Timeline for Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials
Current License: CC BY-SA 4.0
12 events
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| Feb 17, 2021 at 2:03 | comment | added | Nate | I think it is worth pointing out that this sort of dependence on $N$ also happens for $GL_N$ if you consider algebraic, and not just polynomial representations. For example, if we compute the character of the adjoint representation we see it has constant term $N$. | |
| Feb 16, 2021 at 21:29 | history | edited | Marcel | CC BY-SA 4.0 |
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| Feb 12, 2021 at 20:24 | history | edited | LSpice | CC BY-SA 4.0 |
Links to @RichardStanley's answer, articles, and books
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| Feb 12, 2021 at 20:13 | comment | added | Richard Stanley | I should have made my answer just a comment, since as Marcel notes it does not address the actual question. | |
| Feb 12, 2021 at 17:44 | history | edited | Marcel | CC BY-SA 4.0 |
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| Feb 12, 2021 at 17:36 | history | edited | Marcel | CC BY-SA 4.0 |
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| Feb 10, 2021 at 19:43 | comment | added | Marcel | @RichardStanley By the way, according to my calculations, $e_2=o_2+o_{11}+1$ for any $N$. | |
| Feb 10, 2021 at 18:18 | comment | added | Marcel | @RichardStanley and of course this touches on the problem of the right expression for the $o$'s. The OP presents a formula which is incorrect according to the source I mentioned. | |
| Feb 10, 2021 at 18:16 | comment | added | Marcel | @RichardStanley I don't understand what you mean. When we write $p$'s in terms of $s$'s, the coefficients don't depend on $N$. In other words the funcional relation is the same for any number of variables. The question was whether the same is true when $p$'s are written in terms of $o$'s. | |
| Feb 10, 2021 at 18:02 | comment | added | Richard Stanley | By the "character of a classical group," I mean a certain symmetric function that encodes the dimensions of the weight spaces as coefficients of monomials. It is these coefficients that depend on $N$ for $O(N)$ or $\mathrm{Sp}(2N)$. That is because the symmetric functions are being evaluated at $x_1,\dots,x_N,x_1^{-1},\dots,x_N^{-1}$ or $x_1,\dots,x_N,x_1^{-1},\dots,x_N^{-1},1$. For instance, the constant term of $e_2(x_1,\dots,x_N,x_1^{-1},\dots,x_N^{-1})$ is $N$, where $e_2$ is an elementary symmetric function. | |
| Feb 10, 2021 at 15:27 | history | edited | Marcel | CC BY-SA 4.0 |
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| Feb 10, 2021 at 14:15 | history | answered | Marcel | CC BY-SA 4.0 |