Highest Weight Vectors in Plethysms, II

For an irreducible polynomial representation V of the general linear group GL n ( C ) , we realize its symmetric square S 2 ( V ) and its alternating square Λ 2 ( V ) as spaces of polynomial functions. In the case when V is labeled by a Young diagram with at most 2 rows, we describe explicitly all the GL n ( C ) highest weight vectors which occur in V ⊗ V , S 2 ( V ) and Λ 2 ( V ) respectively. In particular, we obtain new description of the multiplicities in S 2 ( V ) and Λ 2 ( V ) .