Dickson invariant

A construction used in the study of quadratic forms over fields of characteristic 2, which allows one, in particular, to introduce analogues of the special orthogonal group over such fields. In fact, a Dickson invariant is an element $D(u)$ of a field $k$ of characteristic 2 associated to any similarity $u$ of a countable-dimensional vector space $E$ over $k$ with respect to the symmetric bilinear form $f$ associated with a non-degenerate quadratic form $Q$ on $E$. Introduced by L.E. Dickson [1].

By virtue of the condition imposed on the characteristic of the field, the form $f$ is alternating and there exists a basis $e_1,\dots,e_{2s}$ in $E$ for which

$$f(e_i,e_j)=f(e_{s+i},e_{s+j})=0,$$

$$f(e_i,e_{s+j})=\delta_{ij},$$

for $1\leq i\leq s$, $1\leq j\leq s$ (cf. Witt decomposition). Let

$$f(u(x),u(y))=\alpha(u)f(x,y)$$

for any vectors $x$ and $y$ from $E$, and let, for each $i=1,\dots,s$,

$$u(e_i)=\sum_{j=1}^sa_{ij}e_j+\sum_{j=1}^sb_{ij}e_{s+j},$$

$$u(e_{s+i})=\sum_{j=1}^sc_{ij}e_j+\sum_{j=1}^sd_{ij}e_{s+j}.$$

Then the following element from $k$:

$$D(u)=\sum_{i,j}(Q(e_i)a_{ij}c_{ij}+Q(e_{s+i})b_{ij}d_{ij}+b_{ij}c_{ij})$$

is called the Dickson invariant of the similarity $u$ with respect to the basis $e_1,\dots,e_{2s}$. For $u$ to be a similarity with respect to $Q$ with similarity coefficient $\alpha(u)$ (i.e. $Q(u(x))=\alpha(u)Q(x)$ for any vector $x\in E$) it is necessary and sufficient that $D(u)=0$ or that $D(u)=\alpha(u)$. Similarities $u$ with respect to $Q$ for which $D(u)=0$ are called direct similarities. The direct similarities form a normal subgroup of index 2 in the group of all similarities with respect to $Q$.

If $Q_1$ is the form defined by $Q_1(x)=Q(u(x))$ for any vector $x\in E$, and if $\Delta(Q)$ and $\Delta(Q_1)$ are the pseudo-discriminants of these forms with respect to the basis $e_1,\dots,e_{2s}$, i.e.

$$\Delta(Q)=Q(e_1)Q(e_{s+1})+\ldots+Q(e_s)Q(e_{2s}),$$

$$\Delta(Q_1)=Q_1(e_1)Q_1(e_{s+1})+\ldots+Q_1(e_s)Q_1(e_{2s}),$$

then

$$\Delta(Q_1)=(\alpha(u))^2\Delta(Q)+(D(u))^2+\alpha(u)D(u).$$

References

[1]	L.E. Dickson, "Linear groups" , Teubner (1901) MR1505871 MR1500573 Zbl 32.0134.03 Zbl 32.0131.03 Zbl 32.0131.01 Zbl 32.0128.01
[2]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[3]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056