The most prominent is the representation theory of groups discussed in this thesis. In representation of groups, elements of group are represented by invertible elements in such a way that the group operation is matrix multiplication. If say R is a representation of a group G on vector space V, then V is called the representation space for G.
That is, representation of a group G on a vector space V is a map φ:G×V →V with property that φg :V →V defined as v 7−→ φ(g, v) linear over the field F is. Furthermore, the Dimension Theorem is proved here which states that the degree of any irreducible representation of a group divides the order of that group. Then, using this theorem, the most fundamental result of group theory known as Burnside's lemma is again proved using representation theory.
In addition to these results, permutation representations, regular representations and the fourth analysis of finite groups are also discussed here in this project. The term 'set representation' is also used in a more general sense to mean any description of a set as a set of transformations of a mathematical object.
Definition and Unitarity
But we can say that the unitary operator is a bounded linear operator U : H → H on the Hilbert space H corresponding to UU∗ =U∗U=I, where I:H→H is the identity operator. A Hilbert space is a real or complex inner product space, which is also a complete metric space with respect to the distance function induced by the inner product. The unitary representation of the group G is the linear representation π of the group G on the complex Hilbert space V, such that π(g) is the unitary operator ∀ g.
A unitary representation of a finite group G is a homomorphism from G to U(V) (when we say representation, we mean unitary representation). A symmetric n×n real matrix M is said to be positive definite if zTM z is strictly positive ∀ nonzero column vector z of n real numbers. Or . an n×n Hermitian matrix M is said to be positive definite if the scalarz∗M z is strictly positive ∀ non-zero column vector z of n complex numbers.
If T=W|T| is a polar decomposition of T, then W is a partial isometry and |T| is positive semidefinite. A partial isometry is a linear mapping between Hilbert spaces such that the isometry is on the orthogonal complement of the kernel.
Irreducibility and Complete Reduction
This means showing that A⊕B is a unity operator. since X⊕Y is a subspace, inner product can be defined as such). A unit representation U of G on X is called irreducible if and only if the only invariant subspaces of U are {0} and X. U is an irrep (we will use this for 'irreducible representation') if and only if it cannot be written as a direct sum of non-trivial (that is, non-zero-dimensional) representations.
Then there exist U1, U2 corresponding to invariant subspaces Y and Y' of X, such that U1 and U2 are representations of G on Y and Y', respectively, and thus U ∼=U1⊕U2, where U1 and U2 are non-trivial representations , which is a contradiction. A representationφ:G→GL(V) is said to be completely reducible if V =V1⊕V2⊕..⊕Vn where Vi are G-invariant subspaces and φ|Vi is irreducible for all i=1,2.n. Gˆ, known as the dual object, is the set of equivalence classes of irreps, each class consisting of uniformly equivalent irreps.
The Group Algebra and the Regular Representations
Now we define a product inherited from the group product for the vector space S, such that,. The complex vector space A(G) of functions on G is called the group algebra if we get the product CONVOLUTION as follows. Conversely, if UA satisfies all the above four conditions, then ∃ is a unitary representation U of G that satisfies (1).
The Group Algebra and the Regular Representations Therefore, we can conclude that there is one correspondence between the representation of G and the *-representation of A(G). Similarly, RA(f)g = g ∗ f∗, the real multiplication, is also a representation of A induced by,. Rxf)(y) =f(yx) known as the right regular representation.
Schur’s Lemma
Tensor Products
To prove this, we only need to show that {ei⊗fj}i= 1, j = 1n,m is linearly independent since we already know that it spans the space X ⊗Y. So, let's examine. This inner product is such that if {ei} and {fj} are orthonormal bases for X and Y respectively, then ei⊗fj is an orthonormal basis for X ⊗ Y. Given that W and V are two representations of a set G in spaces X and Y respectively, we define W ⊗ V to be a representation of G in X ⊗Y with,.
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The Orthogonality Relations
Schur Orthogonality Relations
Characters and Class functions
Without loss of generality, assume that φ:G →Un(C) and ψ :G → Um(C) are unitary (since every representation is equivalent to a unitary one.) Now, if φ and ψ are equivalent, then, χφ =χψ. By previous theorem, we know that unequal irreps have different characters and form an orthonormal set. We also know that the cardinality of an orthonormal set is less than or equal to the dimension of the space.
So there exists at least |Cl(G)| equivalence classes of irreps of G (if we consider equivalence relation to be conjugation). Note that since the character depends only on the equivalence class, then the multiplicity of φi will be the same no matter what the decomposition of ψ is. Letφ1, .., φs be a complete set of representatives of equivalence classes of irreps of G and let.
Consequently, the decomposition of ρ into irreducible components is unique, and ρ is defined up to equivalence by its characters. So the decomposition of ρ into irreducible components is unique, and if the representations are equivalent, then the characters are the same. Therefore we conclude that this theorem helps us check whether a representation is irreducible or not.
Character table: The rows of the character table correspond to the irreducible characters and the columns of the character table correspond to the conjugation classes. It can be observed that the columns of the character table are pairwise rectangular as shown in the table above.
The Regular Representation
The Regular Representation of G is a homomorphism L:G→GL(CG) defined by. Using change of variable, gh=x)Lg is a linear operator acting on a linear combination of basis vectors given the action on basis. The Regular Representation is never irreducible when G is non-trivial, but contains all irreps of G as constituents. Number of equivalence classes of irreps of G is equal to the number of conjugation classes of G.
Therefore, the columns of the Character table form an orthogonal set of non-zero vectors and are therefore linearly independent.
Representation of Finite Abelian Groups
From the above definition, we can conclude that these functions are in one-to-one correspondence with the elements of A(Z/nZ), i.e. the functions f : Z/nZ → C.
Fourier Analysis o finite abelian groups
This shows that T is injective by the Fourier inversion theorem because we can epocrit ˆf and thus f from Tf. Algebraic number: A complex number 'k' is called an algebraic number if it is a root of a polynomial with integer coefficients. Algebraic integer: A complex number 'c' is called an algebraic integer if it is a root of a monic polynomial with integer coefficients.
Following the above definition, we can say that every nth root of unity is an algebraic integer. An element y ∈ C is an algebraic integer if there exists y1, y2, .., yt∈C, not all zero, so that.
The Dimension Theorem
Let g' be conjugate to g, then we show that the number of ways to write g=xy is the same as the number of ways to write g'=xy with x∈Ci and y∈Cj, that is, we shows |Xg|=| Xg0|. Now coming back to the proof of second proposition, let aijk be the value of aijg with g ∈Ck so,. So siχi/d is an algebraic integer with the equivalent condition that a complex number is an algebraic integer stated in the lemma done before.
More precisely, if G' is the commutator subgroup of G, there is a bijection between the degree one representation of G and the irreps of the abelian group G/G'. This shows that ρ =ψπ. So there is bijection between degree one representations of G and irreps of the abelian group G/G'. Let n be the number of degree p representations of G and let m be the number of degree one representations of G.
The Burnside’s Theorem
An action of a finite group G on a finite set X is a homomorphism σ :G→SX where SX is the set of all bijections from X to X. Then, the representation σ˜ :G→GL(CX) is the permutation representation of G and is a unit representation. Burnside's lemma: Let σ :G→SX be a group action and let m be the number of orbits of G on X.
Our goal is to study the set HomG(˜σ,σ), which is the set of all morphisms from ˜˜σ to self. Let λ:G→SG be the regular action of G, and therefore lambda˜ :G→GL(CG) is the regular representation of G say, L. Symmetric Gelfand Pair: Let G be a group and H be its subgroup with corresponding group action σ :G→SG/H.
Equally, G is an inner direct product of N1, N2 if every element of N1 commutes with every element of N2, i.e. N1 ⊂centralizer(N2).
