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Difference between revisions of "Irreducible matrix group"

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A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526001.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526002.png" />-matrices over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526003.png" /> that cannot be brought by simultaneous conjugation in the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526004.png" /> to the semi-reduced form
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A group $G$ of $n \times n$-matrices over a field $k$ that cannot be brought by simultaneous conjugation in the [[general linear group]] $\mathrm{GL}(n,k)$ to the semi-reduced form
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526005.png" /></td> </tr></table>
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$$
 
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\left( \begin{array}{cc} A & \star \\ 0 & B \end{array} \right)
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526007.png" /> are square blocks of fixed dimensions. More accurately, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526008.png" /> is called irreducible over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i0526009.png" />. In the language of transformations: A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260010.png" /> of linear transformations of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260011.png" /> is called irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260012.png" /> is a minimal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260013.png" />-invariant subspace (other than the null space). Irreducible Abelian groups of matrices over an algebraically closed field are one-dimensional. A group of matrices over a field that is irreducible over any extension field is called absolutely irreducible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260014.png" /> is algebraically closed, then for every group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260015.png" /> the following conditions are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260016.png" /> is irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260017.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260018.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260019.png" /> matrices that are linearly independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260020.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260021.png" /> is absolutely irreducible. Thus, absolute irreducibility over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260022.png" /> is equivalent to irreducibility over the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052600/i05260023.png" />.
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$$
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where $A$ and $B$ are square blocks of fixed dimensions. More accurately, $G$ is called irreducible over the field $k$. In the language of transformations: A group $G$ of linear transformations of a finite-dimensional vector space $V$ is called irreducible if $V$ is a minimal $G$-invariant subspace (other than the null space). Irreducible Abelian groups of matrices over an algebraically closed field are one-dimensional. A group of matrices over a field that is irreducible over any extension field is called absolutely irreducible. If $k$ is algebraically closed, then for every group $G \subseteq \mathrm{GL}(n,k)$ the following conditions are equivalent: 1) $G$ is irreducible over $k$; 2) $G$ contains $n^2$ matrices that are linearly independent over $k$; and 3) $G$ is absolutely irreducible. Thus, absolute irreducibility over a field $k$ is equivalent to irreducibility over the algebraic closure of $k$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.I. Merzlyakov,  "Rational groups" , Moscow  (1987)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.I. Merzlyakov,  "Rational groups" , Moscow  (1987)  (In Russian)</TD></TR>
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</table>

Latest revision as of 20:16, 23 December 2014

A group $G$ of $n \times n$-matrices over a field $k$ that cannot be brought by simultaneous conjugation in the general linear group $\mathrm{GL}(n,k)$ to the semi-reduced form $$ \left( \begin{array}{cc} A & \star \\ 0 & B \end{array} \right) $$ where $A$ and $B$ are square blocks of fixed dimensions. More accurately, $G$ is called irreducible over the field $k$. In the language of transformations: A group $G$ of linear transformations of a finite-dimensional vector space $V$ is called irreducible if $V$ is a minimal $G$-invariant subspace (other than the null space). Irreducible Abelian groups of matrices over an algebraically closed field are one-dimensional. A group of matrices over a field that is irreducible over any extension field is called absolutely irreducible. If $k$ is algebraically closed, then for every group $G \subseteq \mathrm{GL}(n,k)$ the following conditions are equivalent: 1) $G$ is irreducible over $k$; 2) $G$ contains $n^2$ matrices that are linearly independent over $k$; and 3) $G$ is absolutely irreducible. Thus, absolute irreducibility over a field $k$ is equivalent to irreducibility over the algebraic closure of $k$.

References

[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] Yu.I. Merzlyakov, "Rational groups" , Moscow (1987) (In Russian)
How to Cite This Entry:
Irreducible matrix group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_matrix_group&oldid=19107
This article was adapted from an original article by Yu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article