App III. Group Algebra & Reduction of Regular Representations

App III. Group Algebra & Reduction of Regular Representations • Group Algebra • Left Ideals, Projection Operators • Idempotents • Complete Reduction of the Regular Representation

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 { | gj } • An inner product can be defined by (we won't be using it): so that

An element r of G also serves as an operator on it via • as follows or so that

Representation of G 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 L r  U(r) that preserves the group algebra structure, i.e., 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

III.2. Left Ideals, Projection Operators ( V of DR of G ) = G Since where D are IRs & nC = 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

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 onto is P

III.3. Idempotents e has a unique decomposition Theorem III.2: Proof: 1. P is linear: Proof left as exercise. 2. 3. 4. 

Definition III.3: { e } are idempotents if { e } are essentiallyidempotents 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.

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

Theorem III.4: Primitive idempotents e1 & e2 generate equivalent IRs iff for some r G Proof () : Let L1 & L2 be minimal left ideals generated by e1 & e2, resp. Assume for some r G Let by   S p = p S  p  G Schur's lemma  L1 = L2 so that IRs on them are equivalent

Proof () : If the IRs D1 & D2 are equivalent, there exists S such that or, equivalently, there exists mapping  Let   i.e. QED

Example: Reduction of DR of G = C3 = { e = a3, a, a2 = a–1 } i) Idempotent e1 for the identity representation 1 : Rearrangement theorem   Theorem III.3  e1 is primitive  1 is irreducible

ii) Let Then  This can be solved using Mathematica. 4 sets of solutions are obtained: ( Discarded ) or or

  e is indeed idempotent  e is not primitive

   e+ is indeed idempotent  e+ is primitive

Changing   –1 & e+  e– gives  e– is a primitive idempotent  e+ & e– generate inequivalent IRs. Ex: Check the Orthogonality theorems Also:

III.4. Complete Reduction of the Regular Representation Summary: 1. 2. 3. primitive  Reduction of DR  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.

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 ) Hence  La and Lb are both in the 2-sided ideal L if either of them is. Let La and Lb be both in the 2-sided ideal L . Then  they generate equivalent rep's. QED Reduction of DR : Decompose G into minimal 2-sided ideals L. Reduce each Linto minimal left ideals La

App III. Group Algebra & Reduction of Regular Representations