Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or *plethysms* of irreducible spaces, this is not in general irreducible.

As an example Let $U \equiv (\bigwedge^3 V) \otimes (Sym^2 V) \subset V^{\otimes 5}$.

The Littlewood-Richardson algorithm can applied in this case to find the irreducible spaces contained in $U$. For and arbitrary element $ X \equiv ( v_1 \wedge v_2 \wedge v_3 )\otimes (w_1 \otimes_s w_2) \in (\bigwedge^3 V) \otimes (Sym^2 V)$. The algorithm generates a standard tableau, for concreteness one of them will be, $T_{\lambda} = (v_1 w_1, v_2, v_3, w_2)$ where *comma* separates the rows. The young symmetriser associated with this is $c_{\lambda}$ and acts in the obvious manner on $( v_1 \wedge v_2 \wedge v_3 )\otimes (w_1 \otimes_s w_2) $. Thus span of $c_\lambda X$ will generate an irreducible subspace of $U$.

**Question 1)** Given a decomposition of a reducible space into irreducibles. Any element can be uniquely decomposed as the sum of elements each living in a irreducible space. My question is what is the projection operator $P_{\lambda}$ associated with a standard young tableau generated by Littlewood-Richardson, for a given vector in the reducible space ?

I understand that young symmetriser is an idempotent and will give one of the elements in the irreducible space, the span of which generates the whole irreducible space. However I don't believe this projects the 'correct' component, given an arbitrary element.

In our running example, $c_{\lambda}X$ would be one of the elements in the irrep. However in general it is not true that $c_\lambda X$ is the orthogonal component of $X$ inside the irreducible rep characterised by the standard tableau $\lambda$ generated by Littlewood-richardson algorithm.

**Question 2)**
In general is there a procedure find out the 'orthogonal components' living in irrep characterised by the standard Young tableau ?