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2 Latex correction

You may also use SAGE , (for example, the Sage online notebook )

Example:

The Riemann curvature tensor $R$ lives in the space $Sym^2(\Lambda^2 V)$ (after identifying $V$ with $V^{\vee}$)

Decomposing it in Sage:

$s = SFASchur(QQ)$

(let s be the Schur functor)

(compute plethysm $Sym^2 \Lambda^2$) Lambda^2 $) s[1, 1, 1, 1] + s[2, 2] -- i.e.,$\Lambda^4 V + s[2,2]$S_{[2,2]}$, as it should be

$s([3])(s([1,1])) s[1, 1, 1, 1, 1, 1] + s[2, 2, 1, 1] + s[3, 3] -- though i understand that the explicit formula is better :) 1 You may also use SAGE , (for example, the Sage online notebook ) Example: The Riemann curvature tensor$R$lives in the space$Sym^2(\Lambda^2 V)$(after identifying$V$with$V^{\vee}$) Decomposing it in Sage:$ s = SFASchur(QQ) $ (let s be the Schur functor) $ s(2)(s([1,1])) $ (compute plethysm$Sym^2 \Lambda^2$)  s[1, 1, 1, 1] + s[2, 2] -- i.e.,$\Lambda^4 + s[2,2]$, as it should be$ s([3])(s([1,1]))

s[1, 1, 1, 1, 1, 1] + s[2, 2, 1, 1] + s[3, 3]