symmetric polynomials in functions
$begingroup$
If I have $N$ variables, $a_i$, $i=1,...,N$, then any symmetric polynomial in these can be expressed as a polynomial in the elementary symmetric polynomials:
$$ e_k = sum_{i_1<...<i_k} a_{i_1} ...a_{i_k}, ;;; k=1,...,N $$
Now suppose I instead have $N$ functions, $f_i(x)$, $i=1,...,N$. Now I'm interested in symmetric polynomials in the functions, which may be evaluated at different arguments, eg, something like (for $N=2$):
$$ {f_1(x)}^2 f_2(y) + {f_2(x)}^2 f_1(y) $$
Is there some analogous set of ``generating functions'' for these symmetric polynomials? Perhaps it is natural to define
$$ e_k(x_1,...x_k) = f_{1}(x_1) ...f_{k}(x_k) + text{permutations}, ;;; k=1,...,N $$
which I may take arbitrary polynomials in and evaluate at arbitrary arguments. But I'm not sure this is enough to capture everything.
abstract-algebra symmetric-polynomials
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 5 '18 at 19:58
user6013user6013
1734
$endgroup$
add a comment |
$begingroup$
If I have $N$ variables, $a_i$, $i=1,...,N$, then any symmetric polynomial in these can be expressed as a polynomial in the elementary symmetric polynomials:
$$ e_k = sum_{i_1<...<i_k} a_{i_1} ...a_{i_k}, ;;; k=1,...,N $$
Now suppose I instead have $N$ functions, $f_i(x)$, $i=1,...,N$. Now I'm interested in symmetric polynomials in the functions, which may be evaluated at different arguments, eg, something like (for $N=2$):
$$ {f_1(x)}^2 f_2(y) + {f_2(x)}^2 f_1(y) $$
Is there some analogous set of ``generating functions'' for these symmetric polynomials? Perhaps it is natural to define
$$ e_k(x_1,...x_k) = f_{1}(x_1) ...f_{k}(x_k) + text{permutations}, ;;; k=1,...,N $$
which I may take arbitrary polynomials in and evaluate at arbitrary arguments. But I'm not sure this is enough to capture everything.
abstract-algebra symmetric-polynomials
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 5 '18 at 19:58
user6013user6013
1734
$endgroup$
$begingroup$
Could you give some more examples, or try to give a more precise formulation of what kind of polynomials you want to consider? Is it just arbitrary, which argument is put into which function? So would $f_1(x)^2f_2(y) + f_2(y)^2f_1(x)$ also be a valid term?
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– red_trumpet
Dec 5 '18 at 20:34
$begingroup$
What I had in mind was that the function should be symmetric under arbitrary permutations of the labels, in this case, {1,2}->{2,1}, so your example would not satisfy this. What I'm really interested in the specific case of functions of the form $sum_{i=1}^N f_i(x_1)^{k_1} f_i(x_2)^{k_2} ...$, which are analogous to power sums in the usual case, and expressing these in terms of some finite generating set, but I'm not sure this is possible.
$endgroup$
– user6013
Dec 5 '18 at 20:54
add a comment |
$begingroup$
If I have $N$ variables, $a_i$, $i=1,...,N$, then any symmetric polynomial in these can be expressed as a polynomial in the elementary symmetric polynomials:
$$ e_k = sum_{i_1<...<i_k} a_{i_1} ...a_{i_k}, ;;; k=1,...,N $$
Now suppose I instead have $N$ functions, $f_i(x)$, $i=1,...,N$. Now I'm interested in symmetric polynomials in the functions, which may be evaluated at different arguments, eg, something like (for $N=2$):
$$ {f_1(x)}^2 f_2(y) + {f_2(x)}^2 f_1(y) $$
Is there some analogous set of ``generating functions'' for these symmetric polynomials? Perhaps it is natural to define
$$ e_k(x_1,...x_k) = f_{1}(x_1) ...f_{k}(x_k) + text{permutations}, ;;; k=1,...,N $$
which I may take arbitrary polynomials in and evaluate at arbitrary arguments. But I'm not sure this is enough to capture everything.
abstract-algebra symmetric-polynomials
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 5 '18 at 19:58
user6013user6013
1734
$endgroup$
If I have $N$ variables, $a_i$, $i=1,...,N$, then any symmetric polynomial in these can be expressed as a polynomial in the elementary symmetric polynomials:
$$ e_k = sum_{i_1<...<i_k} a_{i_1} ...a_{i_k}, ;;; k=1,...,N $$
Now suppose I instead have $N$ functions, $f_i(x)$, $i=1,...,N$. Now I'm interested in symmetric polynomials in the functions, which may be evaluated at different arguments, eg, something like (for $N=2$):
$$ {f_1(x)}^2 f_2(y) + {f_2(x)}^2 f_1(y) $$
Is there some analogous set of ``generating functions'' for these symmetric polynomials? Perhaps it is natural to define
$$ e_k(x_1,...x_k) = f_{1}(x_1) ...f_{k}(x_k) + text{permutations}, ;;; k=1,...,N $$
which I may take arbitrary polynomials in and evaluate at arbitrary arguments. But I'm not sure this is enough to capture everything.
abstract-algebra symmetric-polynomials
abstract-algebra symmetric-polynomials
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 5 '18 at 19:58
user6013user6013
1734
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 5 '18 at 19:58
user6013user6013
1734
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
edited Dec 5 '18 at 21:16
GNUSupporter 8964民主女神 地下教會
12.8k72445
12.8k72445
asked Dec 5 '18 at 19:58
user6013user6013
1734
asked Dec 5 '18 at 19:58
user6013user6013
1734
asked Dec 5 '18 at 19:58
user6013user6013
1734
1734
$begingroup$
Could you give some more examples, or try to give a more precise formulation of what kind of polynomials you want to consider? Is it just arbitrary, which argument is put into which function? So would $f_1(x)^2f_2(y) + f_2(y)^2f_1(x)$ also be a valid term?
$endgroup$
– red_trumpet
Dec 5 '18 at 20:34
$begingroup$
What I had in mind was that the function should be symmetric under arbitrary permutations of the labels, in this case, {1,2}->{2,1}, so your example would not satisfy this. What I'm really interested in the specific case of functions of the form $sum_{i=1}^N f_i(x_1)^{k_1} f_i(x_2)^{k_2} ...$, which are analogous to power sums in the usual case, and expressing these in terms of some finite generating set, but I'm not sure this is possible.
$endgroup$
– user6013
Dec 5 '18 at 20:54
add a comment |
$begingroup$
Could you give some more examples, or try to give a more precise formulation of what kind of polynomials you want to consider? Is it just arbitrary, which argument is put into which function? So would $f_1(x)^2f_2(y) + f_2(y)^2f_1(x)$ also be a valid term?
$endgroup$
– red_trumpet
Dec 5 '18 at 20:34
$begingroup$
What I had in mind was that the function should be symmetric under arbitrary permutations of the labels, in this case, {1,2}->{2,1}, so your example would not satisfy this. What I'm really interested in the specific case of functions of the form $sum_{i=1}^N f_i(x_1)^{k_1} f_i(x_2)^{k_2} ...$, which are analogous to power sums in the usual case, and expressing these in terms of some finite generating set, but I'm not sure this is possible.
$endgroup$
– user6013
Dec 5 '18 at 20:54
$begingroup$
Could you give some more examples, or try to give a more precise formulation of what kind of polynomials you want to consider? Is it just arbitrary, which argument is put into which function? So would $f_1(x)^2f_2(y) + f_2(y)^2f_1(x)$ also be a valid term?
$endgroup$
– red_trumpet
Dec 5 '18 at 20:34
$begingroup$
Could you give some more examples, or try to give a more precise formulation of what kind of polynomials you want to consider? Is it just arbitrary, which argument is put into which function? So would $f_1(x)^2f_2(y) + f_2(y)^2f_1(x)$ also be a valid term?
$endgroup$
– red_trumpet
Dec 5 '18 at 20:34
$begingroup$
What I had in mind was that the function should be symmetric under arbitrary permutations of the labels, in this case, {1,2}->{2,1}, so your example would not satisfy this. What I'm really interested in the specific case of functions of the form $sum_{i=1}^N f_i(x_1)^{k_1} f_i(x_2)^{k_2} ...$, which are analogous to power sums in the usual case, and expressing these in terms of some finite generating set, but I'm not sure this is possible.
$endgroup$
– user6013
Dec 5 '18 at 20:54
$begingroup$
What I had in mind was that the function should be symmetric under arbitrary permutations of the labels, in this case, {1,2}->{2,1}, so your example would not satisfy this. What I'm really interested in the specific case of functions of the form $sum_{i=1}^N f_i(x_1)^{k_1} f_i(x_2)^{k_2} ...$, which are analogous to power sums in the usual case, and expressing these in terms of some finite generating set, but I'm not sure this is possible.
$endgroup$
– user6013
Dec 5 '18 at 20:54
add a comment |
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$begingroup$
Could you give some more examples, or try to give a more precise formulation of what kind of polynomials you want to consider? Is it just arbitrary, which argument is put into which function? So would $f_1(x)^2f_2(y) + f_2(y)^2f_1(x)$ also be a valid term?
$endgroup$
– red_trumpet
Dec 5 '18 at 20:34
$begingroup$
What I had in mind was that the function should be symmetric under arbitrary permutations of the labels, in this case, {1,2}->{2,1}, so your example would not satisfy this. What I'm really interested in the specific case of functions of the form $sum_{i=1}^N f_i(x_1)^{k_1} f_i(x_2)^{k_2} ...$, which are analogous to power sums in the usual case, and expressing these in terms of some finite generating set, but I'm not sure this is possible.
$endgroup$
– user6013
Dec 5 '18 at 20:54