On the modular representations of the general linear group $GL_5$ and the hit problem for the polynomial algebra of five variables
Đặng Võ Phúc
Abstract
The Steenrod ring over the finite field $\mathbb Z/2$ of two elements is denoted by $\mathcal A.$ We knew that the $\mathbb Z/2$-cohomology of the product of $h$ copies of the infinite real projective space $\mathbb RP(\infty)$ can be identified with $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h],$ a polynomial algebra on $h$ generators with the degree of each $t_i$ being one. Moreover, this $P^{\otimes h}$ equipped with the (left) unstable $\mathcal A$-module structure. In the work [Abstracts Amer. Math. Soc. 833 (1987)], Franklin Peterson proposed the "hit" problem of determination a minimal generating set for the $\mathcal A$-module $P^{\otimes h}.$ Equivalently, the hit problem is to find a monomial basis for the unhit space $\mathbb Z/2\otimes_{\mathcal A}P^{\otimes h}$ in each positive degree. This problem, which remains open for arbitrary $h\geq 5,$ plays an important role in discovering some classical problems in homotopy theory. Singer's algebraic transfer of rank $h$ [Math. Z. 202, 493-523 (1989)], which passes from the coinvariants of certain representation of the general linear group $GL_h$ of rank $h$ over $\mathbb Z/2$ to the $\mathbb Z/2$-cohomology group ${\rm Ext}_{\mathcal A}^{h, h+*}(\mathbb Z/2, \mathbb Z/2)$ of the ring $\mathcal A,$ is a relatively efficient tool to describe mysterious Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains unanswered for all ranks $\geq 5.$It is very difficult to calculate both sides of Singer's transfer in each positive degree. So, in the present study, we wish to use techniques of the hit problem to study this transfer for rank 5 in certain internal degrees. The approach is quite effective to determine the transfer. Especially, our results have shown that for each integer $s\geq 0,$ the cohomology group ${\rm Ext}_{\mathcal A}^{5, 67.2^{s}}(\mathbb Z/2, \mathbb Z/2)$ is isomorphic to the image of the dual of the invariant $[\mathbb Z/2\otimes_{\mathcal A}P^{\otimes 5}]^{GL_5}$ under the fifth Singer transfer in general degree $67.2^{s}-5.$ As immediate consequences, we notice that all decomposable elements $h_{s+1}H_1(s)\in {\rm Ext}_{\mathcal {A}}^{6,69.2^{s}}(\mathbb Z/2,\mathbb Z/2),\, s\geq 0$ are in the image the sixth algebraic transfer and that Singer's conjecture is true for rank $5$ in respective degrees. The obtained results are new and important for this rank.