| Description |
An n-dimensional matrix representation of a group G on a vector space V is a homomorphism from G to GL(V). For our purposes, we consider an irreducible representation to be a representation which cannot be decomposed into the direct sum of smaller-dimensional representations. Let H be a subgroup of G. The way in which irreducible representations of G decompose into irreducible representations of H is called branching. In order to calculate such branching, one must first obtain a trace formula for each conjugacy class of H in irreducible representations of G. In this thesis, we calculate this trace formula for each of the sixteen conjugacy classes of G2(2) as a subgroup of the real compact form of G2. The smallest of the five exceptional Lie groups G2 is realized as the group of automorphisms of the real octonians or Cayley numbers. Inside the Cayley numbers, there exists a nice order or in other words a ring of integers much like the ring of Gaussian integers in C. This order, known as the Coxeter order, gives an integral structure or lattice inside the octonions. We define G2(Z) to be the subgroup of G2 preserving the lattice. We define the group G2(p) to be the automorphism group of the octonions taken modulo p. G2(Z) acts on the Coxeter order, and therefore G2(Z) acts on the Coxeter order modulo p. In the case of p = 2, this action yields an isomorphism, thereby establishing that G2(2) is a finite subgroup of G2. The special unitary group SU(3) is also abstractly a subgroup of G2. We show that each of the 16 conjugacy classes of G2(2) is conjugate to a diagonal element of SU(3) inside of G2. Using the character table and power table for G2(2) along with the properties of SU(3), we find a diagonal representative element in SU(3) for each of the 16 conjugacy classes of G2(2). We use theWeyl character formula to write the trace formulas for each of the regular conjugacy classes. Since the Weyl character formula is undefined for irregular elements, we use a formal differential operator together with l'Hospital's rule to develop a general formula for irregular elements. This allows us to calculate the traces of the irregular conjugacy classes of G2(2). This process gives a closed-form formula for the character of each of the 16 conjugacy classes of G2(2). In other words, given only the highest weight l of an irreducible representation Vl of G2, one may use our formulas to calculate the multiplicity of each of the irreducible representations of G2(2) in the direct sum decomposition of Vl . We conclude with Matlab code which implements our formulas along with examples of calculation and verification of our formulas. |
