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App III. Group Algebra & Reduction of Regular Representations 1. Group Algebra 2. Left Ideals, Projection Operators 3. Idempotents 4. Complete Reduction of the Regular Representation
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III.1. Group Algebra Definition III.1: Group Algebra The group algebra { G ;,+, C } of a finite group { G, } is the set Together with the algebraic rules: where Comments { G ;,+ } is a ring with identity { G ; +, C } is a complex linear vector space spanned by { | g j } An inner product can be defined by (we won't be using it): so that
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An element r of G also serves as an operator on it via as follows or so that
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Definition III.2: Representation of G Let L be the space of linear operators on V. A rep of G on V is a homomorphism U: G Lr U(r) that preserves the group algebra structure, i.e., Representation of G U(G) is an irreducible representation (IR) if V has no non-trivial invariant subspace wrt U(G) Theorem III.1: U is rep of G U is rep of G U is IR of G U is IR of G
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III.2. Left Ideals, Projection Operators ( V of D R of G ) = G Since where D are IRs & n C = number of classes ( G is decomposable) L is an invariant subspace: L is a left ideal. If L doesn't contain a smaller ideal, it is minimal ~ irreducible invar subspace
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Minimal left ideals can be found by means of projections (idempotents) A projection P a onto the minimal left ideal L a must satisfy 1.i.e., 2. 3. 4. The projection ontois P
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III.3. Idempotents e has a unique decomposition Theorem III.2: Proof: 1. P is linear: Proof left as exercise. 2. 3. 4.
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Definition III.3: { e } are idempotents if { e } are essentially idempotents if All results remain valid if P & e are replaced by P & e , resp. Definition III.4: A primitive idempotent generates a minimal left ideal.
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Theorem III.3: An idempotent e is primitive iff Proof ( ) : e is primitive is a minimal left ideal & realization of G on L is irreducible Define R by Schur's lemma Proof ( ) : Let If e is not primitive e' & e'' are idempotents e is primitive
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Theorem III.4: Primitive idempotents e 1 & e 2 generate equivalent IRs iff for some r G Proof ( ) : Let L 1 & L 2 be minimal left ideals generated by e 1 & e 2, resp. Assume for some r G Letby S p = p S p G Schur's lemma L 1 = L 2 so that IRs on them are equivalent
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Proof ( ) : If the IRs D 1 & D 2 are equivalent, there exists S such that or, equivalently, there exists mapping Let i.e. QED
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Example: Reduction of D R of G = C 3 = { e = a 3, a, a 2 = a –1 } i) Idempotent e 1 for the identity representation 1 : Rearrangement theorem Theorem III.3 e 1 is primitive 1 is irreducible
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ii) LetThen This can be solved using Mathematica. 4 sets of solutions are obtained: ( Discarded ) or
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e is indeed idempotent e is not primitive
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e + is indeed idempotent e + is primitive
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Changing –1 & e + e – gives e – is a primitive idempotent e + & e – generate inequivalent IRs. Ex: Check the Orthogonality theorems Also:
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III.4. Complete Reduction of the Regular Representation Summary: 1. 2. 3. primitive Reduction of D R Finding all inequivalent e a 's. L is a 2-sided ideal, i.e., A 2-sided ideal is minimal if it doesn't contain another 2-sided ideal.
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If a minimal 2-sided ideal L contains a minimal left-sided ideal L a, then it is a direct sum of all minimal left-sided ideals of the same . Proof: Let L a and L b correspond to equivalent IRs ( belong to same ). Then ( See proof of Theorem III.4 ) L a and L b are both in the 2-sided ideal L if either of them is. Hence Let L a and L b be both in the 2-sided ideal L . Then they generate equivalent rep's. QED Reduction of D R : 1. Decompose G into minimal 2-sided ideals L . 2. Reduce each L into minimal left ideals L a
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