Invariant bilinear forms under the operator group of order p^3 with odd prime p

In this paper we compute the number of n degree representations of a group of order p^3 for p an odd prime and the dimensions of corresponding spaces of invariant bilinear forms over an algebraically closed field F. We explicitly discuss about the existence of non-degenerate invariant bilinear forms. The results are important due to their application in the studies of physical sciences.


Introduction
Representation theory enables the study of a group as operators on certain vector spaces and as an orthogonal group with respect to a corresponding bilinear form.Also since last several decades the search of non-degenerate invariant bilinear forms has remained of great interest.Such type of studies acquire an important place in quantum mechanics and other branches of physical sciences.
Let G be a group and V a vector space over a field F, then we have following.
Definition 1.1.A homomorphism ρ : G → GL(V) is called a representation of the group G. V is also called a representing space of G.The dimension of V over F is called degree of the representation ρ.Definition 1.2.A class function is a map f : G → F so that f (g) = f (h) if g is a conjugate of h in G. Definition 1.3.A bilinear form on a finite dimensional vector space V(F) is said to be invariant under the representation ρ of a finite group G if B(ρ(g)x, ρ(g)y) = B(x, y), ∀ g ∈ G and x,y ∈ V(F).
Let Ξ denotes the space of bilinear forms on the vector space V(F) over F and C F (G) the set of all class functions on G.
Assume that F consists of a primitive |G| th root of unity.The representation (ρ, V) [11] is irreducible of degree if and only if {0} and V are the only invariant subspaces of V under the representation ρ.The class function C F (G) is a vector space over F with dimension r, where r is the number of conjugacy classes of G.By the Frobenius theorem (see pp 319, Theorem (5.9) [1]) there are r irreducible representations ρ i (say), 1 ≤ i ≤ r of G and χ i (say) the corresponding characters of ρ i .Also by Maschke's theorem ( see pp 316, corollary (4.9) [1]) every n degree representation of G can be written as a direct sum of copies of irreducuble representations.For ρ = ⊕ r i=1 k i ρ i an n degree representation of G, the coefficient of ρ i is k i , 1 ≤ i ≤ r, so that r i=1 d i k i = n, and r i=1 d 2 i = |G|, where d i is the degree of ρ i and d i ||G| with d j ≥ d i when j > i.It is already well understood in the literature that the invariant space Ξ G under ρ can be expressed by the set Ξ ′ G = {X ∈ M n (F) | C t ρ(g) XC ρ(g) = X, ∀g ∈ G} with respect to an ordered basis e of V(F), where M n (F) is the set of square matrices of order n with entries from F and C ρ(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 p 3 with p an odd prime, F a field with char(F) = p, which consists of a primitive p 3 th root of unity and (ρ, V) an n degree representation of G over F. Then the corresponding set Ξ G 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 Ξ G 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 = 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 = u ∈ U and 0 = v ∈ V .Using these we define a bilinear form B on U ⊕ V as follows:

Now we may easily check the C invariance as follows
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 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 p 3 with p ≥ 3 a prime, over a field F which consists of a primitive p 3 th 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 p 3 , with p an odd prime is n+p 3 −1 when G is abelian and Theorem 1.2.The space Ξ G of invariant bilinear forms of a group G of order p 3 (p an odd prime), under an n degree representation (ρ, V(F)) is isomorphic to the direct sum of the subspaces , where A G = {(i, j) | ρ i and ρ j are dual to each other} and for every (i, j) , ∀g ∈ G and rest blocks are zeros }.Also for (i, j) ∈ A G , the dimension of W (i,j) = k i k j .
. Theorem 1.3.If G is group of order p 3 , 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 p 3 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.

Preliminaries
The classification of groups of order p 3 , 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, Z p3 , Z p 2 × Z p and Z p × Z p × Z p .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 Z p .If at all there exists any other non-abelian group of this order then it must have a non identity element of order p 2 .Let us consider the 2 × 2 upper triangular matrices with a 11 = 1 + pm, (m ∈ Z p ), a 12 = a ∈ Z p 2 and a 22 = 1.Here the element with entries a 11 = a 12 = a 22 = 1 has order p 2 making it non-isomorphic to the Heisenberg group.We denote this group by G p .Thus upto isomorphism there are five groups of order p 3 with an odd prime p [3].For an abelian group of order p 3 , there are p 3 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  Proof.See p -319, corollary ( 5.13) [1].

Irreducible representation (irrep.
) of group of order p 3 with an odd prime p.
In this section G is a group of order p 3 with p an odd prime, (ρ i , W di (F)) stands for an irreducible representation ∈ Ker(σ s ), so we have χ s (g) = 0. Also we have trivial character χ i (g) = 1, ∀g ∈ G and (χ i , χ s ) = p 2 p 3 = 0, which fails the orthonormality property of the irreducible characters.Thus |Ker(σ s )| = 1 and hence σ s (G) {eG} which is isomorphic to a non-abelian group of order p 3 .Now σ s (G) has subgroups H and K of orders p and p 2 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 H p of order p which is not normal (not contained in Z(G))[see p-188, [4], Theorem 1], thus we have, σ s (G) = KH p and every element of σ s (G) can be expressed uniquely in the form of kh for some k ∈ K and h ∈ H p (this uniqueness follows from the condition GL(p, F), which consists of those elements whose trace is zero and order of every element is either p or p 2 .We will depict all irreducible representations of G in the next subsections.

Heisenberg group.
For the Heisenberg group, center Z(G) = xax −1 a −1 = ((0, 1), 0) .Each of the irreducible representations a scalar matrix say c s I p , c s ∈ F and since order of z is p so }. 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(σ s (G)), it has been recorded in the following table.
Thus the p degree irreducible representations σ s can be expressed as below Non-abelian groups of order p 3 have p 2 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.

The non-abelian group
We have recorded the p ordered irreducible representations of G p in the following table.
Thus the p degree irreducible representations σ s is defined as below In the table 4 we have recorded all 1 degree representations of G p .
Here Z p 3 = a | a p 3 = 1 .The representation tables are given as below.

The group
Here T Z p 2 ×Zp = ,

Now
where for every 1 ≤ i ≤ r, k i ρ i stands for the direct sum of k i copies of the irreducible representation ρ i .
Let χ be the corresponding character of the representation ρ, then where χ i is the character of ρ i , ∀ 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)).
Note 3.5.Equation ( 2) holds in more general situation, which helps us to find all possible distinct r-tuples (k 1 , k 2 , ......, k r ), which correspond to the distinct n degree representations (up to isomorphism) of a given finite group.
Theorem 3.1.Let G be a group of order p 3 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 T D4 and T Q8 in the article [11].For an abelian group there is no irreducible representation of degree p.
An element in the space of invariant bilinear forms under representation of a finite group is either nondegenerate or degenerate.All the elements of the space are degenerate when k 2i = k 2i+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 p 3 , with p an odd prime.
Remark 4.1.The space Ξ ′ G of invariant bilinear forms under an n degree representation ρ contains only those X ∈ M n (F) whose (i, j) th block is a 0 sub-matrix of order ρ i and ρ j are dual to each other } whereas for (i, j) ∈ A G with d i = d j = 1, the block matrix X ij diki×dj kj is given by and k 2i = k 2i+1 , then X must be singular.
Proof.With reference to the above remark and Note 3.4, for every , so the number of rows and columns of X ij diki×djkj is differ hence either rows (or columns) is linearly dependent.Thus the result follows.
In the next lemma we characterise the representations of G each of which admits a non-degenerate invariant bilinear form.To prove the next lemma we will choose only those X ∈ M n (F) whose (i, j) th block is zero for (i, j) / ∈ A G , whereas for (i, j) ∈ A G with k i = k j and the block matrices X ij diki×dj kj , is non-singular.
Lemma 4.2.For n ∈ Z + , an n-degree representation of G has a non-degenerate invariant bilinear form iff Proof.From equation (2) we have .
Suppose k 2i = k 2i+1 , and for (i, j) ∈ A G , the chosen block matrices X ij diki×dj kj is non-singular.Thus the rows (or columns) of the X is linearly independent.Also from remark 4.1 we have G implies that X ij diki×dj kj is non-singular for (i, j) ∈ A G , therefore the corresponding block matrix is square which is possible only when Note that the Lemmas 4.
Now in the abelian case G is either of Z p × Z p × Z p , Z p 2 × Z p and Z p 3 for each of which r = p 3 and d i = 1 for 1 ≤ i ≤ p 3 .Now from (2), we have Thus the number of all distinct p 3 -tuples (k Thus from equation ( 2) and Theorem 2.2 the number of n degree representations (upto isomorphism) of a group G consisting non-degenerate invariant bilinear form is for non-abelian groups and for abelian groups of order p 3 , with p an odd prime.
Definition 4.1.The space Ξ G 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 p 3 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 p 3 , with p an odd prime.
Theorem 5.1.If Ξ G is the space of invariant bilinear forms under an n degree representation ρ , for (i, j) ∈ A G and to generate each of these blocks of X it needs k i k j vectors from Ξ ′ G .Thus the result follows.
Corollary 5.1.The space of invariant symmetric bilinear forms under an n degree representation ρ = ⊕ r i=1 k i ρ i of a group G of order p 3 has dimension = k1(k1+1) 2 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 i=1 k i ρ i of a group G of order p 3 has dimension = k1(k1−1) 2 Proof.Follows from the proof of theorem 5.1 .

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 p 3 , with an odd prime p and degree of the representation where µ = k p 2 +1 + ........ + k p 2 +p−1 , 0 ≤ µ ≤ ⌊ n p ⌋. i.e, we have ⌊ n p ⌋ + 1 equations placed in the chronological order and the µ th equation is given by The number of distinct solutions to equation 6 is On the otherhand if G is either of Z p × Z p × Z p , Z p 2 × Z p and Z p 3 then r = p 3 and d i =1 for 1 ≤ i ≤ p Thus the number of all distinct p 3 -tuples (k Thus from equation (2) and Theorem 2.2 the number of n degree representations (upto isomorphism) of a group G of order p 3 is , when G is non-abelian, whereas it is n+p 3 −1 p 3 −1 , when G is abelian. in the abelian case.
Proof of Theorem 1.2 Let A G = {(i, j) | ρ i and ρ j are dual to each other} and for every (i, j) ∈ A G , W (i,j) = { X ∈ M n (F) | X ij diki×djkj = C t kiρi(g) X ij diki×dj kj C kj ρj (g) , ∀g ∈ G and rest blocks are zeros }.Then for (i, j) ∈ A G , W (i,j) is a subspace of M n (F).Let X be an element of Ξ ′ G , then C t ρ(g) XC ρ(g) = X and X = [X ij diki×djkj ] (i,j)∈AG . Existence: Let X ∈ Ξ ′ G then for every (i, j) ∈ A G , there exists at least one X (i,j) ∈ W (i,j) , such that (i,j)∈AG X (i,j) = X.
Now as for (i, j) ∈ A G , W (i,j) = {X ∈ M n (F) | (i, j) th block 'X ij ' is a sub -matrix of order d i k i × d j k j satisfying X ij = C t kiρi(g) X ij C kj ρj (g) , ∀g ∈ G and rest blocks are zeros }.Now by the remark 4.1, we see that for (i, j) ∈ A G , the sub-matrices X ij in W (i,j) have k i k j free variables & W (i,j) ∼ = M ki×kj (F).Thus Ξ ′ G ∼ = ⊕ (i,j)∈AG M ki×kj (F) and dim(W (i,j) ) = k i k j .
Thus substituting these in equation (7) we get the dimension of Ξ ′ G .
Proof of Theorem 1.3 Follows immediately from Lemmas 4.1 and 4.2 .
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 p 3 has only trivial irreducible representation.Therefore for every g ∈ G, we have ρ(g) = nρ 1 (g), where C g and C(g) are the conjugacy class and the centralizer respectively of g in G.If g / ∈ Z(G) then |C(g)| = p 2 .Therefore there are p 2 − 1 non trivial conjugacy classes of order p.Thus total number of conjugacy classes for a non-abelian group is r = p 2 − 1 + p, which is same as the number of irreducible representations with degree d i since d i ||G| and r i=1 d 2 i = |G|, therefore d i = 1 or p.Thus there are p 2 representations of degree 1 and p − 1 representations of degree p for a non-abelian group.We here formulate every irreducible representation ρ i in such a way that the entries of C ρi(g) are either 0 or p 3 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.
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 p 2 of G) denoted byIm(Z(G)) = {ω sp 2 I p | 1 ≤ s ≤ p }under the irreducible representation σ s (since center elements commute and are scalar matrices).Thus Im(Z(G))( σ s (K)) ∼ = Z p , each non-identity element of Z(G) have p − 1 choices in Z p under σ s and rest p 3 − p elements of G map to rest p 3 − p elements of σ s (G).We decide all p degree representations by the elements of Z(G) and elements of G − Z(G) by mapping to the set σ s (G) − ImZ(G) ⊆

. 3 . 5 .
The group Z p × Z p × Z p .Here Z p × Z p × Z p = a, b, c | a p = b p = c p = 1, ab = ba, ac = ca, bc = cb .The corresponding tables are given by T Zp×Zp×Zp =
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 = ±1.Its action on the dual of V is by multiplication by ζ(= ζ −1

6. 1 . 1 p 3
Degenerate invariant spacesFrom Lemma 4.1, if k 2i = k 2i+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 only degenerate invariant bilinear forms are abelian case and it is n+p 3 −

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

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

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

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

Table 5 :
All irreducible representations of the group Z p 3 .

Table 6 :
All irreducible representations of the group Z 8 .

Table 9 :
All irreducible representations of Zp × Zp × Zp ).Since ζ = ζ, it follows that V is not self-dual.Corollary 4.1.For n ∈ Z + , an n degree representation of a group of order p 3 has a non-degenerate invariant bilinear form if and only if every irreducible representation and its dual have same multiplicity in the representation.Thus the number of all distinct p 2 + p − 1 tuples (k 1 , k 2 , k 3 ......., k p 2 +p−2 , k p 2 +p−1 ) with