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show/hide this revision's text 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)

$ s(2s([2])(s([1,1])) (s([1,1])) $

(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 :)
show/hide this revision's text 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]