D irreducible representations

If function / transforms according to a 1-D irreducible representation, the function is an eigenfunction of aU the symmetiy operators it, with the corresponding eigenvalues x (R). 

If the projections were made for the function b, then we would obtain a trivial repetition of the irreducible representations Ai and A2 and a non-trivial result for the irreducible representation E P b = 2b — a—c + Q-a + O-b + O-c) = 2 b — c) — a — c)]. This is just another linear combination of U2 and M3. These two functions are therefore inseparable and form a basis for a 2-D irreducible representation. 

Let us come back to R . Ry. Rz- Imagine R c, Ry, Rz as oriented circles perpendicular to a rotation axis (i.e., x, y, or z) that symbolize rotations about these axes. For instance, the operation E and the two rotations C3 leave the circle R unchanged, while the operations change its orientation to the opposite one hence R transforms according to the irreducible representation A2. It turns out that Rj and Rj, transform under the symmetry operations into their linear combinations and therefore correspond to a 2-D irreducible representation (E). 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

Irreducible Co-Representations of Nonunitary Croups.—To determine the co-representation matrices D , let us select a set of l functions, which forms a basis for an irreducible representation A (u) of the unitary subgroup H. That is... 

Case (a) a0, reproduces the set of functions The irreducible corepresentation D of 0 corresponds to a single irreducible representation A (u) of H, and has the same dimension. In this case no new degeneracy is introduced by the coset ua0. 

Case (b) aproduces a set of functions t/rj, which is independent of the set but which also forms a basis for A (u) of H. The irreducible co-representation D of G corresponds again to a single irreducible representation of H, but has twice its dimension. In this case the dimension of A (u) is doubled. 

The irreducible representations arising from any strong field configuration for a given dx system may readily be obtained from the listed headings of the columns of the repulsion matrices of Tables 3—6, e.g. d2, (n8) -> 11 + d + 3II + 3. 

To find the irreducible representations of 0(3) it is necessary to find a set of basis functions which transform into their linear combinations on operating with the elements of 0(3). The set of 21 + 1 spherical harmonics Y[m(d, ), where l = 0,1, 2... and —l[Pg.91]

D. Definitions of Young diagrams, tableaux, and operators should be understood, as well as the property of the Young operator of projecting onto an irreducible representation (Theorems 1 and 2). 

Familiarity is also assumed with the concepts of representation and irreducible representation (IR). A representation r of dimension n associates to each group element s an n X n matrix D(s), with matrix elements D(s)y, in such a way that for every s, t, D(s)D(t) =D(st), with the product formed by ordinary matrix multiplication. We will sometimes use the bra-ket notation... 

The matrix D R) is the matrix representative of R. Furthermore, the representation D R) is irreducible, so that to every energy state of 3C, an irreducible representation of G can be assigned. 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

Problem 10-5. In a homonuclear diatomic molecule, taking the molecular axis as z, the pair of LCAO-MO s tpi = 2p A + PxB and tp2 = 2 PyA + 2 PyB forms a basis for a degenerate irreducible representation of D h, as does the pair 3 = 2pxA PxB and 4 = PxA — PxB Identify the symmetry species of these wave functions. Write down the four-by-four matrices for the direct product representation by examining the effect of the group elements on the products 0i 03, 0i 04, V 2 03) and 02 04- Verify that the characters of the direct product representation are the products of the characters of the individual representations. 

It should be reasonably self-evident that the conjugate of a standard tableau is a standard tableau of the conjugate shape. Therefore, fx = fi, and irreducible representations corresponding to conjugate partitions are the same size. In fact, the irreducible representations are closely related. If D p) is one of the irreducible representation matrices for partition X, one has... 

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