Invariant Bilinear Forms Under the Operator Group of Order P3 with Odd

Home , Class function

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where di is the degree of ρi and di||G| with dj ≥ di when j>i. It is alreadyP well understood inP the literature ′ t that the invariant space ΞG under ρ can be expressed by the set ΞG = {X ∈ Mn(F) | Cρ(g)XCρ(g) = X, ∀g ∈ G}

with respect to an ordered basis e of V(F), where Mn(F) is the set of square matrices of order n with entries from

F and Cρ(g) is the matrix representation of the linear transformation ρ(g) with respect to e.

Here we consider G to be a group of order p3 with p an odd prime, F a ﬁeld with char(F) 6= p, which consists of a primitive p3th root of unity and (ρ, V) an n degree representation of G over F. Then the corresponding set

ΞG of invariant bilinear forms on V under ρ, forms a subspace of Ξ. In this paper our investigation is about the following questions.

Question. How many n degree representations (upto isomosphism) of G can be there ? What is the dimension of ΞG for every n degree representation ? What are the necessary and suﬃcient conditions for the existence of a non-degenerate invariant bilinear form. The primary focus is on the existence of a non-degenerate invariant bilinear forms. Over complex numbers it has been seen with positive ﬁndings, as an evidence we present here one.

It is well known that every maximal (proper) subgroup of G has index p and is normal (As finite p groups are nilpotent and any proper subgroup of a nilpotent group is properly contained in its normaliser). Thus there are epimorphisms from G to the cyclic group C of order p. Fix a generator c of C and 1 6= ζ a primitive pth -root of unity. Let U and V be the one-dimensional representations of C on which c acts respectively by ζ and ζ¯. We claim that U ⊕V admits a C-invariant non-degenerate bilinear form. Via some epimorphism to C one can pullback these representations and the forms to G. To prove the claim let us fix the vectors 0 6= u ∈ U and 0 6= v ∈ V . Using these we define a bilinear form B on U ⊕ V as follows: B(u,u)=0= B(v, v), B(u, v)=1= B(v,u) so that B(λu + µv,λ′u + µ′v)= λµ′ + λ′µ. Now we may easily check the C invariance as follows: B(c(λu+µv),c(λ′u+µ′v)) = B(ζ(λu+µv), ζ¯(λ′u+µ′v)) = ζζλµ¯ ′ + λ′µ = B(λu + µv,λ′u + µ′v). The questions in concern have been studied in the literature in several distinct contexts. Gongopadhyay and Kulkarni [6] studied the existence of T -invariant non-degenerate symmetric (resp. skew-symmetric) bilinear forms. Kulkarni and Tanti [10] formulated the dimension of space of T-invariant bilinear forms. Mahto and Tanti [11] formulated the dimensions of invariant spaces and explicitly discussed about the existence of the non-degenerate invariant bilinear forms under n degree representations of a group of order 8. Sergeichuk [15] studied systems of forms and linear mappings by associating with them self-adjoint representations of a category with involution. Frobenius [5] proved that every endomorphism of a finite dimensional vector space V is self-adjoint for at least one non-degenerate symmetric bilinear form on V. Later, Stenzel [13] determined when an endomorphism could be skew- selfadjoint for a non-degenerate quadratic form, or self-adjoint or skew-self adjoint for a symplectic form

2 on complex vector spaces. However his results were later generalized to an arbitrary field [7]. Pazzis [12] tackled the case of the automorphisms of a finite dimensional vector space that are orthogonal (resp. symplectic) for at least one non-degenerate quadratic form (resp. symplectic form) over an arbitrary field of characteristics 2. In this paper we investigate about the counting of n degree representations of a group of order p3 with p ≥ 3 a prime, over a field F which consists of a primitive p3th root of unity, dimensions of their corresponding spaces of invariant bilinear forms and establish a characterization criteria for existence of a non-degenerate invariant bilinear form. Our investigations are stated in the following three main theorems.

Theorem 1.1. The number of n degree representations (upto isomorphism) of a group G of order p3, with p an 3 n 2 n+p −1 ⌊ p ⌋ µ+p−2 n−µp+p −1 odd prime is p3−1 when G is abelian and µ=0 p−2 p2−1 otherwise.

P 3 Theorem 1.2. The space ΞG of invariant bilinear forms of a group G of order p (p an odd prime), under an n

degree representation (ρ, V(F)) is isomorphic to the direct sum of the subspaces W(i,j), (i, j) ∈ AG of Mn(F), i.e., ′ , where A = {(i, j) | ρ and ρ are dual to each other} and for every (i, j) ∈ A , = ΞG = (i,j)∈AG W(i,j) G i j G W(i,j) { X ∈ ( ) | Xij = Ct Xij C , ∀g ∈ G and rest blocks are zeros }. Also for (i, j) ∈ A , LMn F diki×dj kj kiρi(g) diki×dj kj kj ρj (g) G the dimension of W(i,j) = kikj . .

Theorem 1.3. If G is group of order p3, with p an odd prime, then an n degree representation of G consists of a non-degenerate invariant bilinear form if and only if every irreducible representation and its dual have the same multiplicity.

Thus we are able to give the answers to the questions in concern. Here for n ∈ N Theorem 1.1 counts all n degree representations, Theorem 1.2 computes the dimension of the space of invariant bilinear forms and Theorem 1.3 characterises those n degree representations, each of which admits a non-degenerate invariant bilinear form for a group of order p3 with p an odd prime over the ﬁeld F.

Remark 1.1. Thus we get the necessary and suﬃcient condition for the existence of a non-degenerate invariant bilinear form under an n degree representation.

2. Preliminaries

The classiﬁcation of groups of order p3, with p an odd prime has been well understood in the literature. Due to the structure theorem of ﬁnite abelian groups, there are only three abelian groups (upto isomorphism) of this order viz, Zp3, Zp2 × Zp and Zp × Zp × Zp. Amongst non-abelian groups of this order Hisenberg group [3] is well known and named after a German theoretical physicist Werner Heisenberg. In this group every non identity element is of order p. The elements of this group are usualy seen in the form of 3 × 3 upper triangular matrices

whose diagonal entries consist of 1 and other three entries are chosen from the ﬁnite ﬁeld Zp. If at all there exists any other non-abelian group of this order then it must have a non identity element of order p2. Let us consider the 2 × 2 upper triangular matrices with a11 =1+ pm, (m ∈ Zp), a12 = a ∈ Zp2 and a22 = 1. Here the element

3 2 with entries a11 = a12 = a22 = 1 has order p making it non-isomorphic to the Heisenberg group. We denote this 3 group by Gp. Thus upto isomorphism there are ﬁve groups of order p with an odd prime p [3]. For an abelian group of order p3, there are p3 number of irreducible representations each having degree 1 and for non-abelian cases, the number of trivial conjugacy calsses is |Z(G)| = p. To ﬁnd a non-trivial conjugacy class we refer to the theory of group action and class equation

|G| = |Z(G)| + |Cg| g∈ /Z(G) varying over distinctX conjugacy classes |G| with |Cg| = |C(g)| , where Cg and C(g) are the conjugacy class and the centralizer respectively of g in G. If g∈ / Z(G) then |C(g)| = p2. Therefore there are p2 − 1 non trivial conjugacy classes of order p. Thus total number of conjugacy classes for a non-abelian group is r = p2 − 1+ p, which is same as the number of irre- r 2 2 ducible representations with degree di since di||G| and i=1 di = |G|, therefore di =1 or p. Thus there are p representations of degree 1 and p − 1 representations ofP degree p for a non-abelian group. We here formulate ev- 3 ery irreducible representation ρi in such a way that the entries of Cρi(g) are either 0 or p th primitive roots of unity.

Deﬁnition 2.1. The character of ρ is a function χ : G → F, χ(g)= tr(ρ(g)) and is also called character of the group G.

Theorem 2.1. (Maschke’s Theorem): If char(F) does not divide |G|, then every representation of G is a direct sum of irreducible representations.

Proof. See p -316, corollary (4.9) [1].

Theorem 2.2. Two representations (ρ, V(F)) and (ρ′, V(F)) of G are isomorphic iﬀ their character values are same i.e, χ(g)= χ′(g) for all g ∈ G.

Proof. See p -319, corollary ( 5.13) [1].

3. Irreducible representation (irrep.) of group of order p3 with an odd prime p.

3 In this section G is a group of order p with p an odd prime, (ρi, Wdi (F)) stands for an irreducible representation 3 ρi of degree di of G over a ﬁeld F with char(F) 6= p, which consists of ω ∈ F, a primitive p th root of unity.

Let σs = ρp2+s, 1 ≤ s ≤ p − 1 denote the irreducible representations of degree p when G is non-abelian. Since σs is a homomorphism from G to GL( ) ∼ GL(p, ), by the fundamental theorem of homomorphism G ∼ Wp = F Ker(σs) =

σs(G) and the possible value of |Ker(σs)| is 1 or p. If |Ker(σs)| = p then (χs,χs) ≥ 1, therefore (χs,χs) = 1 only 2 p when g∈ / Ker(σs), so we have χs(g) = 0. Also we have trivial character χi(g)=1, ∀g ∈ G and (χi,χs)= p3 6= 0,

which fails the orthonormality property of the irreducible characters. Thus |Ker(σs)| = 1 and hence σs(G) ∼ G which is isomorphic to a non-abelian group of order p3. Now σ (G) has subgroups H and K of orders = {eG} s p and p2 respectively [see p-132, [4], exercise-29]. Since K is a maximal subgroup of G so K must be a normal

subgroup and there exists a subgroup Hp of order p which is not normal (not contained in Z(G))[see p-188, [4],

Theorem 1], thus we have, σs(G) = KHp and every element of σs(G) can be expressed uniquely in the form

of kh for some k ∈ K and h ∈ Hp (this uniqueness follows from the condition K ∩ Hp = {σs(eG)}), therefore

4 ∼ ∼ σs(Heis(Zp)) = (Zp × Zp) ⋊ Zp and σs(Gp) = Zp2 ⋊ Zp . As |Z(G)| = p, we can choose a subgroup of order p from GL(p, F) and say that it is the Image of Z(G) (subset of a normal subgroup of order p2 of G) denoted by 2 sp Im(Z(G)) = {ω Ip| 1 ≤ s ≤ p } under the irreducible representation σs (since center elements commute and ∼ are scalar matrices). Thus Im(Z(G))($ σs(K)) = Zp, each non-identity element of Z(G) have p − 1 choices 3 3 in Zp under σs and rest p − p elements of G map to rest p − p elements of σs(G). We decide all p degree representations by the elements of Z(G) and elements of G − Z(G) by mapping to the set σs(G) − ImZ(G) ⊆ GL(p, F), which consists of those elements whose trace is zero and order of every element is either p or p2. We will depict all irreducible representations of G in the next subsections.

3.1. Heisenberg group. 1 α β

G = Heis(Zp) = ((α, β),γ)= 0 1 γ | α,β,γ ∈ Zp.       0 0 1       Another presentation of Heisenberg group is ([4], pp 179) 

G = hx, a | xp = ap =1, x(a−1xa)x−1 = a−1xa, a(xax−1)a−1 = xax−1i.

For the Heisenberg group, center Z(G) = hxax−1a−1i = h((0, 1), 0)i. Each of the irreducible representations

σs, 1 ≤ s ≤ p − 1 of degree p maps the center of G to the center of GL(p, F), i. e., if z ∈ Z(G) then σs(z) is 2 p p 2 (p−1) a scalar matrix say csIp, cs ∈ F and since order of z is p so cs = 1. Hence cs ∈ {1,ω = ǫ,ǫ , ······ ,ǫ }. Thus each of the p − 1 irreducible representations of degree p maps Z(G) into the Z(GL(p, F)) even maps to the

Z(σs(G)), it has been recorded in the following table.

Table 1: All p degree irreducible representations of Heis(Zp)

All irrep. → σ2η−1 σ2η p−1 Runing variable → 1 ≤ η ≤ 2 , η (p−η) g ∈ Z(G) ǫ Ip ǫ Ip . g∈ / Z(G) See Note 3.1 See Note 3.1

∗ Dual irrep. σ2η−1 = σ2η # irrep. p − 1

−1 −1 −1 −1 Note 3.1. For the p degree representations σs of Heisenberg group G, if xax a ∈ Z(G) then σs(xax a )= −1 −1 m ρp2+s(xax a ) = ǫ Ip, for some m, 1 ≤ m

p 3 σs(G) − σs Z(G) = {Au ∈ GL(p, F) | T r(Au)=0 and Au = Ip for 1 ≤ u ≤ p − p}.

5 Thus the p degree irreducible representations σs can be expressed as below

η 0 10 0 ··· 0 ǫη 0 0 0 ··· 0 0 01 0 ··· 0  0 ǫη+1 0 0 ··· 0     η+2  0 00 1 ··· 0  0 0 ǫ 0 ··· 0  σ2η−1(x)=   andσ2η−1(a)=   ,    η+3  0 00 0 ··· 0  0 0 0 ǫ ··· 0      ......   ......  ......   ......          1 00 0 ··· 0  00 0 0 ··· ǫη+p−1        

−η 0 10 0 ··· 0 ǫp+1−η 0 0 0 ··· 0 0 01 0 ··· 0  0 ǫp+2−η 0 0 ··· 0     p+3−η  0 00 1 ··· 0  0 0 ǫ 0 ··· 0  σ2η(x)=   and σ2η(a)=   .    p+4−η  0 00 0 ··· 0  0 0 0 ǫ ··· 0      ......   ......  ......   ......          1 00 0 ··· 0  0 0 0 0 ··· ǫp−η         3 2 Non-abelian groups of order p have p representations of degree 1. Let σ(s−1,t−1), 1 ≤ s,t ≤ p, denote representations of degree 1. Here we present all 1 degree representations for the Heisenberg group.

Table 2: All irreducible representations of degree 1 for the Heisenberg group.

All irrep. → ρ(t−1)p+2s−2 = ρ(t−1)p+2s−1 = ρ(t−1)p+2s−1 = ρ(t−1)p+2s =

σ(s−1,t−1) σ(p−s+1,p−t+1) σ(s−1,t−1) σ(p−s+1,p−t+1) p+1 p+1 Runing Variable → 1 ≤ s ≤ 2 ,t =1 1 ≤ s ≤ p, 2 ≤ t ≤ 2 1 0 β 0 1 0 ǫp =1 1 1 1   0 0 1 .   1 α 0 0 1 γ ǫα(s−1) ǫα(p−s+1) ǫ{α(s−1)+γ(t−1)} ǫ{α(p−s+1)+γ(p−t+1)}   0 0 1     ∗ ∗ Dual irrep. ρ2s−2 = ρ2s−1 & ρ0 = ρ1 ρ(t−1)p+2s−1 = ρ(t−1)p+2s # irrep. p p2 − p

(α(s−1)+γ(t−1)) Since 1 ≤ s,t ≤ p, and α, γ ∈ Zp, so ǫ is generated by ǫ, thus the representation σ(s−1,t−1) maps an element g ∈ G to ǫm, for some m, 1 ≤ m ≤ p.

3.2. The non-abelian group Gp 1+ pγ δ Gp = (pγ,δ)= | γ ∈ Zp, δ ∈ Zp2 .   0 1      Another presentation of this group is ([4], pp 180) 

6 2 p p p+1 Gp = hx, y | x = y =1, xy = y xi.

1+ p 0 p The center of Gp is Z(Gp)= y = . *  0 1+

  2 p p p ηp Note 3.2. For the p degree representations σs of group Gp, if y ∈ Z(Gp) then σ2η−1(y )= ρp2+2η−1(y )= ω Ip 2 p−1 p 2 p (p−η)p and its dual σ2η(y )= ρp +2η(y )= ω Ip, 1 ≤ η ≤ 2 . Also the elements of Gp − Z(Gp) map bijectively to the set

2 p p 3 σs(Gp) − σs Z(Gp) = {Au ∈ GL(p, F) | T r(Au)=0 and Au = Ip or Au = Ip for 1 ≤ u ≤ p − p}.

We have recorded the p ordered irreducible representations of Gp in the following table.

Table 3: All irreducible representations of degree p for the group Gp.

All irrep. → σ2η−1 σ2η p−1 Runing variable → 1 ≤ η ≤ 2 , 2 2 ηp (p−η)p g ∈ Z(Gp) ω Ip ω Ip . g∈ / Z(Gp) See Note 3.2 See Note 3.2 ∗ Dual irrep. σ2η−1 = σ2η # irrep. p − 1

Thus the p degree irreducible representations σs is deﬁned as below

η 0 1 0 ··· 0 ωηp 0 ··· 0 2 0 0 1 ··· 0  0 ωp +ηp 0 ··· 0  2    2p +ηp  σ2η−1(x)= 0 0 0 ··· 0 andσ2η−1(y)=  0 0 ω ··· 0  ,     ......   ......  . . . . .  . . . . .         2  1 0 0 ··· 0  0 0 0 ··· ω(p−1)p +ηp         −η 2 0 1 0 ··· 0 ωp −ηp 0 ··· 0 2 0 0 1 ··· 0  0 ω2p −ηp 0 ··· 0  2    3p −ηp  σ2η(x)= 0 0 0 ··· 0 andσ2η(y)=  0 0 ω ··· 0  .     ......   ......  . . . . .  . . . . .         3  1 0 0 ··· 0  0 0 0 ··· ωp −ηp     2     There are p representations σ(s−1,t−1), 1 ≤ s,t ≤ p of degree 1. As g ∈ Gp − Z(Gp), we have g = (pγ,δ) =

1+ pγ δ ∗ 2 , δ ∈ Zp2 , |Gp −Z(Gp)| = (p −1)p. In the table 4 we have recorded all 1 degree representations of Gp.  0 1  

7 Table 4: All irreducible representations of degree 1 for the group Gp.

All irrep. → ρ(t−1)p+2s−2 = ρ(t−1)p+2s−1 = ρ(t−1)p+2s−1 = ρ(t−1)p+2s =

σ(s−1,t−1) σ(p−s+1,p−t+1) σ(s−1,t−1) σ(p−s+1,p−t+1) p+1 p+1 Runing Variable → 1 ≤ s ≤ 2 ,t =1 1 ≤ s ≤ p, 2 ≤ t ≤ 2

1+ pγ 0 3 ypγ = ωp =1 1 1 1  0 1 .   1+ pγ δ 2 2 2 ′ 2 ′ ,δ 6=0 ωp γ(s−1) ωp γ(p−s+1) ωp {γ(s−1)+δ (t−1)} ωp {γ(p−s+1)+δ (p−t+1)}  0 1   where δ′ = δ mod p Where δ′ = δ mod p ∗ ∗ Dual irrep. ρ2s−2 = ρ2s−1 & ρ0 = ρ1 ρ(t−1)p+2s−1 = ρ(t−1)p+2s # irrep. p p2 − p

2 ′ 2 p (γ(s−1)+δ (t−1)) p Here ω is generated by ω , thus the representation σ(s−1,t−1) maps an element g ∈ Gp to 2 ωmp , for some m, 1 ≤ m ≤ p.

Note 3.3. As a finite abelian group is finitely generated, here for the abelian groups Zp3 , Zp2 ×Zp and Zp ×Zp ×Zp there exist the finite generating subsets {a}, {b,c} and {d,e,f} respectively. For example we may take a = 1+p3Z, b = (1+ p2Z, 0+ pZ), c = (0+ p2Z, 1+ pZ), d = (1+ pZ, 0+ pZ, 0+ pZ), e = (0+ pZ, 1+ pZ, 0+ pZ) and

f =(0+ pZ, 0+ pZ, 1+ pZ). Further order of an element g under ρi, 1 ≤ i ≤ r is |ρi(g)|.

3.3. The cyclic group Zp3 . 3 p Here Zp3 = ha | a =1i. The representation tables are given as below.

Table 5: All irreducible representations of the group Zp3 .

All irrep. → ρt ρps−1+2t ρps−1+2t+1 Runing variable →

s s−1 3 3−s 3−s p(p −p −2) i =1 i ∈ x | gcd(p , xp )= p , 1 ≤ x ≤ 2(p−1) +1 t =1 t = i′s place in above set Z T p3 = , s |ρi(a)| 1 p , s =1, 2, 3 3 3−s 3 3−s a ωp =1 ωip , ωp −ip

∗ −1 Dual irrep. self ρps−1+2t = ρps +2t+1 # irrep. 1 ps − ps−1

Table 6: All irreducible representations of the group Z8.

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 TZ8 = . a ω8 ω4 ω2 ω6 ω ω7 ω3 ω5

3.4. The group Zp2 × Zp. 2 p p Here Zp2 × Zp =ha,b | a = b =1,ab = bai.

8 Z Z T p2 × p =

× Table 7: All irreducible representations of group Zp2 Zp.

All irrep.→ σ(s,t) σ(p2,t) σ(p2,p−t) σ(s,p) σ(p−s,p) σ(s,t) σ(p−s,p−t) σ(s,t) σ(p2−s,p−t) σ(s,p) σ(p−s,p) Runing s t ∈{1, .., p − 1} t ∈{1, .., p − 1} 2 p−1 p −p variable = t &1 ≤ s ≤ 2 or &1 ≤ s ≤ 2 or 2 p−1 p−1 2 p −p → = p 1 ≤ t ≤ 2 1 ≤ s ≤ 2 s ∈{1, ..., p − 1} s ∈{1, ..., p − p} 1 ≤ s ≤ 2 p−1 p−1 &1 ≤ t ≤ 2 &1 ≤ t ≤ 2 2 2 |ρi(a)| 1 1 p p p p ,

|ρi(b)| 1 p 1 p p 1 2 2 2 2 2 2 a 1 1 1 ωsp ω(p−s)p ωsp ω(p−s)p ωsp ω(p −s)p ωsp ω(p −s)p 2 2 2 2 2 2 b 1 ωtp ω(p−t)p 1 1 ωsp ω(p−t)p ωtp ω(p−t)p 1 1

∗ 2 ∗ ∗ ∗ 2 ∗ Dual irrep. self σ(p2,t) = σ(p ,p−t) σ(s,p) = σ(p−s,p) σ(s,t) = σ(p−s,p−t) σ(s,t) = σ(p −s,p−t) σ(s,p) = σ(p−s,p) # irrep. 1 p − 1 p − 1 (p − 1)(p − 1) (p2 − p)(p − 1) p2 − p

Table 8: All irreducible representations of group Z4 × Z2.

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 8 8 4 4 2 6 2 6 TZ4×Z2 = a ω ω ω ω ω ω ω ω . b ω8 ω4 ω8 ω4 ω4 ω4 ω8 ω8

3.5. The group Zp × Zp × Zp.

p p p Here Zp × Zp × Zp = ha,b,c | a = b = c =1,ab = ba,ac = ca,bc = cbi. The corresponding tables are given by

9 TZp×Zp×Zp =

Table 9: All irreducible representations of Zp × Zp × Zp 3-tuples → All place are same Exactly two place are same All place are distinct (s,t,m) (s,s,s) (s,s,m), (s,m,s), (m,s,s),m 6= s (s,t,m), s 6= t 6= m 6= s

All irrep. → σ(s,s,s) σ(p−s,p−s,p−s) σ(s,s,m) σ(p−s,p−s,p−m) σ(s,t,m) σ(p−s,p−t,p−m) p+1 p−1 Runing 2 ≤ s ≤ p s = p &1 ≤ m ≤ 2 or 1 ≤ s,t,m ≤ p, p−1 variable → 1 ≤ s ≤ 2 & m ∈{1, 2, ··· p} 2 2 2 2 2 2 a ωsp ω(p−s)p ωsp ω(p−s)p ωsp ω(p−s)p 2 2 2 2 2 2 b ωsp ω(p−s)p ωsp ω(p−s)p ωtp ω(p−t)p 2 2 2 2 2 2 , c ωsp ω(p−s)p ωmp ω(p−m)p ωmp ω(p−m)p

Twice σ(p,p,p) = σ(0,0,0) σ(s,t,m) = σ(p−s1,p−t1,p−m1) if

repeated irrep. Only trivial irrep. s + s1 = t + t1 = m + m1 = p or → repeated twice No irrep. repeated twice 2p, Each irrep. repeated twice ∗ ∗ ∗ Dual irrep. σ(s,s,s) = σ(p−s,p−s,p−s) σ(s,s,m) = σ(p−s,p−s,p−m) σ(s,t,m) = σ(p−s,p−t,p−m) Distinct # irrep. p 3p(p − 1) p(p − 1)(p − 2)

Table 10: All irreducible representations of Z2 × Z2 × Z2

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 a ω8 ω4 ω8 ω8 ω8 ω4 ω4 ω4 TZ2×Z2×Z2 = . b ω8 ω4 ω4 ω4 ω8 ω4 ω8 ω8 c ω8 ω4 ω8 ω4 ω4 ω8 ω8 ω4

Note 3.4. An irreducible representation ρ2i is seated together (preceded or succeeded) to its dual ρ2i+1 in ρ.

Now

ρ = k1ρ1 ⊕ k2ρ2 ⊕ ...... ⊕ krρr, (1)

where for every 1 ≤ i ≤ r, kiρi stands for the direct sum of ki copies of the irreducible representation ρi. Let χ be the corresponding character of the representation ρ, then

χ = k1χ1 + k2χ2 + ...... + krχr,

where χi is the character of ρi, ∀ 1 ≤ i ≤ r. Dimension of the character χ is being calculated at the identity element of the group. i.e, dim(ρ)= χ(1) = tr(ρ(1)).

=⇒ d1k1 + d2k2 + ...... + drkr = n. (2)

Note 3.5. Equation (2) holds in more general situation, which helps us to ﬁnd all possible distinct r-tuples

(k1, k2, ...... , kr), which correspond to the distinct n degree representations (up to isomorphism) of a given ﬁnite group.

10 Theorem 3.1. Let G be a group of order p3 with p a prime. If σ is an irreducible representation of G of degree p, then σ is a faithful representation.

Proof. For non-abelian groups it is clear from the tables 1 and 3 in the subsections 3.1 and 3.2 in this paper if p is an odd prime, whereas for p = 2 it follows from the tables TD4 and TQ8 in the article [11]. For an abelian group there is no irreducible representation of degree p.

4. Existence of non-degenerate invariant forms.

An element in the space of invariant bilinear forms under representation of a ﬁnite group is either non- degenerate or degenerate. All the elements of the space are degenerate when k2i 6= k2i+1, such a space is called a degenerate invariant space which has also been discussed in [11] for the groups of order 8. How many such representations exist out of total representations, is a matter of investigation. Some of the spaces contain both non-degenerate and degenerate invariant bilinear forms under a particular representation. In this section we compute the number of such representations of the group G of order p3, with p an odd prime.

′ Remark 4.1. The space ΞG of invariant bilinear forms under an n degree representation ρ contains only th those X ∈ Mn(F) whose (i, j) block is a 0 sub-matrix of order diki × dj kj when (i, j) ∈/ AG = {(i, j) | ij ρi and ρj are dual to each other } whereas for (i, j) ∈ AG with di = dj =1, the block matrix Xdiki×dj kj is given by xij xij ··· xij 1,1 1,2 1,kj  ij ij ij  x2,1 x2,2 ··· x2,k Xij = Xij = j . diki×dj kj ki×kj  . . .. .   . . . .     ij ij ij  x x ··· x   ki,1 ki,2 ki ,kj    And for (i, j) ∈ AGp with di = dj = p it is,

xij L xij L ··· xij L 1,p 1,2p 1,pkj  ij ij ij  xp+1,pL xp+1,2pL ··· xp+1,pk L Xij = Xij = j , diki×dj kj pki×pkj  . . .. .   . . . .     ij ij ij  x L x L ··· x L  (ki−1)p+1,p (ki−1)p+1,2p (ki−1)p+1,pkj    0 ··· 0 1 0 ··· 1 0 where L = . . . . . . . . . .     1 ··· 0 0  p×p   ′ Lemma 4.1. If X ∈ ΞG, and k2i 6= k2i+1, then X must be singular.

′ , we have X = [Xij ] . Proof. With reference to the above remark and Note 3.4, for every X ∈ ΞG diki×dj kj (i,j)∈AG I.e.

11 11 0 0 Xd1k1×d1k1 ···   0 X23 0 d2k2×d3k3 0  ···   X32 0    d3k3×d2k2  X =   ,  .  .  .. .   . . . .     0 r−1 r   X −1 −1   0 0 ··· dr kr ×drkr   r r−1   X 0   drkr ×dr−1kr−1  ij  ij   with X = Ct X C , for (i, j) ∈ A . If k 6= k and since d = d , so the diki×dj kj kiρi(g) diki×dj kj kiρi(g) G 2i 2i+1 2i 2i+1 number of rows and columns of Xij is diﬀer hence either rows (or columns) is linearly dependent. Thus diki×dj kj the result follows.

In the next lemma we characterise the representations of G each of which admits a non-degenerate invariant th bilinear form. To prove the next lemma we will choose only those X ∈ Mn(F) whose (i, j) block is zero for (i, j) ∈/ A , whereas for (i, j) ∈ A with k = k and the block matrices Xij , is non-singular. G G i j diki×dj kj

Lemma 4.2. For n ∈ Z+, an n-degree representation of G has a non-degenerate invariant bilinear form iﬀ r−1 k2i = k2i+1, 1 ≤ i ≤ 2 .

Proof. From equation (2) we have d1k1 + d2k2 + ...... + drkr = n and chose X ∈ Mn(F) such that

11 0 0 Xd1k1×d1k1 ···   0 X23 0 d2k2×d3k3 0  ···   X32 0    d3k3×d2k2  X =   .  .  .  .. .   . . . .     0 r−1 r   X −1 −1   0 0 ··· dr kr ×drkr   r r−1   X 0   drkr ×dr−1kr−1     Suppose k = k , and for (i, j) ∈ A , the chosen block matrices Xij is non-singular. Thus the rows (or 2i 2i+1 G diki×dj kj columns) of the X is linearly independent. Also from remark 4.1 we have Xij = Ct Xij C diki×dj kj kiρi(g) diki×dj kj kiρi(g) t ′ so Cρ(g)XCρ(g) = X. Therefore X ∈ ΞG. ′ ij On other hand X ∈ ΞG non-singular implies that Xdiki×dj kj is non-singular for (i, j) ∈ AG, therefore the r−1 corresponding block matrix is square which is possible only when k2i = k2i+1, ∀ i ∈{1, 2, ..., 2 }.

Note that the Lemmas 4.1 and 4.2 can be covered in a more general situation by stating as ’no non-trivial irreducible representation of a finite p-group can be self dual’. For if L a finite p-group and V a non-trivial irreducible representation of L, replacing L by its image in the matrix group (the general linear group), we may assume that the representation is faithful. Being a non-trivial finite p-group, L has non-trivial centre. Let g be a non-trivial central element in L. The action of g on V is by multiplication by a root of unity ζ 6= ±1. Its action on the dual of V is by multiplication by ζ¯(= ζ−1). Since ζ 6= ζ¯, it follows that V is not self-dual.

Corollary 4.1. For n ∈ Z+, an n degree representation of a group of order p3 has a non-degenerate invariant bilinear form if and only if every irreducible representation and its dual have same multiplicity in the representation.

12 Proof. Follows from the proof of the lemma 4.2 .

Remark 4.2. If F is algebraically closed, it has inﬁnitely many non zero elements, hence if there is one non- degenerate invariant bilinear form in the space ΞG, it has inﬁnitely many.

Lemma 4.3. Let G be a group of order p3 and n ∈ N. Then the number of n degree representations of G n n−2pℓ 2 ⌊ 2p ⌋ p−3 ⌊ 2 ⌋ p −3 ℓ + 2 s + 2 each of which admits a non-degenerate invariant bilinear form is p−3 p2−3 when G ℓ=0 " 2 s=0 2 # n 3 X X ⌊ 2 ⌋ p −3 ℓ + 2 is non-abelian whereas it is p3−3 when G is abelian. 2 Xℓ=0 r Proof. Let ρ = ⊕i=1kiρi be an n degree representation of G which admits a non-degenerate bilinear form. So, we r−1 have k2i = k2i+1, 1 ≤ i ≤ 2 . In the non-abelian case, G is either Gp or Heis(Zp) and for each of these two, we 2 2 2 2 have r = p + p − 1, di =1 for 1 ≤ i ≤ p and di = p for p +1 ≤ i ≤ p + p − 1. Now from equation (2), we have

k1 + 2(k2 + k4 + ··· + kp2−1)+2p(kp2+1 + kp2+3 + ··· + kp2+p−2)= n

p2−1 p−1 2 2 | {z } | {z } =⇒ k1 + 2(k2 + k4 + ··· + kp2−1)= n − 2p(kp2+1 + kp2+3 + ··· + kp2+p−2)

p2−1 p−1 2 2 | {z } | {z }

=⇒ k1 + 2(k2 + k4 + ··· + kp2−1)= n − 2pℓ. (3)

p2−1 2

2 | 2 {z 2} n n To solve the above equation we have (kp +1 + kp +3 + ··· + kp +p−2)= ℓ,0 ≤ ℓ ≤⌊ 2p ⌋. i.e, we have ⌊ 2p ⌋ +1

p−1 2 th equations. The ℓ equation is | {z }

kp2+1 + kp2+3 + ··· + kp2+p−2 = ℓ. (4)

p−1 2

p−1 | {z ℓ+ 2 −}1 n The number of distinct solution to above equations 4 is p−1 , 0 ≤ ℓ ≤⌊ ⌋. 2 −1 2p Further from equation 3 we have

k1 = n − 2pℓ − 2(k2 + k4 + ··· + kp2−1)

p2−1 2 | {z } =⇒ k1 = n − 2pℓ − 2λ,

n−2pℓ n−2pℓ 2 where k2 + k4 + ··· + kp −1 = λ,0 ≤ λ ≤⌊ 2 ⌋. i.e, we have ⌊ 2 ⌋+1 equations and the number of solutions

p2−1 2 2 p −1 | {z } λ+ 2 −1 for every λ to the equation is p2−1 . 2 −1

13 2 Thus the number of all distinct p + p − 1 tuples (k1, k2, k3...... , kp2+p−2, kp2+p−1) with k2i = k2i+1 is

n n−2pℓ 2 ⌊ 2p ⌋ p−1 ⌊ 2 ⌋ p −1 ℓ + 2 − 1 λ + 2 − 1 p−1 p2−1 . " 2 − 1 2 − 1 # Xℓ=0 λX=0

3 Now in the abelian case G is either of Zp × Zp × Zp, Zp2 × Zp and Zp3 for each of which r = p and di = 1 for 1 ≤ i ≤ p3. Now from (2), we have

k1 + 2(k2 + k4 + ...... + kp3−1)= n.

p3−1 2

3 | {z } n p −1 ⌊ 2 ⌋ ℓ+ 2 −1 3 3 Thus the number of all distinct p -tuples (k1, k2, k3, ········· , kp ) with k2i = k2i+1 is ℓ=0 p3−1 . 2 −1 P Thus from equation (2) and Theorem 2.2 the number of n degree representations (upto isomorphism) of a group 2 n p−1 n−2pℓ p −1 ⌊ 2p ⌋ ℓ+ 2 −1 ⌊ 2 ⌋ λ+ 2 −1 G consisting non-degenerate invariant bilinear form is ℓ=0 p−1 λ=0 p2−1 for non-abelian " 2 −1 2 −1 # 3 n p −1 P P ⌊ 2 ⌋ l+ 2 −1 3 groups and ℓ=0 p3−1 for abelian groups of order p , with p an odd prime. 2 −1 P

Deﬁnition 4.1. The space ΞG of invariant bilinear forms is called degenerate if it’s all elements are degenerate.

We will discuss about the degenerate invariant space in the later section.

5. Dimensions of spaces of invariant bilinear forms under the representations of groups of order p3 with prime p > 2.

The space of invariant bilinear forms under an n degree representation is generated by ﬁnitely many vectors, so its dimension is ﬁnite along with its symmetric and the skew-symmetric subspace. In this section we formulate the dimension of the space of invariant bilinear forms under a representation of a group of order p3, with p an odd prime.

r Theorem 5.1. If ΞG is the space of invariant bilinear forms under an n degree representation ρ = ⊕i=1kiρi of 3 a group G of order p , then dim(ΞG)= (i,j)∈AG kikj .

′ P Proof. For every X ∈ ΞG, we have

11 0 0 Xd1k1×d1k1 ···   0 X23 0 d2k2×d3k3 0  ···   X32 0    d3k3×d2k2  X =   ,  .  .  .. .   . . . .     0 r−1 r   X −1 −1   0 0 ··· dr kr ×drkr   r r−1   X 0   drkr ×dr−1kr−1     14 with Xij = Ct Xij C , for (i, j) ∈ A and to generate each of these blocks of X it needs diki×dj kj kiρi(g) diki×dj kj kj ρj (g) G ′ kikj vectors from ΞG. Thus the result follows.

r Corollary 5.1. The space of invariant symmetric bilinear forms under an n degree representation ρ = ⊕i=1kiρi 3 k1(k1+1) kikj of a group G of order p has dimension = 2 + (i,j)∈AG 2 . i6=j P Proof. Follows from the proof of theorem 5.1 .

Corollary 5.2. The space of invariant skew-symmetric bilinear forms under an n degree representation ρ =

r 3 k1(k1−1) kikj ⊕i=1kiρi of a group G of order p has dimension = 2 + (i,j)∈AG 2 . i6=j P Proof. Follows from the proof of theorem 5.1 .

6. Main results

Here we present the proofs of the main theorems stated in the Introduction section.

Proof of Theorem 1.1 Since G is the group of order p3, with an odd prime p and degree of the representation 2 2 ρ is n, if G is either Gp or Heis(Zp), we have r = p + p − 1, di =1 for 1 ≤ i ≤ p and di = p for p2 +1 ≤ i ≤ p2 + p − 1. Now from equation (2), we have

k1 + k2 + ...... + kp2 + pkp2+1 + ...... + pkp2+p−1 = n

Or, k1 + k2 + ...... + kp2 = n − p(kp2+1 + ...... + kp2+p−1).

Or, k1 + k2 + ...... + kp2 = n − pµ, (5)

2 2 n n where µ = kp +1 + ...... + kp +p−1, 0 ≤ µ ≤⌊ p ⌋. i.e, we have ⌊ p ⌋ + 1 equations placed in the chronological order and the µth equation is given by

kp2+1 + kp2+2 + ...... + kp2+p−1 = µ. (6)

µ+p−2 n The number of distinct solutions to equation 6 is p−2 , 0 ≤ µ ≤⌊ p ⌋. n 2 ⌊ p ⌋ µ+p−2 n−µp+p −1 2 2 Thus the number of all distinct p + p − 1 tuples (k1, k2, ...... , kp +p−1) is µ=0 p−2 p2−1 . 3 3 On the otherhand if G is either of Zp × Zp × Zp, Zp2 × Zp and Zp3 then r P= p and di =1 for 1 ≤ i ≤ p . Now from (2), we have

k1 + k2 + ...... + kp + ...... + kp2 + ...... + kp3 = n.

3 n+p −1 3 3 Thus the number of all distinct p -tuples (k1, k2, k3, ········· , kp ) is p3−1 . Thus from equation (2) and Theorem 2.2 the number of n degree representations (upto isomorphism) of a n 2 3 3 ⌊ p ⌋ µ+p−2 n−µp+p −1 n+p −1 group G of order p is µ=0 p−2 p2−1 , when G is non-abelian, whereas it is p3−1 , when G is abelian. P

15 6.1. Degenerate invariant spaces

From Lemma 4.1, if k2i 6= k2i+1 then all the elements of the space are degenerate Thus by the Theorem 1.1 and Lemma 4.2, the number of n degree representations whose corresponding invariant spaces of bilinear forms contain 2 n 2 n p−3 n−2pℓ p −3 ⌊ p ⌋ µ+p−2 n−µp+p −1 ⌊ 2p ⌋ ℓ+ 2 ⌊ 2 ⌋ λ+ 2 only degenerate invariant bilinear forms are µ=0 p−2 p2−1 − ℓ=0 p−3 λ=0 p2−3 in " 2 2 # 3 3 Pn p −3 P P n+p −1 ⌊ 2 ⌋ ℓ+ 2 3 the non-abelian case and it is p3−1 − ℓ=0 p −3 in the abelian case. 2 P Proof of Theorem 1.2 Let A = {(i, j) | ρ and ρ are dual to each other} and for every (i, j) ∈ A , = { X ∈ ( ) | Xij = G i j G W(i,j) Mn F diki×dj kj Ct Xij C , ∀g ∈ G and rest blocks are zeros }. Then for (i, j) ∈ A , is a subspace of ( ). kiρi(g) diki×dj kj kj ρj (g) G W(i,j) Mn F ′ Let X be an element of ΞG, then

t ij Cρ(g)XCρ(g) = X and X = [Xdiki×dj kj ](i,j)∈AG .

Existence: ′ X = X. Let X ∈ ΞG then for every (i, j) ∈ AG, there exists at least one X(i,j) ∈ W(i,j), such that (i,j)∈AG (i,j) P Uniqueness: Y = X = X , For every (i, j) ∈ AG, suppose there are Y(i,j) and X(i,j) ∈ W(i,j), such that (i,j)∈AG (i,j) (i,j)∈AG (i,j)

′ ′ ′ ′ ′ ′ then (i,j)∈A X(i,j) = (i,j)∈A Y(i,j) i.e., Y(i ,j ) − X(i ,j )= (i,j)∈AGP(X(i,j) − Y(i,j)). ThereforeP Y(i ,j ) − G G ′ ′ (i,j)6=(i ,j ) ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ X(i ,jP) ∈ (i,j)∈AG W(Pi,j), hence Y(i ,j ) − X(i ,j ) = O =⇒ Y(Pi ,j ) = X(i ,j ) for all (i , j ) ∈ AG. ′ ′ (i,j)6=(i ,j ) Thus we haveP ′ ′ ΞG = ⊕(i,j)∈AG W(i,j) and dim(ΞG)= dimW(i,j). (7) (i,jX)∈AG th ij Now as for (i, j) ∈ AG, W(i,j) = {X ∈ Mn(F) | (i, j) block ’X ’ is a sub - matrix of order diki × dj kj satisfying Xij = Ct Xij C , ∀g ∈ G and rest blocks are zeros }. Now by the remark 4.1, we see that for kiρi(g) kj ρj (g) ij ∼ ′ ∼ (i, j) ∈ AG, the sub-matrices X in W(i,j) have kikj free variables & W(i,j) = Mki×kj (F). Thus ΞG =

⊕(i,j)∈AG Mki×kj (F) and dim(W(i,j))= kikj .

′ Thus substituting these in equation (7) we get the dimension of ΞG.

Proof of Theorem 1.3 Follows immediately from Lemmas 4.1and4.2 .

7. Representation over a ﬁeld of characteristic p.

Remark 7.1. If characteristic of the ﬁeld F is p then a group G of order p3 has only trivial irreducible representation. Therefore for every g ∈ G, we have

ρ(g)= nρ1(g),

16 where ρ1 is the trivial representation of group G. So the representation ρ is a trivial representation of degree n. i.e,

ρ(g)= In, for all g ∈ G.

t Note 7.1. As X = InXIn, ∀X ∈ Mn(F), therefore every X ∈ Mn(F) gives an invariant bilinear form in the case when characteristic of the ﬁeld is p. This is summarised in the following result.

Proposition 7.1. The space of invariant bilinear forms under an n degree representation of a group G of order 3 p with char(F) = p an odd prime is isomorphic to Mn(F) and contains a non-degenerate invariant bilinear form.

Thus here we have completely characterised the representations of a group of order p3 each of which admits a non-degenerate invariant bilinear form over a ﬁeld of characteristic diﬀerent from p consisting of a primitive p3th root of unity. Authors hope to evaluate these results for a group of higher order in future.

Acknowledgement The first author would like to thank UGC, India for providing the research fellowship and to the Central University of Jharkhand, India for facilitating this research work. The second author would like to express his gratitude towards Babasaheb Bhimrao Ambedkar University, Lucknow, India where he got affiliated while finalizing this paper.

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