Ratio Mathematica Vol. 35, 2018, pp. 101-109 ISSN: 1592-7415 eISSN: 2282-8214 Note on Heisenberg Characters of Heisenberg Groups Alieh Zolfi ∗and Ali Reza Ashrafi † Received: 08-08-2018. Accepted: 21-10-2018. Published: 31-12-2018 doi:10.23755/rm.v35i0.429 c©Alieh Zolfi and Ali Reza Ashrafi Abstract An irreducible character χ of a group G is called a Heisenberg character, if Kerχ ⊇ [G, [G,G]]. In this paper, the Heisenberg characters of the quater- nion Heisenberg, generalized Heisenberg, polarised Heisenberg and three other types of infinite Heisenberg groups are computed. Keywords: Heisenberg character, Heisenberg group. 1 Introduction Suppose G is a finite group and V is a vector space over the complex field C. A representation of G is a homomorphism ϕ : G −→ GL(V ), where GL(V ) denotes the group of all invertible linear transformations V −→ V equipped with ∗Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, I. R. Iran †Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, I. R. Iran; ashrafi@kashanu.ac.ir 101 Alieh Zolfi and Ali Reza Ashrafi . composition of functions. The commutator subgroup [G,G] is the subgroup gen- erated by all the commutators [x,y] = xyx−1y−1 of the group G. An irreducible character χ of a group G is called a Heisenberg character, if kerχ ⊇ [G, [G,G]] [1]. Suppose ϕ : G −→ GL(V ) is an irreducible repre- sentation with irreducible character χ. Since [G,G′] ≤ Kerχ, ϕ : G [G,G′] −→ GL(V ) is an irreducible representation of G [G,G′] . Conversely, we assume that χ ∈ Irr ( G [G,G′] ) and δ : G [G,G′] −→ GL(V ) affords the irreducible character χ. If γ : G −→ G [G,G′] denotes the canonical homomorphism then δoγ : G −→ G [G,G′] is an irreducible representation for G and Ker δoγ ⊇ [G,G′]. This proves that there is a one to one correspondence between Heisenberg characters of G and irreducible characters of G [G,G′] , see [2, 8] for details. Marberg [8] in his interesting paper proved that the number of Heisenberg characters of the group Un(q) is a polynomial in q − 1 with nonnegative integer coefficients, with degree n − 1, and whose leading coefficient is the (n − 1)−th Fibonacci number. The present authors [1], characterized groups with at most five Heisenberg characters. The aim of this paper is to compute all Heisenberg characters of five classes of infinite Heisenberg groups. These are as follows: 1. Suppose T denotes the set of all complex numbers of unit modulus and H = Rn × Rn × T. Define (y1,x1,z1)(y2,x2,z2) = (y1 + y2,x1 + x2,e −2πiy2.x1z1z2). It is easy to see that H is a group under this operation. This group is called the Heisenberg group of second type [5]. 2. The polarised Heisenberg group H3n is defined as the set of all triples in Rn ×Rn ×R under the multiplication (x,y,z)(a,b,c) = (x + a,y + b,z + c + 1 2 (x.b−y.a)), see [3] for details. 3. Suppose a = (a1,a2, . . . ,an) is an n−tuple in Rn, where ai’s are positive real constants, 1 ≤ i ≤ n. Following Tianwu and Jianxun [10], we define a group operation on Han = R n ×Rn ×R given by (x,y,z)(r,s,t) = (x + r,y + s,z + t + 1 2 n∑ j=1 aj(rjyj −sjxj)). 102 Note on Heisenberg Characters of Heisenberg Groups This group is called the generalized Heisenberg group. In the mentioned pa- per, the authors proved that the group operation of the generalized Heisen- berg group can be simplified in the following way: Suppose x = (x1,x2, . . . ,xn) and y = (y1,y2, . . . ,yn). Define x ∗ y = (x1y1,x2y2, . . . ,xnyn) and ab = ∑n j=1 ajbj. If c = (c1,c2, . . . ,cn) and λ ∈ R then we can see that (i) x ∗ (y + c) = x ∗ y + x ∗ c; (ii) (x ∗ y)c = x(y ∗ c); and (iii) (λx) ∗ y = λ(x ∗ y). Therefore, the group operation of the generalized Heisenberg group can be written as (x,y,z)(r,s,t) = (x + r,y + s,z + t + 1 2 ((a∗ r)y − (a∗s)x)). 4. Suppose H denotes the set of all of quaternion numbers with three imagi- nary units i,j and k such that i2 = j2 = k2 = ijk = −1. Following Liu and Wang [7], we define the quaternion Heisenberg group N as a nilpotent Lie group with underlying manifold R4 ×R3. The group structure is given by (q,t)(p,s) = (q + p,t + s + 1 2 Im(pq)), where p,q ∈ R4 and t,s ∈ R3. 5. Following Qingyan and Zunwei [9], the Heisenberg group Hn of third type is a non-commutative nilpotent Lie group, with the underlying manifold R2n ×R. The group operation can be given as: (x1,x2, . . . ,x2n,x2n+1)(x ′ 1,x ′ 2, . . . ,x ′ 2n,x ′ 2n+1) = (x1 + x ′ 1,x2 + x ′ 2, . . . ,x2n + x ′ 2n,x2n+1 + x ′ 2n+1 + 2 n∑ j=1 (x ′ jxn+j −xjx ′ n+j). 6. Suppose Hn = Cn×R with group law defined by (z,t)·(w,s) = (z+w,t+ s + 2Im(z.w)). This is our sixth class of Heisenberg groups. Following Chang et al. [4], this group can be realized as the boundary of the Siegel upper half-space Un+1 in Cn+1, where the group operation gives a group action on the hypersurface. Throughout this paper our notation is standard and can be taken from the fa- mous book of Isaacs [6]. Suppose G is a group and {{e} = A0,A1, . . . ,An = G} is a set of normal subgroups of G such that A0 � A1 � . . . � An = G. (1) 103 Alieh Zolfi and Ali Reza Ashrafi . The sequence (1) is called a central series for G, if [G,Ai+1] ≤ Ai in which [G,H] denotes the subgroup of G generated by all commutators ghg−1h−1, where g ∈ G,h ∈ H. The group G is called nilpotent, if it has a central series. The nilpotency class of G, nc(G), is the length of its central series. The set of all irreducible characters of G is denoted by Irr(G) and the trivial character of G is denoted by 1G. 2 Main Results The aim of this section is to compute the Heisenberg characters of five differ- ent types of Heisenberg groups. To do this, we first note that every linear character of a group G is Heisenberg. This proves that all irreducible characters of abelian groups are Heisenberg. Lemma 2.1. All irreducible characters of a group G are Heisenberg if and only if G is nilpotent of class two. Proof. Suppose nc(G) = 2. Then [G,G′] = 1 and so all irreducible characters are Heisenberg. If all irreducible characters are Heisenberg then [G,G′] ≤ ∩χ∈Irr(G) = {e}, as desired. Theorem 2.2. All irreducible characters of the Heisenberg groups H, H3n, H a n, N , Hn and Hn are Heisenberg. Proof. Apply Lemma 2.1. Our main proof will consider five separate cases as follows: 1. The Heisenberg Group H. We first compute the derived subgroup H′. We have, H′ = 〈 [(x,y,z),(x ′ ,y ′ ,z ′ )] | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ H,z = eiΘ1,z ′ = eiΘ2 〉 = 〈(x + x ′ ,y + y ′ ,ei(Θ1+Θ2−2πx ′ y))(−x−x ′ ,−y −y ′ , ei(Θ1+Θ2+2πxy+2πx ′ y ′ +2πx ′ y)) | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ H,z = eiΘ1,z ′ = eiΘ2〉 = 〈 (0,0,e2πi(x.y ′ −x ′ .y) | x,y,x′,y′ ∈ Rn 〉 . 104 Note on Heisenberg Characters of Heisenberg Groups Therefore, [H,H′] = 〈 (x,y,eiΘ1)(0,0,eiΘ2)(−x,−y,e−iΘ1−2πixy)(0,0,e−iΘ2) | (x,y,eiΘ1) ∈ H,(0,0,eiΘ2) ∈ H′ 〉 = 〈(x,y,eiΘ1+iΘ2)(−x,−y,e−i(Θ1+Θ2+2πixy) | (x,y,eiΘ1) ∈ H,(0,0,eiΘ2) ∈ H′〉 = {(0,0,1)}. So, all irreducible characters of H are Heisenberg. 2. The Heisenberg group H3n. The commutator subgroup of H 3 n can be computed as follows: (H3n) ′ = 〈[(x,y,z),(x ′ ,y ′ ,z ′ )] | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ H3n〉 = 〈(x,y,z)(x ′ ,y ′ ,z ′ )(x,y,z)−1(x ′ ,y ′ ,z ′ )−1 | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ H3n〉 = 〈(0,0,(x.y ′ −y.x ′ ) | x,y,x′,y′ ∈ Rn〉. On the other hand, [H3n,(H 3 n) ′] = {(0,0,0)} and so all irreducible characters of this group are Heisenberg. 3. The Heisenberg group Han. Again, we first compute the commutator subgroup (Han) ′. We have, (Han) ′ = 〈[(x,y,z),(x ′ ,y ′ ,z ′ )] | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ Han,a ∈ R n +〉 = 〈 (x + x ′ ,y + y ′ , 1 2 ((a∗x ′ )y − (a∗y ′ )x))(−x−x ′ ,−y −y ′ , + 1 2 ( (a∗−x ′ )(−y)− (a∗−y ′ )(−x) ) | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ Han,a ∈ R n +〉 = 〈 (0,0,(a∗x ′ )y − (a∗y ′ )x) | x,y,x′,y′ ∈ Rn,a ∈ Rn+ 〉 . Therefore, [Han,(H a n) ′] = {(0,0,0)}. This shows that all irreducible characters are Heisenberg. 4. The Heisenberg group N . By definition of this group, we have N ′ = 〈[(p,t),(q,s)] | (p,t),(q,s) ∈N〉 = 〈(p + q,t + s + 1 2 Im(qp))(−p− q,−t−s + 1 2 Im(qp)) | p,q ∈ R4, t,s ∈ R3〉 = 〈(0,Im(qp)) | p,q ∈ R4〉. Therefore, we have again [N ,N ′] = {(0,0)}. Now apply Lemma 2.1 to deduce that all irreducible characters of this group are Heisenberg. 105 Alieh Zolfi and Ali Reza Ashrafi . 5. The Heisenberg group Hn. By definition of this group, (Hn)′ = 〈 [(x1, . . . ,x2n,x2n+1),(x ′ 1, . . . ,x ′ 2n,x ′ 2n+1)] | (x1, . . . ,x2n,x2n+1),(x ′ 1, . . . ,x ′ 2n,x ′ 2n+1) ∈ H n 〉 = 〈(x1 + x ′ 1, . . . ,x2n + x ′ 2n,x2n+1 + x ′ 2n+1 + 2 ( n∑ j=1 (x ′ jxn+j −xjx ′ n+j) ) (−x1 −x ′ 1, . . . ,−x2n −x ′ 2n,−x2n+1 −x ′ 2n+1 + 2 ( n∑ j=1 (x ′ jxn+j −xjx ′ n+j) ) | (x1, . . . ,x2n,x2n+1),(x ′ 1, . . . ,x ′ 2n,x ′ 2n+1) ∈ H n〉 = 〈 (0, . . . ,4 ( n∑ j=1 (x ′ jxn+j −xjx ′ n+j) ) | x1, . . . ,x2n,x2n+1,x ′ 1, . . . ,x ′ 2n,x ′ 2n+1 ∈ R 〉 . Therefore, [Hn,(Hn)′] = {(0, . . . ,0,0)} and by Lemma 2.1 all irreducible charac- ters of this group are Heisenberg. 6. The Heisenberg group Hn. The derived subgroup of this group can be com- puted as follows: (Hn)′ = 〈[(z,t),(w,s)] | (z,t),(w,s) ∈Hn〉 = 〈(z,t)(w,s)(−z,−t)(−w,−s) | (z,t),(w,s) ∈Hn〉 = 〈(z + w,t + s + 2Im(zw))(−z −w,−t−s + 2Im(zw)) | z,w ∈ Cn, t,s ∈ R〉 = 〈(0,2Im(zw −wz)) | z,w ∈ Cn〉. Therefore, [Hn,(Hn)′] = {(0,0,0)} and by Lemma 2.1, all irreducible characters of this group are again Heisenberg. This completes our argument. In the end of this paper we compute the factor groups of six types of Heisen- berg groups modulus their centers. Theorem 2.3. The factor groups of all Heisenberg groups modulus their centers 106 Note on Heisenberg Characters of Heisenberg Groups can be computed as: H Z(H) ∼= Rn ×Rn, H3n Z(H3n) ∼= Rn ×Rn, Han Z(Han) ∼= Rn ×Rn N Z(N) ∼= R4, Hn Z(Hn) ∼= Rn ×Rn, H Z(H) ∼= Cn. Proof. An easy calculations show that Z(H) = {(0,0,z)|z ∈ T}, Z(H3n) = {(0,0,s)|s ∈ R}, Z(Han) = {(0,0,s)|s ∈ R}, Z(Hn) = {(0, . . . ,0,x2n+1)| x2n+1 ∈ R} and Z(Hn) ∼= R. Therefore, HZ(H) ∼= Rn × Rn, H 3 n Z(H3n) ∼= Rn × Rn, Han Z(Han) ∼= Rn ×Rn, H n Z(Hn) ∼= Rn ×Rn and H Z(H) ∼= Cn. So, it is enough to compute N Z(N) ∼= R4. To do this, we assume that (p,t) ∈ Z(N) is arbitrary. Hence for each pair (q,s), (p,t)(q,s) = (q,s)(p,t). This proves that (p + q,s + t + 1 2 Im(qp)) = (q + p,s + t + 1 2 Im(pq)) and so Im(pq) = 0. Suppose p = p0 + ip1 + jp2 + kp3. Then by considering three different values q = (1,0,0,0),(0,1,0,0),(0,0,1,0), we will have the following system of equations:  p1 + p2 + p3 = 0, p0 + p2 −p3 = 0, p0 −p1 + p3 = 0. Hence Z(N)∼= R3 and N Z(N) ∼= R4 that completes the proof. 3 Concluding Remarks In this paper the Heisenberg characters of six classes of Heisenberg groups were computed. It is proved that all irreducible characters of these Heisenberg groups are Heisenberg. We also compute all factor groups of these Heisenberg groups which show these factor groups are abelian and so all irreducible characters of these factor groups are again Heisenberg. Acknowledgment We are very thankful from the referee for his/her comments and corrections. The authors also would like to express our sincere gratitude to professor Alireza Abdollahi for discussion on the exact form of Lemma 1 during the 48th Annual 107 Alieh Zolfi and Ali Reza Ashrafi . Iranian Mathematics Conference in that was held in the University of Hamedan. The research of the authors are partially supported by the University of Kashan under grant no 364988/127. References [1] A. R. Ashrafi and A. Zolfi, On the number of Heisenberg characters of finite groups, submitted. [2] M. Boyarchenko and V. Drinfeld, A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteris- tic, arXiv:math/0609769v2. [3] A. Brodlie, The representation theory of the Heisenberg group and be- yond, XI-th International Conference Symmetry Methods in Physics, Prague, Czech Republic, June 21-24, 2004. [4] D.-C. Chang, W. Eby and E. Grinberg, Deconvolution for the Pompeiu Prob- lem on the Heisenberg Group I, I. Sabadini, D. C. Struppa (eds.), The Math- ematical Legacy of Leon Ehrenpreis, Springer Proceedings in Mathematics 16, Springer-Verlag Italia, 2012. [5] R. Howe, On the role of the Heisenberg group in harmonic analysis, Ameri- can Mathematical Society. Bulletin. New Series 3 (2) (1980) 821–843. [6] I. M. Isaacs, Character Theory of Finite Groups, AMS Chelsea Publishing, Providence, RI, 2006. [7] H.-P. Liu and Y.-Z. Wang, A restriction theorem for the quaternion Heisen- berg group, Applied Mathematics. A Journal of Chinese Universities. Ser. B 26 (1) (2011) 86–92. [8] E. Marberg, Heisenberg characters, unitriangular groups, and Fibonacci numbers, Journal of Combinatorial Theory, Ser. A 119 (2012) 882–903. [9] W. Qingyan and F. Zunwei, Sharp estimates for Hardy operators on Heisen- berg group, Frontiers of Mathematics in China 11 (1) (2016) 155–172. 108 Note on Heisenberg Characters of Heisenberg Groups [10] L. Tianwu and H. Jianxun, The radon transforms on the generalized Heisen- berg Group, ISRN Mathematical Analysis (2014) Art. ID 490601, 7 pp. 109