I just want to consider the simplest case:


Let S=[0,0,0,0,1], how to derive the general formula for $Sym^k$ S?


My conjectured formula based on the results of LiE program for finite k values is:


$Sym^k(S)=\overset{\left[k/2\right]}{\underset{i=0}{\oplus}}[i,0,0,0,k-2i]$


But I have no clue how to prove it or even a derivation for the simplest case when k=2.


Any reference or tips would be greatly appreciated.


PS. Sorry for confusion. Now I changed the spinor representation "V" to "S".



How to calculate symmetric tensor products of SO(10) representations?
Based on comments from Eric Rowell and José Figueroa-O'Farrill, I assume that your $V$ is a half-spinor representation of $Spin\_{10}$. The key issue is that your hypothetical decomposition of $S(V)$ is *multiplicity-free*, i.e. any module that appears in it has multiplicity one. There is a beautiful theory of multiplicity-free actions that allows you to do the following.


(1) Test that $V$ is a multiplicity-free $G$-module.


(2) Find the highest weights of the simple summands in $S(V)$.


For (1), the action is multiplicity-free if and only if the Borel subgroup $B$ has an open orbit on $V$. For (2), the highest weights form a free commutative semigroup whose generators are determined from the stabilizer in $B$ of a point in the open orbit. 


An excellent survey is Roger Howe's *Schur lectures.*


All multiplicity-free linear actions (i.e. modules) of reductive algebraic groups have been classified. A convenient reference is 


Howe and Umeda, *The Capelli identity, the double commutant
theorem and multiplicity-free-actions*, Mathematische Annalen, 290, 565 - 620 (freely available through [GDZ](http://gdz.sub.uni-goettingen.de/), but tricky to link to).


Item (x) of the list (11.0.1) on p.583 is your action $Spin\_{10}\times GL\_1.$ This module arises from the action of a Levi factor of a maximal parabolic of $E\_6$ on its abelian nilradical (remove last node from the Dynkin diagram of $E\_6$ to get $D\_5$). The decomposition is described in 11.10 on p.602: there are two fundamental heighest weights, one in degree 1 corresponding to the Spin module itself and one in degree 2 corresponding to the standard 10-dimensional representation of $SO\_{10}.$ This leads to the decomposition you have stated.


