Subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras

Abstract

Let A( t) be the subalgebra of the Steenrod algebra generated by Sq i , i ⩽ 2 t . We determine which A-module indecomposable summands of F 2[ x 1 …, x n ] are free over A( t) by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of M n( F 2) (or Gl n( F 2)) occur as composition factors in the truncated polynomial algebra F 2[ x 1, …, x n] (x 1 2′, …, x n 2′) . The second question is then answered.