Conjugate Real Classes in General Linear Groups
Krishnendu Gongopadhyay, Sudip Mazumder, Sujit Kumar Sardar

TL;DR
This paper characterizes $c$-real elements in general linear groups over fields with involution, showing they correspond exactly to elements representable in some unitary group over the same field.
Contribution
It establishes a precise equivalence between $c$-reality of elements and their representability in unitary groups over fields with involution.
Findings
$c$-real elements are conjugate to their $c$-inverse
Characterization holds for all $n geq 2$
Connection between $c$-reality and unitary group representations
Abstract
Let be a field with a non-trivial involution . An element is called -real if it is conjugate to . We prove that for , is -real if and only if it has a representation in some unitary group of degree over .
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Conjugate Real Classes in General Linear Groups
Krishnendu Gongopadhyay
Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, SAS Nagar, Punjab 140306, India
[email protected], [email protected]
,
Sudip Mazumder
Department of Mathematics, Jadavpur University, Jadavpur, Kolkata 700032
and
Sujit Kumar Sardar
Department of Mathematics, Jadavpur University, Jadavpur, Kolkata 700032
Abstract.
Let be a field with a non-trivial involution . An element is called -real if it is conjugate to . We prove that for , is -real if and only if it has a representation in some unitary group of degree over .
Key words and phrases:
real elements, general linear group
2010 Mathematics Subject Classification:
Primary 20E45; Secondary 20G15, 15A04
1. Introduction
Let be a (commutative) field. Let be a fixed involution on , i.e. is an automorphism of satisfying for all in . The involution is called non-trivial if it is different from the identity automorphism. Let be a finite dimensional vector space over of dimension at least .
Recall that a -sesquilinear form on is a bi-additive map such that for all , , and , . A -sesquilinear form is called -hermitian if for all , . The isometry group consisting of linear transformations that preserve a -hermitian form is called a unitary group.
An element in an abstract group is called real or reversible if is conjugate to in . Real elements appear naturally in representation theory, geometry and dynamics. There have been an ongoing activity to understand real elements from several point of views, for example see the recent articles, [SFV16], [DGN15], [GS11], [Gon11], [GP13], [HL06], [KN05], [KK15], [KS11], [OR10], [ST08], [TZ05], [R1̈1], [VG10]. For an up to date exposition of real elements from different point of views, we refer to the recent monograph [OS15].
The notion of real elements has a natural extension to linear groups over fields with involutions as follows. Let has dimension over . Let be the general linear group, i.e. group of all invertible linear maps on . Let be a -invariant linear subgroup of . For an element in , let . An element in is called conjugate real or, simply -real if is conjugate to in . When is identity, it matches with the notion of reality in groups. It would be curious to investigate -reality in several classes of linear groups over a field with involution and, to compare it with the notion of reality. Some notion of twisted reality, mostly transpose-reality, is implicit in some recent works related to group representations, for example see [Vin05], [RS12]. Investigation of the notion of -reality seems missing in the literature. We consider -reality in the general linear group in this paper.
The -reality of a linear map has close connection with the self-duality of its invariant factors. Let be the algebraic closure of . Let , , be a monic polynomial of degree over such that [math], or are not its roots. The dual of is defined to be the polynomial , where . Thus, . In other words, if in is a root of with multiplicity , then is a root of with the same multiplicity. The polynomial is said to be self-dual if . For notational convenience, we shall slightly extend this notion and will call a polynomial self-dual if is either a power of , or self-dual in the above sense.
Our main theorem is the following.
Theorem 1.1**.**
Let be a field with a non-trivial involution . Let be an element in , . Then is -real if and only if it has a unitary representation of dimension .
We prove this theorem in Section 3. In Section 2, we have proved a result related to the existence of invariant form under a linear map. This result is crucial for the proof of the main theorem, and is of independent interest as well.
2. Existence of Invariant Forms
In this section we investigate the following problem that has seen some attention in the literature: given an invertible linear map , when does the vector space over admit a -invariant non-degenerate -hermitian form? The solution to this problem will be crucial in the proof of Theorem 1.1.
Sergeichuk [Ser87, Ser08, HS08] solved the above problem assuming that the characteristic of is different from two. Using a different approach, Gongopadhyay and Kulkarni [GK11] obtained conditions for an invertible linear map to admit an invariant non-degenerate quadratic and symplectic form assuming that the underlying field is of large characteristic and, this was extended in [GMar] over fields of characteristic different from two to derive conditions for an invertible linear map to admit an invariant non-degenerate -hermitian form. All these works assumed that the characteristic of the underlying field is different from two. When is trivial, the characteristic two case has been addressed by de Seguins Pazzis [dSP12]. In this section we work out the remaining case of non-trivial in characteristic two.
Recall that a -invariant subspace of is said to be indecomposable with respect to , or simply -indecomposable if it can not be expressed as a direct sum of proper -invariant subspaces. If is -indecomposable, is called cyclic. It is well-known from the structure theory of linear operators that can be written as a direct sum , where each is -indecomposable for , and each pair is isomorphic to , where is an irreducible monic factor of the minimal polynomial of , and is the operator , for eg. see [Jac75, Kul08]. Such is an elementary divisor of . If occurs times in the decomposition, we call the multiplicity of the elementary divisor .
With the notions as above, we prove the following result in this section that will be used in the proof of the main theorem.
Theorem 2.1**.**
Let be a field with a non-trivial involution . Then admits a -invariant non-degenerate -hermitian form if and only if an elementary divisor of is either self-dual, or its dual is also an elementary divisor with the same multiplicity.
When characteristic of is different from two, the above theorem was proved in [GMar]. In this section, we shall extend the proof to arbitrary characteristic. We shall prove the following result that combining with [GMar, Theorem 1.1], gives the above theorem.
Theorem 2.2**.**
Let be a field of characteristic 2, and let be an involution on . Then admits a -invariant non-degenerate -hermitian form if and only if the following conditions hold.
- (i)
An elementary divisor of is either self-dual, or its dual is also an elementary divisor with the same multiplicity. 2. (ii)
If is identity and is an elementary divisor, then either is even, or if is odd, then multiplicity of the elementary divisor must be an even number.
To prove Theorem 2.2, we first prove Lemma 2.3. When characteristic of is different from two, a version of the lemma was proved by Sergeichuk [Ser08, HS08], and an easy proof was sketched in [GMar]. Our proof below is motivated by a computational idea following [KT], where it has been used to investigate dimensions of the spaces of bilinear forms.
Lemma 2.3**.**
Let be a finite dimensional vector space over a field . Let be a non-unipotent, cyclic, self-dual linear transformation. Then there exists a -invariant non-degenerate -hermitian form on .
Proof.
Suppose . Since is cyclic, . Let be the minimal polynomial of , where is irreducible over , . Let . As is self-dual then , . Further since is cyclic, there is a vector in such that the -orbit of spans , i. e. the set
[TABLE]
forms a basis of and .
The linear transformation of with respect to is given by the matrix:
[TABLE]
Note that when characteristic of is , above for all .
Suppose is a sesquilinear -invariant form. Then, with respect to the basis , it is given by the matrix . The -invariance of is equivalent to the relation . Thus,
[TABLE]
Let , then it follows that for all . In view of this, for simplicity of notation, let and . Clearly, . So,
[TABLE]
Now for , we get
[TABLE]
We get similar expression for , and we have, for ,
[TABLE]
[TABLE]
Suppose is even: . For , using , the above equations can be written as:
[TABLE]
[TABLE]
Thus, are expressible in terms of .
Suppose is odd: . For , using , the above equations can be written as:
[TABLE]
[TABLE]
Thus, are expressible in terms of , ,,. Thus the vector , , , completely determines .
Now, choose a vector such that . Choose, as in (2.0.1) such that is the -th row of . Then , are the rows of . We claim that they must be linearly independent. If not, then assume is a linear combination of the other rows. This would imply that for . This would give a contradiction. So, we can choose to be non-degenerate.
Choose , , and in , and we get the required -hermitian form. ∎
Corollary 1**.**
Let be a finite dimensional vector space over a field with non-trivial involution . Let is a non-unipotent, cyclic, self-dual linear transformation. Then there exists a -invariant non-degenerate -hermitian form on . If , there also exists a -invariant -skew hermitian form.
Proof.
If , the existence of skew-hermitian form is similar noting that there always exists an element in such that . ∎
2.1. Invariant form under unipotent linear maps
Let be a field with a non-trivial involution . Then is the quadratic extension of a field
[TABLE]
Let . If , we can take and . If , then we can take and . We prove it for the characteristic case as that is relevant to us.
Let . Let , for , in . Clearly, cannot be zero, hence we can assume . Then implies , and hence, . Thus and , that implies .
Lemma 2.4**.**
Let be a field of characteristic with a non-trivial involution . Let be an -dimensional vector space of dimension over . Let be a unipotent linear map with minimal polynomial . Suppose is -indecomposable. Then admits a -invariant -hermitian form over .
Proof.
Let be an unipotent linear map. Suppose the minimal polynomial of is . Without loss of generality we can assume that is of the form
[TABLE]
Suppose preserves a -sesquilinear form . In matrix form, let . Then, . This gives the following relations: For ,
[TABLE]
[TABLE]
[TABLE]
From the above 2 equations, we have for and ,
[TABLE]
This implies that is a triangular matrix of the form
[TABLE]
where,
[TABLE]
In particular, it follows that , where . Thus must be chosen non-zero for to be non-singular. It also follows from this relation and (2.1.4) that , and since the characteristic of the base field is , we have . Thus . For to be hermitian we shall further have .
Case 1: Suppose is even, . Then by (2.1.4) one can find the following relations as follows:
[TABLE]
It follows that for , . Now expanding with respect to the final row the determinant of would be . Thus choosing to be , it provides a non-singular hermitian form.
Case 2: Suppose . In this case using (2.1.4), we have in particular
[TABLE]
So, we can choose and , for . Next we choose and for , thus , and so on. By doing this recursively in (2.1.4), we get a non-singular hermitian matrix . This completes the proof. ∎
2.2. Proof of Theorem 2.2
Suppose that the linear map admits an invariant non-degenerate hermitian form . Then the necessary condition follows from existing literatures, for example see [Wal63, Ser87, Ser08].
Conversely, let be a vector space of dim over the field and an invertible map such that the an elementary divisors of is either self-dual or, its dual is also an elementary divisor. For an elementary divisor , let denote the -indecomposable subspace isomorphic to . From the theory of elementary divisors it follows that has the decomposition, see [Wal63, Jac75],
[TABLE]
where, for each , is either self-dual with no root , or and, for each , , , are dual to each other, ; note that . Here denotes the direct sum. To prove the theorem it is sufficient to induce a -invariant hermitian form on each of the summands. After we have Lemma 2.3 and Lemma 2.4, the proof goes similarly as in the proof of [GMar, Theorem 1.1] or [GK11, Theorem 1.1].
3. Proof of Theorem 1.1
In this section, we shall prove the main theorem. We reformulate the statement of the main theorem as follows and will prove it in the rest of the section.
Theorem 3.1**.**
Let be a field with a non-trivial involution . Let be an invertible linear map. Then admits a -invariant non-degenerate -hermitian form if and only if is -real.
First we shall prove the following lemma.
Lemma 3.2**.**
Let be a field with involution and , . Then is -real if and only if an elementary divisor of is either self-dual, or its dual is also an elementary divisor with the same multiplicity.
Proof.
Let be a vector space of dimension over and be an invertible linear map. Suppose, is -real, that is is conjugate to . By the structure theory of linear maps, they have the same elementary divisors, and hence the same primary decomposition, see [Jac75]. is conjugate to on each of the summands in this decomposition, and that implies the necessary condition.
Suppose, an elementary divisor of is either self-dual or, its dual is also an elementary divisor. Then, has the primary decomposition as in (2.2.1). Hence, it is sufficient to prove the lemma on each of the summands in the decomposition. So, without loss of generality, we assume to be one of these summands and prove that is -real.
Case (i): Suppose , for some self-dual monic polynomial that has no root . Then belongs to , . As is self-dual then , . The set
[TABLE]
form a basis for and . The linear transformation with respect to the basis is given by: , and this is represented by the matrix:
[TABLE]
It is easy to see that
[TABLE]
Now the transformation given by conjugates to . In matrix representation with respect to the basis ,
[TABLE]
Case (ii): Suppose , where , are powers of an irreducible polynomial over , and are dual to each other. In this case, with respect to the cyclic basis , if , then with respect to the dual basis . Thus, with respect to the basis , has the representation:
[TABLE]
where each block has order . Then
[TABLE]
Now the matrix , where
[TABLE]
denote the identity matrix of order .
Case (iii): Let , where . Then and hence, by the theory of Jordan canonical form it follows that is real, see [OS15, Chapter 4].
This completes the proof. ∎
3.1. Proof of Theorem 3.1
Suppose is non-trivial. Then it follows from Theorem 2.1 that admits a -invariant hermitian form if and only if an elementary divisor of is either self-dual or its dual is also an elementary divisor. In view of Lemma 3.2, this is equivalent to the -reality of . This proves the theorem. Hence, Theorem 1.1 follows.
Remark 3.3*.*
The classical notion of real elements in a group is closely related to the notion of strongly real elements. An element in a group is strongly real if it is a product of two involutions in , see [OS15]. We can define strong -reality in a -invariant linear group as follows: an element is called strongly -real if is conjugate to by an involution in , i.e. there is an involution such that . When is identity, a strongly -real element is strongly real. The proof of Lemma 3.2 shows that:
Corollary 2**.**
Let be a field with involution . An element of is -real if and only if it is strongly -real.
Acknowledgement
The authors are thankful to Dipendra Prasad, Anupam Singh and Ian Short for their comments on this work. The second-named author acknowledges a UGC non-NET fellowship from Jadavpur University. It is a pleasure to thank the referee for helpful remarks.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[DGN 15] Silvio Dolfi, David Gluck, and Gabriel Navarro. On the orders of real elements of solvable groups. Israel J. Math. , 210(1):1–21, 2015.
- 2[d SP 12] Clément de Seguins Pazzis. When does a linear map belong to at least one orthogonal or symplectic group? Linear Algebra Appl. , 436(5):1385–1405, 2012.
- 3[GK 11] Krishnendu Gongopadhyay and Ravi S. Kulkarni. On the existence of an invariant non-degenerate bilinear form under a linear map. Linear Algebra Appl. , 434(1):89–103, 2011.
- 4[G Mar] Krishnendu Gongopadhyay and Sudip Mazumder. Existence of an invariant form under a linear map. Indian J. Pure Appl. Math , to appear.
- 5[Gon 11] Krishnendu Gongopadhyay. Conjugacy classes in Möbius groups. Geom. Dedicata , 151:245–258, 2011.
- 6[GP 13] Krishnendu Gongopadhyay and John R. Parker. Reversible complex hyperbolic isometries. Linear Algebra Appl. , 438(6):2728–2739, 2013.
- 7[GS 11] Nick Gill and Anupam Singh. Real and strongly real classes in PGL n ( q ) subscript PGL 𝑛 𝑞 {\rm PGL}_{n}(q) and quasi-simple covers of PSL n ( q ) subscript PSL 𝑛 𝑞 {\rm PSL}_{n}(q) . J. Group Theory , 14(3):461–489, 2011.
- 8[HL 06] Ale Jan Homburg and Jeroen S. W. Lamb. Symmetric homoclinic tangles in reversible systems. Ergodic Theory Dynam. Systems , 26(6):1769–1789, 2006.
