How can I split a polynomial into parts which are symmetric and antisymmetric under exchange of the variables? I have an explicit polynomial, which is a function of two variables (and some further constants). In the following image you can see the polynomial as well as further explanations. Thanks for your help!
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Thomas
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the polynomial $xy^2$ is not symmetric nor antsymmetric in $x,y$. What do you exactly mean by "splitting"? – user126154 Mar 26 '15 at 11:57
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1$(f(x,y)+f(y,x))/2,(f(x,y)-f(y,x))/2$ – Gerry Myerson Mar 26 '15 at 12:07
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My answer to your other question (http://math.stackexchange.com/questions/1206372/splitting-a-polynomial-into-parts-which-are-symmetric-and-antisymmetric-under-ex) is also an answer to this question; it works in any number of variables. – Qiaochu Yuan Mar 26 '15 at 16:35
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@GerryMyerson. What if I have three variables and want to make my polynomial anti-symmetric under cyclic permutations? – Thomas Mar 27 '15 at 22:19
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If you're asking what I think you're asking, it doesn't make sense. It sounds like you want a polynomial with $f(x,y,z)=-f(y,z,x)$ and $f(x,y,z)=-f(z,x,y)$, but then you get $f(y,z,x)=f(z,x,y)$, a contradiction unless $f$ is identically zero. But maybe that's not what you were asking. – Gerry Myerson Mar 28 '15 at 05:08
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I'm sorry, I asked the wrong question. I have three variables and would like my polynomial to be anti-symmetric under exchange of any two of them. – Thomas Mar 28 '15 at 14:04