Math 480, Spring 2014, Assignment 4

Carefully define the following terms, then give one example and one non-example of each:

1. Invariant polynomial (for a given group $G$ of $n\times n$ matrices).
2. Symmetric polynomial (in $n$ variables).
3. Elementary symmetric polynomial (in $n$ variables).

Carefully state the following theorems (you need not prove them):

1. Theorem identifying generators of the algebra of symmetric polynomials.

Solve the following problems:

1. Working in $\mathsf{k}[x,y,z]$, write the symmetric polynomial $$f = x^2y + x^2z + xy^2 + 6xyz + xz^2 + y^2z + yz^2$$ as a polynomial in the elementary symmetric polynomials $\sigma_1,\sigma_2,\sigma_3$. (Hint: working with lex order, find a product of the form $c\sigma_1^{d_1}\sigma_2^{d_2}\sigma_3^{d_3}$ whose leading term matches that of $f$. Form the "remainder" $r = f - c\sigma_1^{d_1}\sigma_2^{d_2}\sigma_3^{d_3}$, which is again a symmetric polynomial. Then repeat this process with $r$ in place of $f$.)

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Questions:

Following up on the discussion of symmetric polynomials, I was interested in counting the total number of symmetric polynomials in n variables. Let \(\sigma(k,n)\) be the number of symmetric polynomials in n variables where each monomial is of degree k. Let's look at \(\sigma(4,4)\) and just list one representative monomial (lex order) for simplicity's sake.

\(wxyz, w^2xy, w^2x^2, w^3x, w^4\)

It should be clear that \(\sigma(k,n) = P(k)\) where \(P(k)\) is the number of ways of partitioning k into addends, where order isn't relevant. This is independent of n, as long as \(k \leqq n\). Starting with \(k=1\), the first ten terms are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42 and is listed in the Online Encyclopedia of Integer Sequences (OEIS) as [1] (beginning with \(k=0\)). Then the total number of symmetric polynomials in n variables \( = \sum_{k=1}^{n}P(k)\), which is the partial sum of the partition numbers. This new sequence (or series of prior sequence) begins with 1, 3, 6, 11, 18, 29, 44, 66, 96, 138 and is listed in the OEIS as [2] "number of sums S of positive integers satisfying S <= n". However, none of the comments thus far say "number of symmetric polynomials of degree n", so unless someone corrects me, or if Prof. Jackson agrees with the result, I'll submit this interpretation as a comment to the sequence.--Matthew.Lehman (talk) 02:54, 22 February 2014 (EST)

Originally, I wrote symmetric polynomial, then changed it to elementary symmetric polynomial before I understood what that meant. That's clearly wrong, there's only one elementary symmetric polynomial for each \((k,n)\). For example, let's take \(n=3\).

\(\sigma_1 = x+y+z\)

\(\sigma_2 = xy+yz+xz\), (but I also had \(x^2+y^2+z^2\))

\(\sigma_3 = xyz\), (but I also had \(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2\) and \(x^3+y^3+z^3\))

However, \(x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+xz) = (\sigma_1)^2-2\sigma_2\).

Also, \(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2 = \sigma_1\sigma_2-3\sigma_3\).

And \(x^3+y^3+z^3 = (\sigma_1)^3-3(\sigma_1\sigma_2+\sigma_3)\).

I encourage you to check my work, it's possible that I might have messed up on these calculations and it might help understand what's going on if you're confused (it helped me). However, each symmetric polynomial can be expressed as sums and products of elementary symmetric polynomials, if not these precise ones.

I changed elementary symmetric polynomial back to symmetric polynomial, so hopefully my original claim still stands.--Matthew.Lehman (talk) 02:36, 23 February 2014 (EST)