Symmetric and antisymmetric powers

The symmetric and antisymmetric powers of group representations have been identified as important in the analysis of several physical problems subject to group theoretical algebra since the appearance of the classic paper by Tisza.  

Group Theory Calculator, Quinn, Fowler and Redmond, Academic Press, New York 2005 

The GT Calculator includes options for the calculation of the symmetric and antisymmetric powers in the range 1 to 6 for a character input as a direct sum. Since the operation of the calculator for both applications is identical, only the instructions for the determination of symmetric powers is given in this section. 

EH Thpnry rfllrikJafnr, tjiiinn, Fnwlw anri Rpdrvoorl, ATiVfpnUr Prps , New rnrir nns 

The traces of the antisymmetric square character for the same input follow from the relation (see equation 4.13), 

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With the GT Calculator you can perform a variety of standard group theory calculations simply by entering the appropriate structure details for the molecular geometry. In addition, on the various worksheets of the calculator files, it is straightforward to determine more advanced group theoretical results, such as the numbers of isomers generated for a given structure by decoration, or to calculate and decompose the symmetric and antisymmetric powers of permutation representations. 

It is presumed in the calculations for symmetric and antisymmetric powers that sufficient cell areas are available to display the results of most calculations of this kind without the need for alternative displays. Difficulties with resolution can be remedied using the Zoom command, accessible via the Setup subsidiary command bar. 

Figure 1.23 The sequence of worksheet displays in the calculation of the square symmetric and antisymmetric powers of the IHg irreducible character of Ih- Note that Figure 1.23c is generated from the action of the antisymmetric powers command button in the options menu displayed in Figure 1.6b.

While cubes, fourth and fifth powers of degenerate irreducible characters of the common point groups are available in tables, these results are extended here for both symmetrized and antisymmetrized powers of reducible and irreducible characters up to p = 6. For one dimensional Fa, the powers are trivially already permutationally symmetric, with [Fa] = Fq for even p and [Ff] = Fa for odd p. 

The thermal conductivity tensor may likewise be split into symmetric and antisymmetric parts, with expansions in powers of B as in eqs. (35) and (36). But Z is not necessarily a symmetric tensor at B = 0, and so the expansion of the antisymmetric part of Z in an equation like eq. (36) is not applicable. Instead,... 

As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

Equation (3-47) is called a recursion relation. If we knew co, we could produce C2, C4, C6, etc., by continued application of Eq. (3-47). Similarly, knowledge of ci would lead to C3, C5, etc. Thus, it appears that the coefficients for even powers of y and those for odd powers of y form separate sets. Choosing co determines one set, and choosing ci determines the other, and the choices for co and ci seem independent. This separation into two sets is reasonable when we recall that our final solutions must be symmetric or antisymmetric in x, hence also in y. The asymptotic part of exp(—/Jx /2), is symmetric about X = 0, and so we expect the remainder of f(y), to be either symmetric (even powers of y = V Jc) or antisymmetric (odd powers). Thus, we can anticipate that some of our solutions will have co 0, C2 0, C4 7 0,... and Cl = C3 = C5 = = 0. This will produce symmetric solutions. The remaining solutions will have co = C2 04 = = 0 and ci 7 0, C3 yt 0,..., and be antisymmetric. 

The wavefunctions for this system are symmetric or antisymmetric for reflection through X = 0. This symmetry alternates as n increases and is related to the presence of even or odd powers of y in the Hermite polynomial in... 

The CO2 laser is a near-infrared gas laser capable of very high power and with an efficiency of about 20 per cent. CO2 has three normal modes of vibration Vj, the symmetric stretch, V2, the bending vibration, and V3, the antisymmetric stretch, with symmetry species (t+, ti , and (7+, and fundamental vibration wavenumbers of 1354, 673, and 2396 cm, respectively. Figure 9.16 shows some of the vibrational levels, the numbering of which is explained in footnote 4 of Chapter 4 (page 93), which are involved in the laser action. This occurs principally in the 3q22 transition, at about 10.6 pm, but may also be induced in the 3oli transition, at about 9.6 pm. 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

The Young operator Y in (1) antisymmetrizes with respect to permutations of sites in the same column in its tableau. The monomial yi, therefore, cannot be symmetrical with respect to any two sites in the same column, i.e., it cannot contain the same power of A for any two such sites. The powers of A for the sites in a given column, therefore, must all be different. The lowest possible choice consistent with this is that they be 0, 1, 2,. .., p, for a column of length /t. Thus, y can be chosen to be independent of A for sites in the first row of the tableau, and to contain A for sites in the second row, A2 for those in the third, etc. The total order is therefore... 

The expansions of C (t) and C/(t) in terms of a power series of Planck s constant show a dependence on even powers of h only, but Cr(0 is symmetric in t and Cfit) is antisymmetric. 

Ibrahim [19] developed a power series mathematical solutiOTi for the problem of instability of an inviscid liquid sheet of parabolic velocity profile emanated from a nozzle into an inviscid gas. The results show that for both antisymmetrical and symmetrical disturbances departure from uniformity of the velocity profile causes the instability to be reduced. It has been suggested that jet instability may be affected by the relaxation of the velocity profile that takes place once the liquid exits the nozzle and is no longer constrained by its wall. The variation of the growth rate with wave number at We = 10,000, p = 0.01, = 0,0.1,0.3,0.4, and = 96 are shown in Fig. 3.8 for antisymmetrical disturbances. The results of Fig. 3.8 indicate... 

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