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

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Invariant Bilinear Forms Under the Operator Group of Order P3 with Odd Invariant bilinear forms under the operator group of order p3 with odd prime p Dilchand Mahtoa,, Jagmohan Tantib, aDepartment of Mathematics, Central University of Jharkhand, Ranchi, India bDepartment of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India Abstract In this paper we formulate the number of n degree representations of a group of order p3 with p an odd prime and the dimensions of corresponding spaces of invariant bilinear forms over a field F which contains a primitive p3 root of unity. We explicitly discuss about the existence of a non-degenerate invariant bilinear form. Keywords: Bilinear forms, Representation theory, Vector spaces, Direct sums, Semi direct product. 2010 MSC: 15A63, 11E04, 06B15, 15A03 1. 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 arXiv:2006.05307v2 [math.RT] 23 Jun 2021 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 CF(G) the set of all class functions on G. Definition 1.4. The set of invariant bilinear forms under the representation ρ is given by ΞG = {B ∈ Ξ | B(ρ(g)x, ρ(g)y)= B(x, y), ∀ g ∈ G and x,y ∈ V}. 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 CF(G) Email addresses: [email protected] (Dilchand Mahto ), [email protected] (Jagmohan Tanti) Preprint submitted to Journal of LATEX Templates June 24, 2021 is a vector space over F with dimension r, where r is the number of conjugacy classes of G. By the Frobenius the- orem (see pp 319, Theorem (5.9) [1]) there are r irreducible representations ρi (say), 1 ≤ i ≤ r of G and χi (say) the corresponding characters of ρi. Also by Maschke’s theorem ( see pp 316, corollary (4.9) [1]) every n degree r representation of G can be written as a direct sum of copies of irreducuble representations. For ρ = ⊕i=1kiρi an r r 2 n degree representation of G, the coefficient of ρi is ki, 1 ≤ i ≤ r, so that i=1 diki = n, and i=1 di = |G|, 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 repre- sentations 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.
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