Editing Orthogonal group (section)

== Group structure ==
The groups {{math|O(''n'')}} and {{math|SO(''n'')}} are real [[Compact space|compact]] [[Lie group]]s of [[dimension (mathematics)|dimension]] {{math|''n''(''n'' − 1) / 2}}. The group {{math|O(''n'')}} has two [[Connected space|connected components]], with {{math|SO(''n'')}} being the [[identity component]], that is, the connected component containing the [[identity matrix]].

=== As algebraic groups ===
The orthogonal group {{math|O(''n'')}} can be identified with the group of the matrices {{mvar|A}} such that {{math|1=''A''{{sup|T}}''A'' = ''I''}}. 
Since both members of this equation are [[symmetric matrices]], this provides {{math|''n''(''n'' + 1) / 2}} equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.

This proves that {{math|O(''n'')}} is an [[algebraic set]]. Moreover, it can be proved{{cn|date=July 2022}} that its dimension is
: <math>\frac{n(n - 1)}{2} = n^2 - \frac{n(n + 1)}{2},</math> 
which implies that {{math|O(''n'')}} is a [[complete intersection]]. This implies that all its [[irreducible component]]s have the same dimension, and that it has no [[embedded prime|embedded component]].
In fact, {{math|O(''n'')}} has two irreducible components, that are distinguished by the sign of the determinant (that is {{math|1=det(''A'') = 1}} or {{math|1=det(''A'') = −1}}). Both are [[singular point of an algebraic variety|nonsingular algebraic varieties]] of the same dimension {{math|''n''(''n'' − 1) / 2}}. The component with {{math|1=det(''A'') = 1}} is {{math|SO(''n'')}}.

=== Maximal tori and Weyl groups===
A [[maximal torus]] in a compact [[Lie group]] ''G'' is a maximal subgroup among those that are isomorphic to {{math|'''T'''<sup>''k''</sup>}} for some {{mvar|k}}, where {{math|1='''T''' = SO(2)}} is the standard one-dimensional torus.<ref>{{harvnb|Hall|2015}} Theorem 11.2</ref>

In {{math|O(2''n'')}} and {{math|SO(2''n'')}}, for every maximal torus, there is a basis on which the torus consists of the [[block matrix#Block diagonal matrices|block-diagonal matrices]] of the form 
: <math>\begin{bmatrix}
  R_1 &        & 0    \\
      & \ddots &      \\
  0   &        & R_n
\end{bmatrix},</math>
where each {{math|''R''<sub>''j''</sub>}} belongs to {{math|SO(2)}}. 
In {{math|O(2''n'' + 1)}} and {{math|SO(2''n'' + 1)}}, the maximal tori have the same form, bordered by a row and a column of zeros, and {{math|1}} on the diagonal.

The [[Maximal torus#Weyl group|Weyl group]]<!--Do not change the link (unless the target atricle(s) have changed: Presently, the article [[Weyl group]] does not talk of the Weyl group relatively to a maximal torus) --> of {{math|SO(2''n'' + 1)}} is the [[semidirect product]] <math>\{\pm 1\}^n \rtimes S_n</math> of a normal [[elementary abelian group|elementary abelian]] [[p-group|2-subgroup]] and a [[symmetric group]], where the nontrivial element of each {{math|{{mset|±1}}}} factor of {{math|{{mset|±1}}<sup>''n''</sup>}} acts on the corresponding circle factor of {{math|''T'' × {1}}} by [[multiplicative inverse|inversion]], and the symmetric group {{math|''S<sub>n</sub>''}} acts on both {{math|{{mset|±1}}<sup>''n''</sup>}} and {{math|''T'' × {1}}} by permuting factors. The elements of the Weyl group are represented by matrices in {{math|O(2''n'') × {{mset|±1}}}}.
The {{math|''S<sub>n</sub>''}} factor is represented by block permutation matrices with 2-by-2 blocks, and a final {{math|1}} on the diagonal. The {{math|{{mset|±1}}<sup>''n''</sup>}} component is represented by block-diagonal matrices with 2-by-2 blocks either
: <math>\begin{bmatrix}
    1 & 0 \\
    0 & 1
  \end{bmatrix} \quad \text{or} \quad
  \begin{bmatrix}
    0 & 1 \\
    1 & 0
  \end{bmatrix},
</math>
with the last component {{math|±1}} chosen to make the determinant {{math|1}}.

The Weyl group of {{math|SO(2''n'')}} is the subgroup <math>H_{n-1} \rtimes S_n < \{\pm 1\}^n \rtimes S_n</math> of that of {{math|SO(2''n'' + 1)}}, where {{math|''H''<sub>''n''−1</sub> < {{mset|±1}}<sup>''n''</sup>}} is the [[kernel (algebra)#Group homomorphism|kernel]] of the product homomorphism {{math|{{mset|±1}}<sup>''n''</sup> → {{mset|±1}}}} given by <math>\left(\varepsilon_1, \ldots, \varepsilon_n\right) \mapsto \varepsilon_1 \cdots \varepsilon_n</math>; that is, {{math|''H''<sub>''n''−1</sub> < {{mset|±1}}<sup>''n''</sup>}} is the subgroup with an even number of minus signs. The Weyl group of {{math|SO(2''n'')}} is represented in {{math|SO(2''n'')}} by the preimages under the standard injection {{math|SO(2''n'') → SO(2''n'' + 1)}} of the representatives for the Weyl group of {{math|SO(2''n'' + 1)}}. Those matrices with an odd number of <math>\begin{bmatrix}
  0 & 1 \\
  1 & 0
\end{bmatrix}</math> blocks have no remaining final {{math|−1}} coordinate to make their determinants positive, and hence cannot be represented in {{math|SO(2''n'')}}.