Young symmetrizers and additive theory
Cristina Caldeira, J.A. Dias da Silva · Linear and Multilinear Algebra · 2003
For λ partition of m and A finite nonempty subset of a field we define the set of λ-restricted sums of m-tuples of elements of A, ∧ m λA, and using Additive Number Theory results from [J.A. Dias da Silva and Y.O. Hamidoune ( 1990 ). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune ( 1994 ). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain a lower bound for its cardinality. Next, using results and techniques from [J.A. Dias da Silva and Y.O. Hamidoune ( 1990 ). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune ( 1994 ). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain lower bounds for the degrees of minimal polynomials of restrictions of derivations to ranges of Young symmetrizers and to the symmetry class of tensors V λ, and we show that the lower bound for the cardinality of ∧ m λA can also be obtained from these lower bounds.