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Show THE UNIVERSITY OF UTAH HONORS COLLEGE ATRACE FORMULA FOR G Steven Sullivan (Gordan Savin) Department of Mathematics University of Utah An n-dimensional matrix representation of a group G is a homomorphism from G to a subgroup of GL(n). An irreducible representation is one 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, w e 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 octonions or Cayley numbers. Inside the Cayley numbers, there exists a nice order or in other words a ring of integers. This order, known as the Coxeter order, gives an integral structure or lattice inside the octonions. W e define G2(Z) to be the subgroup of G2 preserving the Coxeter order. The subgroup G2(2) acts on the Coxeter order and therefore on the Coxeter order mod 2. In the m o d 2 case, the action of G2(2) yields an isomorphism, thereby establishing that G2(2) is a subgroup of G2. The special unitary group SU[3) is also 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 G2. Using the character table and power table for G2(2) along with the properties of SU{3), w e find a diagonal representative element in SU{3) for each of these classes. W e use the Weyl character formula to write the trace formulas for each of the regular conjugacy classes. Since the Weyl character formula is undefined for irregular elements, w e use a formal differential operator together with I'Hospital's rule to develop a general formula for irregular elements. This allows us to calculate the traces for the irregular conjugacy classes of G2(2), which completes the trace formula for the restriction from G2 to G2(2). 249 |
