#### Document Type

Technical Report

#### Abstract

Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k : pi_*^S((B[doublestrike V]_k)_+) --> pi_*^S(S^0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3 and that T_r = ⊕_k(Tr_k) is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq^0 repeatedly at most (k- 2) times, then we get into the region, in which all the iterated squaring operations are isomorphisms on the coinvariants of the GL_k-representation. As a consequence, every finite Sq^0-family in the coinvariants has at most (k - 2) non zero elements. Two applications are exploited.

The first main theorem is that Tr_k is not an isomorphism for k gte 5. Furthermore, Tr_k is not an isomorphism in infinitely many degrees for each k > 5. We also show that if Tr_ell detects a nonzero element in certain degrees of Ker(Sq^0), then it is not a monomorphism and further, Tr_k is not a monomorphism in infinitely many degrees for each k > ell.

The second main theorem is that the elements of any Sq^0-family in the cohomology of the Steenrod algebra, except at most its first (k - 2) elements, are either all detected or all not detected by Tr_k, for every k. Applications of this study to the cases k = 4 and 5 show that Tr_4 does not detect the three families g, D_3, p' and Tr_5 does not detect the family {h_(n+1)g_n|n gte 1}.

#### Number in Series

2003.02

#### Disciplines

Algebraic Geometry | Applied Mathematics | Geometry and Topology | Mathematics

#### AMS Subject Classification

Primary 55P47, 55Q45, 55S10, 55T15

#### Recommended Citation

Hu'ng, Nguyen H. V., "The Cohomology of the Steendrod Algebra and Representations of the General Linear Groups" (2003). *Mathematics Research Reports*. 8.

https://digitalcommons.wayne.edu/math_reports/8

## Comments

The work was supported in part by the National Research Program, Grant N°140801