View source for Dickson algebra

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Let $\mathbf{F} _ { q }$ denote the field with $q$ elements (cf. [[Finite field|Finite field]]) and $V$ an $n$-dimensional $\mathbf{F} _ { q }$-vector space (cf. also [[Vector space|Vector space]]). Let $S ( V )$ denote the [[symmetric algebra]] generated by $V$ over $\mathbf{F} _ { q }$. Since the [[general linear group]], $\operatorname{GL} ( V ) = \operatorname { Aut } _ { \mathbf{F} _ { q } } ( V )$, acts on $V$, there is an induced action on the algebra $S ( V )$.  L.E. Dickson determined the structure of the $\operatorname{GL} ( V )$-fixed subalgebra, $S ( V ) ^ { G L ( V ) }$, now known as the ''Dickson algebra''. In [[#References|[a5]]] (see also [[#References|[a2]]], p. 90) $S ( V ) ^ { \operatorname{GL} ( V ) }$ was shown to be a polynomial algebra of the form $\mathbf{F}_q[c_{n,n-1},c_{n,n-2},\ldots,c_{n,0}]$ where the $\{ c _ { n,j }  \}$ are homogeneous polynomials in $S ( V )$, called the Dickson invariants (sometimes this term is used to refer to any element of the Dickson algebra), and are constructed in the following manner. Let $K$ be a field extension of $\mathbf{F} _ { q }$ which contains $V$ (cf. also [[Extension of a field|Extension of a field]]). Then the monic separable polynomial whose roots are precisely the elements of $V$ has the form

\begin{equation*} 
f ( X ) = X ^ { q ^ { n } } + \sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ { n - i } c _ { n , i } X ^ { q ^ { i } } \in K [ X ].
\end{equation*}

Suppose that $W$ is an $n$-dimensional $\mathbf{F} _ { p }$-vector space ($p$ a prime number). When $p = 2$, the cohomology algebra $H ^ { * } ( W ; \mathbf{F} _ { 2 } )$ is isomorphic to $S ( H ^ { 1 } ( W ; \mathbf{F} _ { 2 } ) )$. Therefore, over $\mathbf{F} _ { p }$ it is natural to endow $S ( V )$ with the structure of a [[graded algebra]] with $V$ of dimension one if $p = 2$ and of dimension two if $p$ is odd.

In this topological manifestation, the Dickson algebra has proved very useful ([[#References|[a1]]], [[#References|[a4]]], [[#References|[a8]]], [[#References|[a10]]]). For example, if $\rho : W \rightarrow O _ { 2^{n} }  ( \mathbf{R} )$ is the real [[Regular representation|regular representation]] of $W$ and $V = H ^ { 1 } ( W ; \mathbf{F} _ { 2 } )$, then the [[Stiefel–Whitney class|Stiefel–Whitney class]] satisfies $w _ { 2 ^ { n } - 2 ^ { i } } ( \rho ) = c _ { n , i }$ [[#References|[a7]]]. When $p$ is odd, $c _ { n , i }$ is related in a similar manner to Chern classes of the regular representation. The observation shows that the Dickson algebra becomes a graded algebra together with an action by the [[Steenrod algebra]] of cohomology operations. In [[algebraic topology]], several other algebras of this type occur, among these the lambda algebra and the Dyer–Lashof algebra of homology operations and are related to the Dickson algebra ([[#References|[a6]]], [[#References|[a9]]], [[#References|[a10]]]).

The corresponding algebras of invariants have been computed when $\operatorname{GL} ( V )$ is replaced by a special linear group, an orthogonal group, a unitary group or a symplectic group ([[#References|[a2]]], p. 92, [[#References|[a3]]]).

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  J.F. Adams,   C.W. Wilkerson,   "Finite H-spaces and algebras over the Steenrod algebra"  ''Ann. Math.'' , '''111'''  (1980)  pp. 95–143</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  D. Benson,   "Polynomial invariants of finite groups" , ''London Math. Soc. Lecture Notes'' , '''190''' , Cambridge Univ. Press  (1993)</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top">  D. Carlisle,   P. Kropholler,   "Rational invariants of certain orthogonal and unitary groups"  ''Bull. London Math. Soc.'' , '''24''' :  1  (1992)  pp. 57–60</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top">  H.A.E. Campbell,   I. Highes,   R.D. Pollack,   "Rings of invariants and $p$-Sylow subgroups"  ''Canad. Bull. Math.'' , '''34'''  (1991)  pp. 42–47</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top">  L.E. Dickson,   "A fundamental system of invariants of the general modular linear group with a solution of the form problem"  ''Trans. Amer. Math. Soc.'' , '''12'''  (1911)  pp. 75–98</td></tr>
<tr><td valign="top">[a6]</td> <td valign="top">  I. Madsen,   "On the action of the Dyer–Lashof algebra on $H ^ { * } ( G )$"  ''Pacific J. Math.'' , '''60'''  (1975)  pp. 235–275</td></tr>
<tr><td valign="top">[a7]</td> <td valign="top">  D.G. Quillen,   "The mod two cohomology rings of extra-special $2$-groups and the Spinor groups"  ''Math. Ann.'' , '''194'''  (1971)  pp. 197–212</td></tr>
<tr><td valign="top">[a8]</td> <td valign="top">  L. Smith,   R. Switzer,   "Realizability and nonrealizability of Dickson algebras as cohomology rings"  ''Proc. Amer. Math. Soc.'' , '''89'''  (1983)  pp. 303–313</td></tr>
<tr><td valign="top">[a9]</td> <td valign="top">  W.M. Singer,   "Invariant theory and the lambda algebra"  ''Trans. Amer. Math. Soc.'' , '''280'''  (1983)  pp. 673–693</td></tr>
<tr><td valign="top">[a10]</td> <td valign="top">  C.W. Wilkerson,   "A primer on the Dickson invariants"  ''Contemp. Math.'' , '''19'''  (1983)  pp. 421–434</td></tr>
</table>