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Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
It is useful to write down the characters for all irreps of a given group in terms of a character table, where each row represents an irrep and the columns represent the conjugacy classes of the group. The entries in the table contain the values of the character of the representation for the given conjugacy class. It is customary to write down the conjugacy classes together with the number of elements in them as the top row. We will also make sure to remember which group the character table belongs to by writing it down in the upper left entry of the table.
Group Representations
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
The character table of the alternating group A4 is (1)(12)(34)(123)(132)χ01111χ111ωω2χ211ω2ωχ33−100
Nuclear spin symmetry conservation studied by cavity ring-down spectroscopy of ammonia in a seeded supersonic jet from a pulsed slit nozzle
Published in Molecular Physics, 2020
G. Wichmann, E. Miloglyadov, G. Seyfang, M. Quack
is a symmetric top molecule of point group symmetry in its equilibrium geometry in the electronic ground state, as shown in Figure 10, with the z-axis along the symmetry axis. As is well known, has a low barrier to inversion through the transition structure, resulting in easily resolved large tunnelling splittings of about , depending upon the rotational and vibrational state. Therefore, the rotation-vibration-tunnelling sublevels can be classified in the molecular symmetry group following Longuet–Higgins [67], which in the present case is identical to the full permutation–inversion group . This is the direct product of the symmetric group of the permutations of the three protons and the inversion group following the notation of [1,2]. is isomorphous to the point group of the planar transition structure, which in this particular case can be used equivalently to classify the rotation-vibration-tunnelling levels. This has in fact been widely used in this context [68]. We note, however, that this procedure is not generally applicable. It would not work for methane , for example (see [2] for a detailed discussion). Table 2 provides the character table and various notations used for the irreducible representations of the isomorphous groups. We use the systematic notation of [1] (see also [2]), which assigns a unique symbol for the irreducible representations of (either as a partition or a letter symbol) and gives parity by a superscript ‘+’ for positive parity or ‘−’ for negative parity. The nuclear spin functions for the three protons (fermions) generate a reducible representation where the four functions correspond to the total nuclear spin (with ) and the two functions correspond to (with ). We shall write also the total spin multiplets as .