Permutational Colour Groups

The permutational colour groups of P-type G^(P), isomorphic to the crystallographic group G,

$$\xi : G^{(P)} \leftrightarrow G,$$ (3)

are of special interest for the Landau theory of phase transitions. The reason is that all group-theoretical criteria known in the Landau theory have been reformulated and effectively used in the symmetry analysis of phase transitions.

Any isomorphic to G Permutational colour group is defined, as

$$G^{(P)} = \left\{ \langle p; g \rangle \\ gt \bigl\vert \\ g... ...G \longrightarrow P \subseteq S_n \right\} \subset P \times G$$ (4)

Here P is a transitive permutational group of degree n, a subgroup of the symmetric group S_n. The main important point is the pairing of the elements $g \in G$ with the permutations $p \in P$. It is determined by the homomorphism $\pi : G \rightarrow P$. The kernel of the homomorphism is a invariant subgroup H of G,

$$\mbox{ \rm Ker }\pi = H \triangleleft G.$$ (5)

All the elements $h \in H$ are paired with the identity permutation of P. Consequently, the maximal subgroup of G^(P) which preserves unchanged all the colours, is isomorphic to H and is invariant subgroup of the colour group:

$$H^{(1)} = \left\{ <e_P;h\gt \\ gt \bigl\vert \\ gt e_P = \mb... ...y of P}, h \in H \triangleleft G \right\} \triangleleft G^{(P)}$$ (6)

Consider now coset decomposition if G with respect to H and of G^(P) with respect to H⁽¹⁾. The corresponding cosets are elements of two factor-groups, both isomorphic to the same abstract group A. Any homomorphism $\pi : G \rightarrow P$ with kernel H can be considered as a canonical one from G to the factor-group $G/H \cong A$ followed by the isomorphism $A \cong P$. Hence, the transitive group P of permutations is isomorphic to the abstract group A, as well as to both factor-groups:

$$P \cong A \cong G/H \cong G^{(P)}/H^{(1)}.$$ (7)

By this isomorphism there is a one-to-one correspondence between the permutations $p_i \in P$ and the cosets g_iH as elements of G/H or the cosets $\langle p_i;g_i \rangle H^{(1)}$as elements of G^(P)/H⁽¹⁾.

Each permutational colour group G^(P) contains a very important set of subgroups, the maximal subgroups of G^(P), preserving one of the colours unchanged. This set forms an equivalence class of conjugate subgroups and one of its members, say the subgroup H'^(P') preserving colour number 1, is chosen as a representative of class,

$$H'^{(P')} = \left\{ \langle p'; h' \rangle \\ gt \bigl\vert ... ...set G, p' \in P' = \pi (H') \subset P \right\} \subset G^{(P)}$$ (8)

Obviously, as each of conjugated with H'^(P') subgroup preserves at least one of the other colours, the set of common elements if all the subgroups of the class (the intersection) preserves all the colours and coincides with the subgroup H⁽¹⁾.

Due to isomorphism $\xi$, there are the same relationship between the group G and its subgroups H' and H, where

$$H' \cong H'^{(P')} \subset G^{(P)}$$ (9)

and

$$H \cong H^{(1)} \triangleleft G^{(P)}$$ (10)

Here H is the intersection of all subgroups of G conjugate with H', the so called core of H',

$$\bigcap\limits_{g \in G} g H' g^{-1} \equiv \mbox{ \rm Core }H' = H \triangleleft G.$$ (11)

It is important that the core of H' is the maximal invariant subgroup of G which is contained in H'. In this case

$$\mbox{ \rm Ker }\pi = H = \mbox{ \rm Core }H'.$$ (12)

The factor group H'/H is isomorphic with the subgroup A' of the abstract group $A \cong G/H$ and

$$\mbox{ \rm Core }A' \cong \mbox{ \rm Core }H'/H \cong C_1.$$ (13)

So, the homomorphism $\pi$ maps the subgroups chain $G \supset H' \supset H$ of the (infinite) crystallographic group G onto the subgroup chain $A \supset A' \supset C_1$ of the (finite) abstract group $A \cong G/H$.

The homomorphism $\pi$ in the definition of colour groups realizes the representation of G, known as transitive permutational representation.

All the above considered relations between the colour group G^(P), its subgroups, the factor groups, the correspondent isomorphism groups and images are collected in the following diagram:

 

$$\begin{array} {lclcl} G^{(P)} & \supseteq & {H'}^{(P')} & \... ...left(a A' a^{-1}\right) \equiv \mbox{ \rm Core }A' \end{array}$$ (14)

The main information from this diagram is given in compact form in the so called ``full symbol''[7,8] of the colour group G^(P)

G/H'/H (A,A')_n

Below are listed the basic conclusions for the colour group and theirs interpretation in the phase transitions:

• This is isomorphic to G n-colour group, subgroup of $G \times (A,A')_n$. In the phase transitions: The high symmetry phase is G.
• The preserving at least one colour subgroup H'^(P') is isomorphic to H'. In the phase transitions: The lower-symmetry phase group is H'.
• The preserving all n colours subgroup H⁽¹⁾ is isomorphic to H. In the phase transitions: The kernel of the order parameter irreducible representation D_G^j of G is H.
• The order parameter irreducible representation must be one of the constituents of the permutational representation with image (A,A')_n.

All colour groups with same group of permutations (A,A')_n forms a ``chromomorphic'' class. It is important to note that symmetry properties of the phase transition are determined from the class (A,A')_n. The results of analysis made in the frame (A,A')_n is valid for all the space groups G from class (A,A')_n. Hence, from mathematical point of view, it is enough to analyze all possible transitions $A \longrightarrow A'$ for the finite groups A. All phase transitions without change of translational symmetry are described with colour groups which translational subgroup preserves all colours. Total number of these groups is 2571[6].

In case of change of the translational symmetry, the number of colour groups increases. The main object of the diploma thesis was investigation in this more complicated case. We restrict our research to colour groups describing transitions to commensurate phases i.e. with irreducible representations which wave vector is from the surface of Brillouin zone. We use images of the irreps from book [9].