Products of Young symmetrizers and ideals in the generic tensor algebra

We describe a formula for computing the product of the Young symmetrizer of a Young tableau with the Young symmetrizer of a subtableau, generalizing the classical quasi-idempotence of Young symmetrizers. We derive some consequences to the structure of ideals in the generic tensor algebra and its partial symmetrizations. Instances of these generic algebras appear in the work of Sam and Snowden on twisted commutative algebras, as well as in the work of the author on the defining ideals of secant varieties of Segre-Veronese varieties, and in joint work of Oeding and the author on the defining ideals of tangential varieties of Segre-Veronese varieties.