Transitive Permutational Representation

The Transitive Permutational Representations (TPR) of the crystallographic groups are of great importance for the application of the colour groups in a physical problems. It follows from three main features of TPR:

1.

Any TPR is equivalent to an induced representation of G;

2.

Any TPR can be constructed as engendered representation by TPR of the factor-group;

3.

The Frobenius Reciprocity Theorem gives important relationship between the irreducible representations of the group G, contained in the induced representation, and the irreps of its subgroup H'.

These features open the possibilities for an effective application of the colour groups in the Landau theory of phase transitions.

The permutational representation can be written in a matrix form  

$$\left[ D_G^{H'}(g) \right]_{ij} = \left\{ \begin{array} {c... ...n H' \\ 0, & \mbox{in an opposite case} \end{array} \right.$$ (16)

Here $g_i,g_j \in G, i,j=1,\ldots,n$ are fixed coset representatives in the coset decomposition of G with respect to H'. But the above equation is exactly the definition of the induced representation of G from the identity representation D_H'¹ of the subgroup $H' \subset G$, i.e.

$$D_G^{H'} = D_G^{\mbox{(induced)}} \equiv D_{H'}^1 \uparrow G.$$ (17)

The same relation is valid for permutational representation $\Pi_A^{A'}$ of A generated by its subgroup $A' \subset A$

$$D_A^{A'} = D_{A'}^1 \uparrow A.$$ (18)

It can be shown that D_A^A' is an image of D_G^H' by the homomorphism $\pi$

$$\mbox{ \rm Image }D_G^{H'} = D_{A}^{A'} = \pi \left( D_G^{H'} \right)$$ (19)

So, D_G^H' is built as an inverse image of D_A^A':

$$D_G^{H'} = \pi^{-1} \left( D_A^{A'} \right)$$ (20)

In this case the decomposition to the irreps are:

$$D_G^{H'} = \sum_j \left( D_G^{H'} \\ gt \bigl\vert \\ gt D_G^j \right) D_G^j$$ (21)

$$D_A^{A'} = \sum_j \left( D_A^{A'} \\ gt \bigl\vert \\ gt D_A^j \right) D_A^j$$ (22)

and all the irreducible components D_G^j in the first equation are engendreded by the irreps of the second one. The corresponding multiplicity coefficients in the both equations are identical. Now with the help of Frobenius Reciprocity Theorem it can be shown that

$$\left( D_G^j \downarrow H' \\ gt \bigl\vert \\ gt D_{H'}^1 \... ...\right) = \left( D^{H'}_G \\ gt \bigl\vert \\ gt D_G^j \right)$$ (23)

This means that the trivial representation D_H'¹ is contained in the subduction of D_G^j to H', if and only if D_G^j is contained in the transitive permutation representation D_G^H' of G.

Not of less importance is the next step. Because the same equations can be typed for the image groups A, A' and their representations, so we find:

$$\left( D_G^j \downarrow H' \\ gt \bigl\vert \\ gt D_{H'}^1 \right) = \left( D_A^{A'} \\ gt \bigl\vert \\ gt D_A^j \right)$$ (24)

The multiplicity coefficient in the right-hand side is for one of the transitive groups (A,A')_n, while the multiplicity coefficient in the left-hand side is is justified for all the groups belonging to the chromomorphic class (A,A')_n.