DIMENSIONS OF THE POLYNOMIAL ALGEBRA ?AS A MODULE OVER THE STEENROD ALGEBRA
Let ?be the polynomial algebra in n-variables ?over the field ?of two elements and let ?be the mod-2 Steenrod algebra. ?is identified with the mod-2 cohomology group of the n-fold product of ?with itself and thereby receives a module structure over ?A major problem is to determine a basis for the quotient vector space ?of the polynomial algebra by the image of the positive part ?of the Steenrod algebra. ?has natural grading by degree dso that ?Let ?denote the number of digits 1 in the binary expansion of m. In this paper, we conjecture that when ?the dimension of ? ?and verify the conjecture in a few cases. We also give lower bounds for the dimension of ?when ?and improve upon them in one special case.
dimension, polynomial algebra, Steenrod algebra.