Checking if a poly. is sym.

Question

Hello I have a general question about find elementary symmetric functions for symmetric polynomial.

My classmate suggested that you could just try to find the elementary symmetric functions for ANY polynomial given and if you could find one, then it is symmetric.

But my perspective is: You first have to check that the polynomial satisfies the definition of symmetric first, correct?

For example, if we have (a^2)(b)+ (b^2)(c) + (c^2)(a). Since the highest degree is 3 = n, so we are considering the symmetric group of 3 and all their elements. Their elements as we know are (1,2),(1,3),(2,3),(1,2,3) and (1,3,2) and so we need to use these elements to try to permute the given formula.

Clearly, we see that the polynomial won't satisfy (1,2), (1,3) and (2,3) (yet it satisfies (1,2,3). So we can conclude that this polynomial is partially symmetric but aren't symmetric for all elements of S3, so technically we CANNOT express this poylnomial in terms of elementary symmetric function, correct?

I would really appreciate it if you could confirm this with me.

My other question is the polynomial I stated is different from the one in the picture below correct? Since the picture below is a sum of variables for ALL i and j, which if we have three variables, we should have 6 terms. Yet the formula I stated only has 3. Thanks for confirming this as well!

Answer 1

My classmate suggested that you could just try to find the elementary symmetric functions for ANY polynomial given and if you could find one, then it is symmetric.

If you can write a polynomial in terms of the elementary symmetric polynomials then, yes, the given polynomial is symmetric.

However, if you don't succeed, then you don't know whether such an expression exists but you failed to find it, or the polynomial is in fact not symmetric.

But my perspective is: You first have to check that the polynomial satisfies the definition of symmetric first, correct?

Indeed, this is the recommended way in general.

For example, if we have (a^2)(b)+ (b^2)(c) + (c^2)(a) [...] technically we CANNOT express this poylnomial in terms of elementary symmetric function, correct?

Correct. The polynomial is not symmetric since for example $P(a,b,0)=a^2b \ne b^2a = P(b,a,0)$.

My other question is the polynomial I stated is different from the one in the picture below correct? Since the picture below is a sum of variables for ALL i and j, which if we have three variables, we should have 6 terms. Yet the formula I stated only has 3.

That's correct, the other polynomial would be $a^2b+a^2c+b^2c+b^2a+c^2a+c^2b$ in your notation, and it is a symmetric polynomial.