Symmetric functions and the symmetric group Sn
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
A simple method for the embedding On → Sn of ordinary and spin irreps in both n-dependent notation and an n-independent reduced notation is given. Basic spin irreps and ordinary irreps are combined using the properties of Q-functions and raising operators in order to give a complete set of branching rules of On → Sn for spin irreps. The modification rules for Q-functions given by Morris are redefined to yield a complete and unambiguous set of rules. Properties of shifted tableaux have been explored in order to improve the algorithm for the calculation of Q-function outer products. A simple technique has been established for finding out the highest and lowest partitions in the expansion of Q-function outer products. Using these techniques and Young's raising operators, the Kronecker product for Sn spin irreps has been completed. A number of properties of Young's raising operator as applied to S-functions and Schur's Q-functions are noted. The order of evaluating the action of inverse raising operators is found to require careful specification and the maximum power of the operators δij is determined. The operation of inverse raising operator on a partition λ is found to be the same as for its conjugate λ. A new definition of Shifted Lattice Property that can efficiently remove all the dead tableaux in the Q-function analogue of the Littlewood-Richardson rule is introduced. A simple combinatorial analogue of raising and inverse raising operators is given. The q-deformation of symmetric functions is introduced leading to q-analogues of many well-known relationships in the theory of symmetric functions. A q-analogue of the spin and ordinary characters of Sn is given by making use of a method that closely parallels that of quantum groups. This formalism leads to a very simple technique for the construction of twisted and untwisted q-vertex operators. An isomorphism between the space of q-vertex operators and the ring of q-deformed Hall-Littlewood symmetric functions has been found.