Realizability and nonrealizability of Dickson algebras as cohomology rings

Abstract:

Fix a prime $p$ and let $V$ be an $n$-dimensional vector space over ${\mathbf {Z}}/p$. The general linear group ${\text {GL(}}V{\text {)}}$ of $V$ acts on the polynomial ring $P(V)$ on $V$. The ring of invariants $P{(V)^{{\text {GL}}(V)}}$ has been computed by Dickson, and we denote it by ${D^*}(n)$. If we grade $P(V)$ by assigning the elements of $V$ the degree 2, then ${D^*}(n)$ becomes a graded polynomial algebra on generators ${Y_1}, \ldots ,{Y_n}$ of degrees $2{p^n} - 2{p^{n - 1}}, \ldots ,2{p^n} - 2$. The $\mod p$ Steenrod algebra acts on $P(V)$ in a unique way compatible with the unstability condition and the Cartan formula. The ${\text {GL}}(V)$ action commutes with the Steenrod algebra action, and so ${D^*}(n)$ inherits the structure of an unstable polynomial algebra over the Steenrod algebra. In this note we determine explicit formulas for the action of the Steenrod algebra on the polynomial generators of ${D^*}(n)$. As a consequence we are able to decide exactly which Dickson algebras can be ${\mathbf {Z}}/p$ cohomology rings.