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]

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 V + 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 :)