 years ago

Here are some immanants:

n = 3
G = SymmetricGroup(n)
R = PolynomialRing(ZZ, n, n, var_array=('x',))
x = {(i, j) : R.gen(n*i + j) for i in range(n) for j in range(n)}
for lamb in SymmetricGroupRepresentations(n):
    chi = lamb.to_character()
    imm = sum(chi(sigma)*prod(x[i,sigma(i+1)-1] for i in range(n)) for sigma in G) 
    print(imm)

Output:

x02*x11*x20 + x01*x12*x20 + x02*x10*x21 + x00*x12*x21 + x01*x10*x22 + x00*x11*x22
-x01*x12*x20 - x02*x10*x21 + 2*x00*x11*x22
-x02*x11*x20 + x01*x12*x20 + x02*x10*x21 - x00*x12*x21 - x01*x10*x22 + x00*x11*x22

To do it for a single partition, use e.g. SymmetricGroupRepresentation([3,2]) to create the representation; see SymmetricGroupRepresentation.

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Asked: 3 years ago

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Last updated: Apr 23 '21