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Can anyone guide me to a reference where I can find a table of some Schur polynomials? Schur polynomials of degree $m$ are indexed by partitions of $m$ and form a basis of symmetric polynomials of degree $m$. If I could find a table with even some Schur polynomials corresponding to partitions of 3,4 and perhaps 5, that would be great.

Thanks!

Bernard
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Eugene
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2 Answers2

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There's a table on page 315 (Figure 7-4) of Stanley's Enumerative Combinatorics Vol. II expressing the first few in terms of the monomial symmetric functions:

Schur functions

Qiaochu Yuan
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  • Thank you very much!! To make sure I have this correct, is it true that $m_{21} = x_1^2x_2 + x_2x_1^2$, whilst $m_{111}=x_1x_2x_3+x_2x_3x_1+x_3x_2x_1+x_1x_3x_2+x_2x_1x_3+x_3x_1x_2 = 6x_1x_2x_3$? It seems strange to me that in the second example I should need to apply these permutations even though the polynomial is symmetric already. – Eugene Sep 03 '20 at 13:00
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    @Herb: the sum defining monomial symmetric functions is just a sum over orbits (the definition is in section 7.3 of Stanley), but on the other hand we might have more than $3$ variables. $m_{111}$ is the elementary symmetric function $e_3$, and in general $m_{1^k}$ is the elementary symmetric function $e_k$. Interpreted as a Schur function this corresponds to the character of the representation of $GL(V)$ given by $\Lambda^k(V)$. – Qiaochu Yuan Sep 04 '20 at 02:47
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You can get these polynomials with:

Here is an example with R:

library(jack)
SchurPol(n = 3, lambda = c(2, 1))
# x^(0, 2, 1) + x^(2, 0, 1) + x^(2, 1) + x^(1, 2) + 2*x^(1, 1, 1) + x^(1, 0, 2) + x^(0, 1, 2)

These packages are able to compute the Jack polynomials, the Schur polynomials, and the zonal polynomials.

Stéphane Laurent
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