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

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

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

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

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

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.  The projection ontois P 

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

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.

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 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

Proof (  ) : If the IRs D 1 & D 2 are equivalent, there exists S such that or, equivalently, there exists mapping  Let   i.e. QED

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

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

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