# Group Theoretical Methods in Physics by Robert Shar (Eds.)

By Robert Shar (Eds.)

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From a macroscopic to a B. From an averaged to a refined microscopic des- microscopic description cription (i) Basic symmetry (approximated) macroscopic crystal form point group K Q main reflexions only C 0(3) (or basic structure) average space group G (ii) C E(3) New admitted transformations space translations: Τ 3 d "internal" translations T^ requiring the extension to: E(3) = Τ (iii) Q 3 requiring the extension to: Λ 0(3) Pure new symmetries E(3) x E(d) (associated with the identity of the old ones) group of "primitive" (or group of "internal" lattice lattice) translations A ^ (iv) New symmetry group G microscopic symmetry group: 3-dim space group (v) translations refined symmetry group (3+d)dim superspace group Mathematical structure of G (group extension (in general non trivial)) O + A + G + K+ l 3 or A _ < G; G / A * Κ ο non-symmorphic case (vi) O + D + G + G + l d Ε D, < G ; G/D, Ξ Μ - G α d h modulated case C Relation with the basic (approximated) symmetry Κ C κ — ο Κ: point group of G = "homogeneous" parts of the space group G G C Ε — C G ο G : space group of £ G = "external" parts of the superspace group G 29 Nuclei, Atoms, Solids (vii) Physical properties (like systematic extinctions) (in general) - explained by G but - explained by G but not by Κ only - K q not by G ^ only symmetry elements - G easily recognized symmetry elements q easily recognized.

5) that / vw dx = X / w dx. 2 2 If w is normalized, we get r 2 2 λ = (X ,v) = J vw dx = (w , v ) . 6) v J X = w . u 2 To build Hamiltonian systems we need to choose some operator J. 7) J = D 7 was suggested by Clifford Gardner . metric and independent of u. 16) asso ciated with the Hamiltonians F and F 1 given in ( 3 . 8). 8) 2 2 Equation ( 3 . 8 ) 2 = JF- = Du = u 1 χ u u. = JF. 8) t 1 is linear and easily solved: u = u(x+t). Equation on the other hand is nonlinear and far from being easy to solve; it is the celebrated Korteweg-de Vries (KdV) equation.

Note that Σ , (Λ ,D,) and n+d η d 6 ^n+d^n'*\P ^ a v e t ^ i e s a m e v holohedry H. J For the corresponding representations Γ(Η) and f(Η) obtained by referring Η to bases Β and B, respectively, given by (1) (which are standard) one veri fies using (127) that Γ(Η) and f(H) are conjugated subgroups of Gl(n,d,Z). More generally the arbitrariness in the choice of bases (1) is as follows: (i) One may change the description of the main reflections by choosing another basis of Λ*: k (ii) * X % Ehk> h=l S Sje l(n,Z).