Unitary representations of finite groups over finite fields












6












$begingroup$


I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $operatorname{GU}(n,mathbb{F}_{q^2})$ consists of all invertible transformations of $mathbb{F}_{q^2}^n$ that preserve the Hermitian form $langle x, y rangle = sum_{i in [n]} x_i y_i^q$, and "unitary representation" means a group homomorphism $rho colon G to operatorname{GU}(n,mathbb{F}_{q^2})$.
This is a special case of the usual notion of a representation $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$.



Over the complex numbers, every representation $rho colon G to operatorname{GL}(n,mathbb{C})$ of a finite group $G$ is similar to a unitary representation $rho' colon G to operatorname{GU}(n,mathbb{C})$, in the sense that there is an invertible operator $M$ such that $rho'(g) = Mrho(g) M^{-1}$ for every $g in G$.
In this sense and others, the theory of unitary representations over $mathbb{C}$ is essentially the same as that of ordinary representations.



However, over finite fields the notions are distinct.
If $G$ is a finite group and $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$ is a representation, there might not be an invertible operator $M$ such that $M rho(g) M^{-1} in operatorname{GU}(n,mathbb{F}_{q^2})$ for every $g in G$.
For example, $mathbb{Z}_5$ has a faithful 2-dimensional representation over $mathbb{F}_{3^2}$ that is not similar to any unitary representation, since 5 divides $|operatorname{GL}(2,mathbb{F}_{3^2})|$ but not $|operatorname{GU}(2,mathbb{F}_{3^2})|$.



Question:
Have unitary representations of finite groups over finite fields been systematically studied, and if so where can I learn the basics?



Here is one example of what I want to learn to do:




  1. Describe all the unitary representations of the dihedral group of order 8 when $q=11$.


At the moment I do not even know how to:




  1. Describe all the unitary representations of $mathbb{Z}_2 times mathbb{Z}_2$ when $q=3$.


Some other things I want to learn include:




  1. Where Maschke's Theorem holds (i.e. $(|G|,q) = 1$ so that $mathbb{F}_{q^2}[G]$ is semisimple), does every unitary representation decompose as an orthogonal direct sum of irreducible unitary subrepresentations?


  2. Again where Maschke's Theorem holds, is there any analogue of the Peter-Weyl Theorem to give the space $L^2(G)$ of functions $f colon G to mathbb{F}_{q^2}$ an orthogonal basis consisting of matrix elements for irreducible unitary representations?


  3. What are some conditions for a modular representation to be similar to a unitary representation? (i.e. which subgroups of $operatorname{GL}(n,mathbb{F}_{q^2})$ are conjugate with subgroups of $operatorname{GU}(n,mathbb{F}_{q^2})$?)



Bonus for answers understandable to a humble analyst.










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Joey Iverson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • $begingroup$
    Notice that finite-dimensional unitary representations are automatically semisimple; a minimal stable non-$0$ subspace is irreducible, and its orthogonal complement has smaller dimension. A similar argument shows that every semisimple unitary representation is an orthogonal direct sum of irreducibles.
    $endgroup$
    – LSpice
    5 hours ago












  • $begingroup$
    I think that a subgroup of $operatorname{GL}(n, mathbb F_{q^2})$ is conjugate to a subgroup of $operatorname{GU}(n, mathbb F_{q^2}/mathbb F_q)$ if and only if it commutes with a torus of the form $(mathbb F_{q^2}^times)^n$.
    $endgroup$
    – LSpice
    5 hours ago










  • $begingroup$
    $mathbb F_3[C_2 times C_2]$ is the orthogonal direct sum $mathbb F_3(1, 1) oplus mathbb F_3(1, -1) oplus mathbb F_3(-1, 1) oplus mathbb F_3(-1, -1)$, where $(a, b)$ denotes the homomorphism $C_2 times C_2 to mathbb F_3^times$ given by $(m, n) mapsto a^m b^n$. The decomposition of the group algebra is guaranteed to capture all irreducible unitaries since, as usual, for an irreducible unitary representation $V$ of $G$, $V otimes V^*$ embeds in $mathbb F_{q^2}[G]$ by $v otimes v^* mapsto g mapsto langle v^*, gcdot vrangle$.
    $endgroup$
    – LSpice
    5 hours ago


















6












$begingroup$


I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $operatorname{GU}(n,mathbb{F}_{q^2})$ consists of all invertible transformations of $mathbb{F}_{q^2}^n$ that preserve the Hermitian form $langle x, y rangle = sum_{i in [n]} x_i y_i^q$, and "unitary representation" means a group homomorphism $rho colon G to operatorname{GU}(n,mathbb{F}_{q^2})$.
This is a special case of the usual notion of a representation $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$.



Over the complex numbers, every representation $rho colon G to operatorname{GL}(n,mathbb{C})$ of a finite group $G$ is similar to a unitary representation $rho' colon G to operatorname{GU}(n,mathbb{C})$, in the sense that there is an invertible operator $M$ such that $rho'(g) = Mrho(g) M^{-1}$ for every $g in G$.
In this sense and others, the theory of unitary representations over $mathbb{C}$ is essentially the same as that of ordinary representations.



However, over finite fields the notions are distinct.
If $G$ is a finite group and $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$ is a representation, there might not be an invertible operator $M$ such that $M rho(g) M^{-1} in operatorname{GU}(n,mathbb{F}_{q^2})$ for every $g in G$.
For example, $mathbb{Z}_5$ has a faithful 2-dimensional representation over $mathbb{F}_{3^2}$ that is not similar to any unitary representation, since 5 divides $|operatorname{GL}(2,mathbb{F}_{3^2})|$ but not $|operatorname{GU}(2,mathbb{F}_{3^2})|$.



Question:
Have unitary representations of finite groups over finite fields been systematically studied, and if so where can I learn the basics?



Here is one example of what I want to learn to do:




  1. Describe all the unitary representations of the dihedral group of order 8 when $q=11$.


At the moment I do not even know how to:




  1. Describe all the unitary representations of $mathbb{Z}_2 times mathbb{Z}_2$ when $q=3$.


Some other things I want to learn include:




  1. Where Maschke's Theorem holds (i.e. $(|G|,q) = 1$ so that $mathbb{F}_{q^2}[G]$ is semisimple), does every unitary representation decompose as an orthogonal direct sum of irreducible unitary subrepresentations?


  2. Again where Maschke's Theorem holds, is there any analogue of the Peter-Weyl Theorem to give the space $L^2(G)$ of functions $f colon G to mathbb{F}_{q^2}$ an orthogonal basis consisting of matrix elements for irreducible unitary representations?


  3. What are some conditions for a modular representation to be similar to a unitary representation? (i.e. which subgroups of $operatorname{GL}(n,mathbb{F}_{q^2})$ are conjugate with subgroups of $operatorname{GU}(n,mathbb{F}_{q^2})$?)



Bonus for answers understandable to a humble analyst.










share|cite|improve this question









New contributor




Joey Iverson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    Notice that finite-dimensional unitary representations are automatically semisimple; a minimal stable non-$0$ subspace is irreducible, and its orthogonal complement has smaller dimension. A similar argument shows that every semisimple unitary representation is an orthogonal direct sum of irreducibles.
    $endgroup$
    – LSpice
    5 hours ago












  • $begingroup$
    I think that a subgroup of $operatorname{GL}(n, mathbb F_{q^2})$ is conjugate to a subgroup of $operatorname{GU}(n, mathbb F_{q^2}/mathbb F_q)$ if and only if it commutes with a torus of the form $(mathbb F_{q^2}^times)^n$.
    $endgroup$
    – LSpice
    5 hours ago










  • $begingroup$
    $mathbb F_3[C_2 times C_2]$ is the orthogonal direct sum $mathbb F_3(1, 1) oplus mathbb F_3(1, -1) oplus mathbb F_3(-1, 1) oplus mathbb F_3(-1, -1)$, where $(a, b)$ denotes the homomorphism $C_2 times C_2 to mathbb F_3^times$ given by $(m, n) mapsto a^m b^n$. The decomposition of the group algebra is guaranteed to capture all irreducible unitaries since, as usual, for an irreducible unitary representation $V$ of $G$, $V otimes V^*$ embeds in $mathbb F_{q^2}[G]$ by $v otimes v^* mapsto g mapsto langle v^*, gcdot vrangle$.
    $endgroup$
    – LSpice
    5 hours ago
















6












6








6





$begingroup$


I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $operatorname{GU}(n,mathbb{F}_{q^2})$ consists of all invertible transformations of $mathbb{F}_{q^2}^n$ that preserve the Hermitian form $langle x, y rangle = sum_{i in [n]} x_i y_i^q$, and "unitary representation" means a group homomorphism $rho colon G to operatorname{GU}(n,mathbb{F}_{q^2})$.
This is a special case of the usual notion of a representation $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$.



Over the complex numbers, every representation $rho colon G to operatorname{GL}(n,mathbb{C})$ of a finite group $G$ is similar to a unitary representation $rho' colon G to operatorname{GU}(n,mathbb{C})$, in the sense that there is an invertible operator $M$ such that $rho'(g) = Mrho(g) M^{-1}$ for every $g in G$.
In this sense and others, the theory of unitary representations over $mathbb{C}$ is essentially the same as that of ordinary representations.



However, over finite fields the notions are distinct.
If $G$ is a finite group and $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$ is a representation, there might not be an invertible operator $M$ such that $M rho(g) M^{-1} in operatorname{GU}(n,mathbb{F}_{q^2})$ for every $g in G$.
For example, $mathbb{Z}_5$ has a faithful 2-dimensional representation over $mathbb{F}_{3^2}$ that is not similar to any unitary representation, since 5 divides $|operatorname{GL}(2,mathbb{F}_{3^2})|$ but not $|operatorname{GU}(2,mathbb{F}_{3^2})|$.



Question:
Have unitary representations of finite groups over finite fields been systematically studied, and if so where can I learn the basics?



Here is one example of what I want to learn to do:




  1. Describe all the unitary representations of the dihedral group of order 8 when $q=11$.


At the moment I do not even know how to:




  1. Describe all the unitary representations of $mathbb{Z}_2 times mathbb{Z}_2$ when $q=3$.


Some other things I want to learn include:




  1. Where Maschke's Theorem holds (i.e. $(|G|,q) = 1$ so that $mathbb{F}_{q^2}[G]$ is semisimple), does every unitary representation decompose as an orthogonal direct sum of irreducible unitary subrepresentations?


  2. Again where Maschke's Theorem holds, is there any analogue of the Peter-Weyl Theorem to give the space $L^2(G)$ of functions $f colon G to mathbb{F}_{q^2}$ an orthogonal basis consisting of matrix elements for irreducible unitary representations?


  3. What are some conditions for a modular representation to be similar to a unitary representation? (i.e. which subgroups of $operatorname{GL}(n,mathbb{F}_{q^2})$ are conjugate with subgroups of $operatorname{GU}(n,mathbb{F}_{q^2})$?)



Bonus for answers understandable to a humble analyst.










share|cite|improve this question









New contributor




Joey Iverson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $operatorname{GU}(n,mathbb{F}_{q^2})$ consists of all invertible transformations of $mathbb{F}_{q^2}^n$ that preserve the Hermitian form $langle x, y rangle = sum_{i in [n]} x_i y_i^q$, and "unitary representation" means a group homomorphism $rho colon G to operatorname{GU}(n,mathbb{F}_{q^2})$.
This is a special case of the usual notion of a representation $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$.



Over the complex numbers, every representation $rho colon G to operatorname{GL}(n,mathbb{C})$ of a finite group $G$ is similar to a unitary representation $rho' colon G to operatorname{GU}(n,mathbb{C})$, in the sense that there is an invertible operator $M$ such that $rho'(g) = Mrho(g) M^{-1}$ for every $g in G$.
In this sense and others, the theory of unitary representations over $mathbb{C}$ is essentially the same as that of ordinary representations.



However, over finite fields the notions are distinct.
If $G$ is a finite group and $rho colon G to operatorname{GL}(n,mathbb{F}_{q^2})$ is a representation, there might not be an invertible operator $M$ such that $M rho(g) M^{-1} in operatorname{GU}(n,mathbb{F}_{q^2})$ for every $g in G$.
For example, $mathbb{Z}_5$ has a faithful 2-dimensional representation over $mathbb{F}_{3^2}$ that is not similar to any unitary representation, since 5 divides $|operatorname{GL}(2,mathbb{F}_{3^2})|$ but not $|operatorname{GU}(2,mathbb{F}_{3^2})|$.



Question:
Have unitary representations of finite groups over finite fields been systematically studied, and if so where can I learn the basics?



Here is one example of what I want to learn to do:




  1. Describe all the unitary representations of the dihedral group of order 8 when $q=11$.


At the moment I do not even know how to:




  1. Describe all the unitary representations of $mathbb{Z}_2 times mathbb{Z}_2$ when $q=3$.


Some other things I want to learn include:




  1. Where Maschke's Theorem holds (i.e. $(|G|,q) = 1$ so that $mathbb{F}_{q^2}[G]$ is semisimple), does every unitary representation decompose as an orthogonal direct sum of irreducible unitary subrepresentations?


  2. Again where Maschke's Theorem holds, is there any analogue of the Peter-Weyl Theorem to give the space $L^2(G)$ of functions $f colon G to mathbb{F}_{q^2}$ an orthogonal basis consisting of matrix elements for irreducible unitary representations?


  3. What are some conditions for a modular representation to be similar to a unitary representation? (i.e. which subgroups of $operatorname{GL}(n,mathbb{F}_{q^2})$ are conjugate with subgroups of $operatorname{GU}(n,mathbb{F}_{q^2})$?)



Bonus for answers understandable to a humble analyst.







reference-request gr.group-theory rt.representation-theory finite-groups harmonic-analysis






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Joey Iverson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











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Check out our Code of Conduct.









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edited 5 hours ago









YCor

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asked 5 hours ago









Joey IversonJoey Iverson

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New contributor





Joey Iverson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






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Check out our Code of Conduct.












  • $begingroup$
    Notice that finite-dimensional unitary representations are automatically semisimple; a minimal stable non-$0$ subspace is irreducible, and its orthogonal complement has smaller dimension. A similar argument shows that every semisimple unitary representation is an orthogonal direct sum of irreducibles.
    $endgroup$
    – LSpice
    5 hours ago












  • $begingroup$
    I think that a subgroup of $operatorname{GL}(n, mathbb F_{q^2})$ is conjugate to a subgroup of $operatorname{GU}(n, mathbb F_{q^2}/mathbb F_q)$ if and only if it commutes with a torus of the form $(mathbb F_{q^2}^times)^n$.
    $endgroup$
    – LSpice
    5 hours ago










  • $begingroup$
    $mathbb F_3[C_2 times C_2]$ is the orthogonal direct sum $mathbb F_3(1, 1) oplus mathbb F_3(1, -1) oplus mathbb F_3(-1, 1) oplus mathbb F_3(-1, -1)$, where $(a, b)$ denotes the homomorphism $C_2 times C_2 to mathbb F_3^times$ given by $(m, n) mapsto a^m b^n$. The decomposition of the group algebra is guaranteed to capture all irreducible unitaries since, as usual, for an irreducible unitary representation $V$ of $G$, $V otimes V^*$ embeds in $mathbb F_{q^2}[G]$ by $v otimes v^* mapsto g mapsto langle v^*, gcdot vrangle$.
    $endgroup$
    – LSpice
    5 hours ago




















  • $begingroup$
    Notice that finite-dimensional unitary representations are automatically semisimple; a minimal stable non-$0$ subspace is irreducible, and its orthogonal complement has smaller dimension. A similar argument shows that every semisimple unitary representation is an orthogonal direct sum of irreducibles.
    $endgroup$
    – LSpice
    5 hours ago












  • $begingroup$
    I think that a subgroup of $operatorname{GL}(n, mathbb F_{q^2})$ is conjugate to a subgroup of $operatorname{GU}(n, mathbb F_{q^2}/mathbb F_q)$ if and only if it commutes with a torus of the form $(mathbb F_{q^2}^times)^n$.
    $endgroup$
    – LSpice
    5 hours ago










  • $begingroup$
    $mathbb F_3[C_2 times C_2]$ is the orthogonal direct sum $mathbb F_3(1, 1) oplus mathbb F_3(1, -1) oplus mathbb F_3(-1, 1) oplus mathbb F_3(-1, -1)$, where $(a, b)$ denotes the homomorphism $C_2 times C_2 to mathbb F_3^times$ given by $(m, n) mapsto a^m b^n$. The decomposition of the group algebra is guaranteed to capture all irreducible unitaries since, as usual, for an irreducible unitary representation $V$ of $G$, $V otimes V^*$ embeds in $mathbb F_{q^2}[G]$ by $v otimes v^* mapsto g mapsto langle v^*, gcdot vrangle$.
    $endgroup$
    – LSpice
    5 hours ago


















$begingroup$
Notice that finite-dimensional unitary representations are automatically semisimple; a minimal stable non-$0$ subspace is irreducible, and its orthogonal complement has smaller dimension. A similar argument shows that every semisimple unitary representation is an orthogonal direct sum of irreducibles.
$endgroup$
– LSpice
5 hours ago






$begingroup$
Notice that finite-dimensional unitary representations are automatically semisimple; a minimal stable non-$0$ subspace is irreducible, and its orthogonal complement has smaller dimension. A similar argument shows that every semisimple unitary representation is an orthogonal direct sum of irreducibles.
$endgroup$
– LSpice
5 hours ago














$begingroup$
I think that a subgroup of $operatorname{GL}(n, mathbb F_{q^2})$ is conjugate to a subgroup of $operatorname{GU}(n, mathbb F_{q^2}/mathbb F_q)$ if and only if it commutes with a torus of the form $(mathbb F_{q^2}^times)^n$.
$endgroup$
– LSpice
5 hours ago




$begingroup$
I think that a subgroup of $operatorname{GL}(n, mathbb F_{q^2})$ is conjugate to a subgroup of $operatorname{GU}(n, mathbb F_{q^2}/mathbb F_q)$ if and only if it commutes with a torus of the form $(mathbb F_{q^2}^times)^n$.
$endgroup$
– LSpice
5 hours ago












$begingroup$
$mathbb F_3[C_2 times C_2]$ is the orthogonal direct sum $mathbb F_3(1, 1) oplus mathbb F_3(1, -1) oplus mathbb F_3(-1, 1) oplus mathbb F_3(-1, -1)$, where $(a, b)$ denotes the homomorphism $C_2 times C_2 to mathbb F_3^times$ given by $(m, n) mapsto a^m b^n$. The decomposition of the group algebra is guaranteed to capture all irreducible unitaries since, as usual, for an irreducible unitary representation $V$ of $G$, $V otimes V^*$ embeds in $mathbb F_{q^2}[G]$ by $v otimes v^* mapsto g mapsto langle v^*, gcdot vrangle$.
$endgroup$
– LSpice
5 hours ago






$begingroup$
$mathbb F_3[C_2 times C_2]$ is the orthogonal direct sum $mathbb F_3(1, 1) oplus mathbb F_3(1, -1) oplus mathbb F_3(-1, 1) oplus mathbb F_3(-1, -1)$, where $(a, b)$ denotes the homomorphism $C_2 times C_2 to mathbb F_3^times$ given by $(m, n) mapsto a^m b^n$. The decomposition of the group algebra is guaranteed to capture all irreducible unitaries since, as usual, for an irreducible unitary representation $V$ of $G$, $V otimes V^*$ embeds in $mathbb F_{q^2}[G]$ by $v otimes v^* mapsto g mapsto langle v^*, gcdot vrangle$.
$endgroup$
– LSpice
5 hours ago












1 Answer
1






active

oldest

votes


















2












$begingroup$

We had to deal with this problem when classifying maximal subgroups of the finite classical groups, which is the aim of our book:



The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, by
John N. Bray,
Derek F. Holt,
Colva M. Roney-Dougal.



The most difficult maximal subgroups to classify, are those in the so-called Aschbacher class ${mathscr S}$, consisting of absolutely irreducible subgroups that are almost simple mod scalars. Many of these arise as reductions of complex representations over finite fields. Tables of complex representations of groups that are close to simple are available up to dimension about $250$, but we needed to know which classical group the reduction lies in, which means identifying the fixed form.



We generally relied on Lemma 4.4.1 of the book, which says:



For a given absolutely irreducible representation over ${mathbb F}_{q^2}$ of a group $G$,
with Frobenius-Schur indicator $circ$, the image of $G$ under the representation consists of
isometries of a unitary form if and only if the action of the field automorphism
$sigma :x to x^q$ on the Brauer character is the same as complex conjugation.



In many cases, such as when $q$ is coprime to the group order, the Brauer character is just the ordinary complex character.



As an example, the reduction of the complex representation of degree $3$ of the $3$-fold cover $3.A_6$ of $A_6$ lies in ${rm PSL}(3,p)$ for primes $p equiv 1,4 bmod 15$, in ${rm PSU}(3,p)$ (as a subgroup of ${rm PSL}(3,p^2)$) when $p equiv 11,14 bmod 15$ (or when $p=5$), and in ${rm PSL}(3,p^2)$ without preserving a unitary form when $p equiv 2,3 bmod 5$.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
    $endgroup$
    – LSpice
    2 hours ago










  • $begingroup$
    OK, I have reworded it, but I was quoting the lemma directly from the book!
    $endgroup$
    – Derek Holt
    1 hour ago












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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

We had to deal with this problem when classifying maximal subgroups of the finite classical groups, which is the aim of our book:



The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, by
John N. Bray,
Derek F. Holt,
Colva M. Roney-Dougal.



The most difficult maximal subgroups to classify, are those in the so-called Aschbacher class ${mathscr S}$, consisting of absolutely irreducible subgroups that are almost simple mod scalars. Many of these arise as reductions of complex representations over finite fields. Tables of complex representations of groups that are close to simple are available up to dimension about $250$, but we needed to know which classical group the reduction lies in, which means identifying the fixed form.



We generally relied on Lemma 4.4.1 of the book, which says:



For a given absolutely irreducible representation over ${mathbb F}_{q^2}$ of a group $G$,
with Frobenius-Schur indicator $circ$, the image of $G$ under the representation consists of
isometries of a unitary form if and only if the action of the field automorphism
$sigma :x to x^q$ on the Brauer character is the same as complex conjugation.



In many cases, such as when $q$ is coprime to the group order, the Brauer character is just the ordinary complex character.



As an example, the reduction of the complex representation of degree $3$ of the $3$-fold cover $3.A_6$ of $A_6$ lies in ${rm PSL}(3,p)$ for primes $p equiv 1,4 bmod 15$, in ${rm PSU}(3,p)$ (as a subgroup of ${rm PSL}(3,p^2)$) when $p equiv 11,14 bmod 15$ (or when $p=5$), and in ${rm PSL}(3,p^2)$ without preserving a unitary form when $p equiv 2,3 bmod 5$.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
    $endgroup$
    – LSpice
    2 hours ago










  • $begingroup$
    OK, I have reworded it, but I was quoting the lemma directly from the book!
    $endgroup$
    – Derek Holt
    1 hour ago
















2












$begingroup$

We had to deal with this problem when classifying maximal subgroups of the finite classical groups, which is the aim of our book:



The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, by
John N. Bray,
Derek F. Holt,
Colva M. Roney-Dougal.



The most difficult maximal subgroups to classify, are those in the so-called Aschbacher class ${mathscr S}$, consisting of absolutely irreducible subgroups that are almost simple mod scalars. Many of these arise as reductions of complex representations over finite fields. Tables of complex representations of groups that are close to simple are available up to dimension about $250$, but we needed to know which classical group the reduction lies in, which means identifying the fixed form.



We generally relied on Lemma 4.4.1 of the book, which says:



For a given absolutely irreducible representation over ${mathbb F}_{q^2}$ of a group $G$,
with Frobenius-Schur indicator $circ$, the image of $G$ under the representation consists of
isometries of a unitary form if and only if the action of the field automorphism
$sigma :x to x^q$ on the Brauer character is the same as complex conjugation.



In many cases, such as when $q$ is coprime to the group order, the Brauer character is just the ordinary complex character.



As an example, the reduction of the complex representation of degree $3$ of the $3$-fold cover $3.A_6$ of $A_6$ lies in ${rm PSL}(3,p)$ for primes $p equiv 1,4 bmod 15$, in ${rm PSU}(3,p)$ (as a subgroup of ${rm PSL}(3,p^2)$) when $p equiv 11,14 bmod 15$ (or when $p=5$), and in ${rm PSL}(3,p^2)$ without preserving a unitary form when $p equiv 2,3 bmod 5$.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
    $endgroup$
    – LSpice
    2 hours ago










  • $begingroup$
    OK, I have reworded it, but I was quoting the lemma directly from the book!
    $endgroup$
    – Derek Holt
    1 hour ago














2












2








2





$begingroup$

We had to deal with this problem when classifying maximal subgroups of the finite classical groups, which is the aim of our book:



The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, by
John N. Bray,
Derek F. Holt,
Colva M. Roney-Dougal.



The most difficult maximal subgroups to classify, are those in the so-called Aschbacher class ${mathscr S}$, consisting of absolutely irreducible subgroups that are almost simple mod scalars. Many of these arise as reductions of complex representations over finite fields. Tables of complex representations of groups that are close to simple are available up to dimension about $250$, but we needed to know which classical group the reduction lies in, which means identifying the fixed form.



We generally relied on Lemma 4.4.1 of the book, which says:



For a given absolutely irreducible representation over ${mathbb F}_{q^2}$ of a group $G$,
with Frobenius-Schur indicator $circ$, the image of $G$ under the representation consists of
isometries of a unitary form if and only if the action of the field automorphism
$sigma :x to x^q$ on the Brauer character is the same as complex conjugation.



In many cases, such as when $q$ is coprime to the group order, the Brauer character is just the ordinary complex character.



As an example, the reduction of the complex representation of degree $3$ of the $3$-fold cover $3.A_6$ of $A_6$ lies in ${rm PSL}(3,p)$ for primes $p equiv 1,4 bmod 15$, in ${rm PSU}(3,p)$ (as a subgroup of ${rm PSL}(3,p^2)$) when $p equiv 11,14 bmod 15$ (or when $p=5$), and in ${rm PSL}(3,p^2)$ without preserving a unitary form when $p equiv 2,3 bmod 5$.






share|cite|improve this answer











$endgroup$



We had to deal with this problem when classifying maximal subgroups of the finite classical groups, which is the aim of our book:



The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, by
John N. Bray,
Derek F. Holt,
Colva M. Roney-Dougal.



The most difficult maximal subgroups to classify, are those in the so-called Aschbacher class ${mathscr S}$, consisting of absolutely irreducible subgroups that are almost simple mod scalars. Many of these arise as reductions of complex representations over finite fields. Tables of complex representations of groups that are close to simple are available up to dimension about $250$, but we needed to know which classical group the reduction lies in, which means identifying the fixed form.



We generally relied on Lemma 4.4.1 of the book, which says:



For a given absolutely irreducible representation over ${mathbb F}_{q^2}$ of a group $G$,
with Frobenius-Schur indicator $circ$, the image of $G$ under the representation consists of
isometries of a unitary form if and only if the action of the field automorphism
$sigma :x to x^q$ on the Brauer character is the same as complex conjugation.



In many cases, such as when $q$ is coprime to the group order, the Brauer character is just the ordinary complex character.



As an example, the reduction of the complex representation of degree $3$ of the $3$-fold cover $3.A_6$ of $A_6$ lies in ${rm PSL}(3,p)$ for primes $p equiv 1,4 bmod 15$, in ${rm PSU}(3,p)$ (as a subgroup of ${rm PSL}(3,p^2)$) when $p equiv 11,14 bmod 15$ (or when $p=5$), and in ${rm PSL}(3,p^2)$ without preserving a unitary form when $p equiv 2,3 bmod 5$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 1 hour ago

























answered 4 hours ago









Derek HoltDerek Holt

27.4k464112




27.4k464112








  • 1




    $begingroup$
    Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
    $endgroup$
    – LSpice
    2 hours ago










  • $begingroup$
    OK, I have reworded it, but I was quoting the lemma directly from the book!
    $endgroup$
    – Derek Holt
    1 hour ago














  • 1




    $begingroup$
    Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
    $endgroup$
    – LSpice
    2 hours ago










  • $begingroup$
    OK, I have reworded it, but I was quoting the lemma directly from the book!
    $endgroup$
    – Derek Holt
    1 hour ago








1




1




$begingroup$
Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
$endgroup$
– LSpice
2 hours ago




$begingroup$
Unless you think there's no risk of confusion, it may be worth it, for people like me who read "representation of a group over $mathbb F_{q^2}$" as "representation of (a group $G$ over $mathbb F_{q^2}$)", seeming to consider representations of algebraic groups, to re-phrase as "representation over $mathbb F_{q^2}$ of a group $G$", so that it's clear we're specifying the field of definition of the representation and not of the group (the latter meaningless anyway in the abstract setting considered).
$endgroup$
– LSpice
2 hours ago












$begingroup$
OK, I have reworded it, but I was quoting the lemma directly from the book!
$endgroup$
– Derek Holt
1 hour ago




$begingroup$
OK, I have reworded it, but I was quoting the lemma directly from the book!
$endgroup$
– Derek Holt
1 hour ago










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