Schur polynomials form a basis for the space of symmetric polynomials
Let $lambda = (lambda_1 geq lambda_2 geq ... geq lambda_n)$ be a partition of $lambda$ (or Young diagram). Schur polynomial for given partition is
$$ s_lambda = frac{det begin{pmatrix} x_1^{lambda_n} & x_2^{lambda_n} & ldots & x_n^{lambda_n} \ x_1^{lambda_{n-1}+1} & x_2^{lambda_{n-1}+1} & ldots & x_n^{lambda_{n-1}+1} \ vdots & vdots & ddots & vdots \ x_1^{lambda_1 + n - 1} & x_2^{lambda_1 + n - 1} & ldots & x_n^{lambda_1 + n - 1} end{pmatrix}}{displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)},
$$
where $displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)$ is a $det$ of Vandermonde matrix.
Prove that Schur polynomials form a basis for the space of symmetric polynomials of $n$ variables.
I know that any symmetric polynomial can be represented as sum and product of elementary symmetric polynomials. So is there any way how any elementary symmetric polynomials can be represented in Schur polynomials and thus any symmetric polynomial?
Thanks!
polynomials symmetric-polynomials
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
asked Jun 6 '17 at 8:45
Gleb Chili
43228
add a comment |
Let $lambda = (lambda_1 geq lambda_2 geq ... geq lambda_n)$ be a partition of $lambda$ (or Young diagram). Schur polynomial for given partition is
$$ s_lambda = frac{det begin{pmatrix} x_1^{lambda_n} & x_2^{lambda_n} & ldots & x_n^{lambda_n} \ x_1^{lambda_{n-1}+1} & x_2^{lambda_{n-1}+1} & ldots & x_n^{lambda_{n-1}+1} \ vdots & vdots & ddots & vdots \ x_1^{lambda_1 + n - 1} & x_2^{lambda_1 + n - 1} & ldots & x_n^{lambda_1 + n - 1} end{pmatrix}}{displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)},
$$
where $displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)$ is a $det$ of Vandermonde matrix.
Prove that Schur polynomials form a basis for the space of symmetric polynomials of $n$ variables.
I know that any symmetric polynomial can be represented as sum and product of elementary symmetric polynomials. So is there any way how any elementary symmetric polynomials can be represented in Schur polynomials and thus any symmetric polynomial?
Thanks!
polynomials symmetric-polynomials
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
asked Jun 6 '17 at 8:45
Gleb Chili
43228
add a comment |
Let $lambda = (lambda_1 geq lambda_2 geq ... geq lambda_n)$ be a partition of $lambda$ (or Young diagram). Schur polynomial for given partition is
$$ s_lambda = frac{det begin{pmatrix} x_1^{lambda_n} & x_2^{lambda_n} & ldots & x_n^{lambda_n} \ x_1^{lambda_{n-1}+1} & x_2^{lambda_{n-1}+1} & ldots & x_n^{lambda_{n-1}+1} \ vdots & vdots & ddots & vdots \ x_1^{lambda_1 + n - 1} & x_2^{lambda_1 + n - 1} & ldots & x_n^{lambda_1 + n - 1} end{pmatrix}}{displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)},
$$
where $displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)$ is a $det$ of Vandermonde matrix.
Prove that Schur polynomials form a basis for the space of symmetric polynomials of $n$ variables.
I know that any symmetric polynomial can be represented as sum and product of elementary symmetric polynomials. So is there any way how any elementary symmetric polynomials can be represented in Schur polynomials and thus any symmetric polynomial?
Thanks!
polynomials symmetric-polynomials
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
asked Jun 6 '17 at 8:45
Gleb Chili
43228
Let $lambda = (lambda_1 geq lambda_2 geq ... geq lambda_n)$ be a partition of $lambda$ (or Young diagram). Schur polynomial for given partition is
$$ s_lambda = frac{det begin{pmatrix} x_1^{lambda_n} & x_2^{lambda_n} & ldots & x_n^{lambda_n} \ x_1^{lambda_{n-1}+1} & x_2^{lambda_{n-1}+1} & ldots & x_n^{lambda_{n-1}+1} \ vdots & vdots & ddots & vdots \ x_1^{lambda_1 + n - 1} & x_2^{lambda_1 + n - 1} & ldots & x_n^{lambda_1 + n - 1} end{pmatrix}}{displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)},
$$
where $displaystyle{prod_{1 leq i < j leq n}} (x_i - x_j)$ is a $det$ of Vandermonde matrix.
Prove that Schur polynomials form a basis for the space of symmetric polynomials of $n$ variables.
I know that any symmetric polynomial can be represented as sum and product of elementary symmetric polynomials. So is there any way how any elementary symmetric polynomials can be represented in Schur polynomials and thus any symmetric polynomial?
Thanks!
polynomials symmetric-polynomials
polynomials symmetric-polynomials
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
asked Jun 6 '17 at 8:45
Gleb Chili
43228
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
asked Jun 6 '17 at 8:45
Gleb Chili
43228
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
edited Jun 6 '17 at 9:25
Mee Seong Im
2,7551517
2,7551517
asked Jun 6 '17 at 8:45
Gleb Chili
43228
asked Jun 6 '17 at 8:45
Gleb Chili
43228
asked Jun 6 '17 at 8:45
Gleb Chili
43228
43228
add a comment |
add a comment |
1 Answer
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Too long for a comment: It is given e.g. in Mike Zabrocki's Notes, page 75, part 4 that
$$
s_lambda= sum_{lambda vdash|mu|}K_{mulambda}m_lambda,
$$
with $K_{mulambda}$ being Kostka numbers. Further on page 40, part 3 a table how to convert power, elementary, monomial, homogenous and forgotten symmetric polynomials is presented including inverses.
edited Nov 26 at 5:58
darij grinberg
10.2k33061
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
add a comment |
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1 Answer
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1 Answer
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active
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Too long for a comment: It is given e.g. in Mike Zabrocki's Notes, page 75, part 4 that
$$
s_lambda= sum_{lambda vdash|mu|}K_{mulambda}m_lambda,
$$
with $K_{mulambda}$ being Kostka numbers. Further on page 40, part 3 a table how to convert power, elementary, monomial, homogenous and forgotten symmetric polynomials is presented including inverses.
edited Nov 26 at 5:58
darij grinberg
10.2k33061
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
add a comment |
Too long for a comment: It is given e.g. in Mike Zabrocki's Notes, page 75, part 4 that
$$
s_lambda= sum_{lambda vdash|mu|}K_{mulambda}m_lambda,
$$
with $K_{mulambda}$ being Kostka numbers. Further on page 40, part 3 a table how to convert power, elementary, monomial, homogenous and forgotten symmetric polynomials is presented including inverses.
edited Nov 26 at 5:58
darij grinberg
10.2k33061
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
add a comment |
Too long for a comment: It is given e.g. in Mike Zabrocki's Notes, page 75, part 4 that
$$
s_lambda= sum_{lambda vdash|mu|}K_{mulambda}m_lambda,
$$
with $K_{mulambda}$ being Kostka numbers. Further on page 40, part 3 a table how to convert power, elementary, monomial, homogenous and forgotten symmetric polynomials is presented including inverses.
edited Nov 26 at 5:58
darij grinberg
10.2k33061
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
Too long for a comment: It is given e.g. in Mike Zabrocki's Notes, page 75, part 4 that
$$
s_lambda= sum_{lambda vdash|mu|}K_{mulambda}m_lambda,
$$
with $K_{mulambda}$ being Kostka numbers. Further on page 40, part 3 a table how to convert power, elementary, monomial, homogenous and forgotten symmetric polynomials is presented including inverses.
edited Nov 26 at 5:58
darij grinberg
10.2k33061
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
edited Nov 26 at 5:58
darij grinberg
10.2k33061
edited Nov 26 at 5:58
darij grinberg
10.2k33061
edited Nov 26 at 5:58
darij grinberg
10.2k33061
10.2k33061
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
answered Jun 6 '17 at 10:31
draks ...
11.6k644128
11.6k644128
add a comment |
add a comment |
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