1. Mathematical Foundations of Crystallography Bill Stein Bill Stein Adam Cloues Adam Cloues David Bauer David Bauer Britt Wooldridge Britt Wooldridge Mang Yang Mang Yang Chris Bouzek Chris Bouzek

2. What are Crystals? “Regular Arrangements” of Atoms/Molecules in Solids. “Regular Arrangements” of Atoms/Molecules in Solids. Question: What kinds of arrangements are possible? Question: What kinds of arrangements are possible? Answer: Use Group Theory to describe the possibilities. Answer: Use Group Theory to describe the possibilities.

3. Applied Group Theory “Groups” model concepts of symmetry. “Groups” model concepts of symmetry. Set of objects w/ a binary operation such that: Set of objects w/ a binary operation such that:  An identity element e exists: ge = eg = g  There exist inverses: g(g -1 ) = (g -1 )g = e  Operation is associative: (gh)k = g(hk)

4. Group Examples Z = “the integers” Z = “the integers” = {…, -3, -2, -1, 0, 1, 2, 3, …..} = {…, -3, -2, -1, 0, 1, 2, 3, …..} Orthogonal Group O(n) Orthogonal Group O(n) Matrix Groups: Matrix Groups:  GL(n,C) = {A a complex invertible matrix}  SL(n,C) = {A  GL(n,C) | det(A) = 1}

5. Orthogonal Group Definition: Definition:  A 2x2 real matrix A is orthogonal if: AA T = A T A = I (i.e. if A T = A -1 ).  Let ||x|| be the length of vector x. A distance preserving linear transformation T is said to be orthogonal. Such transformations form the orthogonal group O(n).

6. Subgroups of the Orthogonal Group An example of a subgroup of the orthogonal group is rotation about the origin on the plane: An example of a subgroup of the orthogonal group is rotation about the origin on the plane:

7. Subgroups Non-empty subsets H are subgroups if: Non-empty subsets H are subgroups if:  h,k  H  hk  H (closure)  h  H  (h -1 )  H (inverses) Examples: Examples:  Z 3 = {0,1,2} is a subgroup of the integers  SL(n,R) = {A  GL(n,R) | det(A) = 1} is a subgroup of the General Linear group GL(n,R).

8. Equivalence Classes “~” is an equivalence relation if it is: “~” is an equivalence relation if it is:  Reflexive: h  G, h ~ h  Symmetric: h, k  G, h ~ k  k ~ h  Transitive: h ~ k, k ~ g  h ~ g

9. Equivalence Classes cont’d. An equivalence class is defined as a subset of the form {x  G | x~y}, where y is an element of G and the notation "x~y" is used to mean that there is an equivalence relation between x and y. An equivalence class is defined as a subset of the form {x  G | x~y}, where y is an element of G and the notation "x~y" is used to mean that there is an equivalence relation between x and y. Equivalence classes are maximal sets of equivalent elements. Equivalence classes are maximal sets of equivalent elements.

10. Example of Equivalence Class Z 3 (Z “mod” 3) Z 3 (Z “mod” 3) In this world, two elements are equivalent if their difference is divisible by three. In this world, two elements are equivalent if their difference is divisible by three.

11. Coset Partitioning The concept of partitioning a group G allows a better understanding of the structure of G by breaking the group into parts (e.g. left cosets). The concept of partitioning a group G allows a better understanding of the structure of G by breaking the group into parts (e.g. left cosets). A subset of G of the form gH for some g  G is said to be a left coset of H. A subset of G of the form gH for some g  G is said to be a left coset of H.  gH = {gh | h  H}

12. Example of Coset Partitioning Example: G can be partitioned by cosets, also by conjugacy classes. Example: G can be partitioned by cosets, also by conjugacy classes.

13. Conjugacy Classes Given G (any group), we say that for h, k, g  G, h~k ( “h is conjugate to k” ) if k = ghg -1. This is known as conjugation. Given G (any group), we say that for h, k, g  G, h~k ( “h is conjugate to k” ) if k = ghg -1. This is known as conjugation. A conjugacy class is an equivalence class. A conjugacy class is an equivalence class. In this case, equivalence classes are called conjugacy classes. In this case, equivalence classes are called conjugacy classes.

14. Conjugacy Classes cont’d. Each element in a group belongs to exactly one class, and the identity element ‘e’ is always its own conjugacy class. Each element in a group belongs to exactly one class, and the identity element ‘e’ is always its own conjugacy class.

15. The Full Symmetric Group S X Let X be a set. Let X be a set. S X is the set of all 1:1 and onto maps of the set X onto itself. S X is the set of all 1:1 and onto maps of the set X onto itself. The permutation groups are subgroups under S X. The permutation groups are subgroups under S X.

16. Permutation Groups Definition: A permutation group is a subgroup of the group S X. Definition: A permutation group is a subgroup of the group S X. Permutation groups are otherwise known as transformation groups. Permutation groups are otherwise known as transformation groups.

17. Examples of Permutation Groups The Symmetric Group for the natural numbers (S N ). The Symmetric Group for the natural numbers (S N ). The Dihedral Group (D n ). The Dihedral Group (D n ). The Alternating Group (A n ). The Alternating Group (A n ).

18. Symmetric Group S N S N is the full symmetric group for the natural numbers. S N is the full symmetric group for the natural numbers. Example: S 3 is the set of all self-bijections of X = {1,2,3}. Example: S 3 is the set of all self-bijections of X = {1,2,3}.

19. Permutation A permutation is a rearrangement of an ordered list, let’s say S, into a 1:1 correspondence with S itself. A permutation is a rearrangement of an ordered list, let’s say S, into a 1:1 correspondence with S itself. The number of different permutations in a set of order n is n factorial (written n!). The number of different permutations in a set of order n is n factorial (written n!).

20. An Example of Permutation Notation

21. Permutation Example Let’s look at the set S = {A, B, C}. Let’s look at the set S = {A, B, C}. The order of its permutation is: n! = 3 * 2 * 1 = 6. The order of its permutation is: n! = 3 * 2 * 1 = 6.

22. Permutation Example cont’d. The elements of S’s permutation are: The elements of S’s permutation are:  ABC  ACB  BAC  BCA  CAB  CBA

23. Cycle Notation First introduced by the great French mathematician Cauchy in 1815. First introduced by the great French mathematician Cauchy in 1815. Has theoretical advantages in that certain important properties of the permutation can be readily determined when cycle notation is used. Has theoretical advantages in that certain important properties of the permutation can be readily determined when cycle notation is used.

24. Permutation The act of changing the linear order of objects in a group. The act of changing the linear order of objects in a group. ABC BCACAB

25. Examples of Elements of S 6

26. Permutation Notation () 1 2 3 4 5 6 2 3 4 5 6 1

27. 1-Cycle Notation (1 2 3 4 5 6)

28. Permutation Notation () 1 2 3 4 5 6 4 3 2 5 6 1

29. 2-Cycle Notation (1 4 5 6) (2 3)

30. Concepts of Cycle Notation Don’t need to write cycles that have only one entry. The missing element is mapped to itself. Don’t need to write cycles that have only one entry. The missing element is mapped to itself. In a sense, the “order” does not matter. In a sense, the “order” does not matter.

31. G-Equivalence Permutation groups induce G-Equivalence. Permutation groups induce G-Equivalence. Suppose we have a permutation group G, where G  S X. Suppose we have a permutation group G, where G  S X. If x, y  X, we say that x and y are “G-Equivalent” (x~y) if gx = y for some g  G. If x, y  X, we say that x and y are “G-Equivalent” (x~y) if gx = y for some g  G.

32. G-Equivalence cont’d. Fact: “G-Equivalence” is an equivalence relation. Fact: “G-Equivalence” is an equivalence relation. Definition: The equivalence classes of X under the equivalence relation “~” are called G-orbits (or orbits). Definition: The equivalence classes of X under the equivalence relation “~” are called G-orbits (or orbits).

33. Orbits Definition of orbit: Gx = {y  X | y = gx for some g  G}. Definition of orbit: Gx = {y  X | y = gx for some g  G}. This is also called G-equivalence. This is also called G-equivalence. All elements in X are G-equivalent if there is only one orbit. In this case, we say that the action of G is transitive. All elements in X are G-equivalent if there is only one orbit. In this case, we say that the action of G is transitive.

34. Examples of Orbits Example of an element of the full symmetric group on the plane:  = R 2  R 2 (1:1, onto). Example of an element of the full symmetric group on the plane:  = R 2  R 2 (1:1, onto). Rotation group about the origin. Rotation group about the origin.

35. Rotation Group About the Origin  (0,0) * ** cos  sin  sin  cos  xyxy The orbits are the concentric circles.

36. Invariance and Symmetry Groups

37. Groups as Models of Geometric Symmetry Crystals are formed through repetition or clustering of crystal atoms. Crystals are formed through repetition or clustering of crystal atoms. Symmetry groups of invariant permutations are the foundation of crystallography because it models symmetry of objects (specifically crystal atoms and molecules) using group theory. Symmetry groups of invariant permutations are the foundation of crystallography because it models symmetry of objects (specifically crystal atoms and molecules) using group theory.

38. What is Invariance of Objects Under Permutation? Definition: Definition:  Recall that S X is the set of all permutations on X.  Suppose we have a set Y  X and a group G  S X. Then the subset Y is “G-invariant” if gY  Y, where gY = g(Y), the image of Y under G,  g  G.

39. Invariance of objects under permutation What does that mean? What does that mean?  For any object on a plane, symmetric permutations (rotation, translation, and reflection) leave an object invariant if it preserves the motif of that object.

40. Invariance of objects under permutation Here y is acted upon by g(y) under reflection. As we can see although the position of y has changed, its pattern remains unchanged. Here y is acted upon by g(y) under reflection. As we can see although the position of y has changed, its pattern remains unchanged.

41. Specifically what are symmetry groups of objects? Symmetry groups are the consequence of an invariant permutation on a plane. Symmetry groups are the consequence of an invariant permutation on a plane. e.g. Triangle permutations (D 3 ) e.g. Triangle permutations (D 3 )  R 0, R 120, R 240  Reflection: R 1, R 2, R 3  Example of a permutation that is not invariant on D 3 is R 90

42. Symmetry Group of a Square Or, the dihedral group of a square

43. Motivation Knowing conjugacy classes and valid geometric transforms allows easier modeling for computer applications by reducing the number of transformations that the computer needs to deal with. Knowing conjugacy classes and valid geometric transforms allows easier modeling for computer applications by reducing the number of transformations that the computer needs to deal with.

44. Why is the Dihedral Group a Group? The dihedral group is a subset of the full symmetric group. The dihedral group is a subset of the full symmetric group. Show each element has an inverse. Show each element has an inverse. Show that the group is closed under the operation. Show that the group is closed under the operation.

45. Elements of the dihedral group Group elements: {e, r, r 2, r 3, v, h, m, n} Group elements: {e, r, r 2, r 3, v, h, m, n} e = identity (no transformation) e = identity (no transformation) r = 90º CCW rotation r = 90º CCW rotation r 2 = 180º CCW rotation r 2 = 180º CCW rotation r 3 = 270º CCW rotation r 3 = 270º CCW rotation h = horizontal reflection h = horizontal reflection v = vertical reflection v = vertical reflection m = reflection across main diagonal m = reflection across main diagonal n = reflection across minor diagonal n = reflection across minor diagonal

46. Cayley Table for the Dihedral Group

47. Inverses of the Elements of the Dihedral Group r -1 = r 3 r -1 = r 3 r -2 = r 2 r -2 = r 2 r -3 = r 1 r -3 = r 1 h -1 = h h -1 = h v -1 = v v -1 = v m -1 = m m -1 = m n -1 = n n -1 = n e -1 = e e -1 = e

48. Conjugacy Classes Recall that an equivalence class is a subset of a group whose members are equivalent under some operation. Recall that an equivalence class is a subset of a group whose members are equivalent under some operation. A conjugacy class is a set whose members are equivalent under conjugation (ghg -1 ) for g, h  G. A conjugacy class is a set whose members are equivalent under conjugation (ghg -1 ) for g, h  G. Thus, a conjugacy class is a way to partition a group into equivalence classes. Thus, a conjugacy class is a way to partition a group into equivalence classes.

49. Determining Conjugacy Classes Technique 1: Pick h, k  G. h ~ k if k = ghg -1 for some g  G. Technique 1: Pick h, k  G. h ~ k if k = ghg -1 for some g  G. Technique 2: Consider the conjugacy class as an orbit. Pick h  G, then find the conjugation  g  G. This will give the conjugacy class for h. Technique 2: Consider the conjugacy class as an orbit. Pick h  G, then find the conjugation  g  G. This will give the conjugacy class for h.

50. Dihedral Group Conjugacy Classes Conjugacy classes of the dihedral group: Conjugacy classes of the dihedral group: { { e }, { r, r 3 }, { r 2 }, { h, v }, { m, n } } { { e }, { r, r 3 }, { r 2 }, { h, v }, { m, n } } To find these, we will use Technique 2: select an h  G, then find the conjugation. To find these, we will use Technique 2: select an h  G, then find the conjugation.

51. Finding the Conjugacy Classes Pick r. I claim that { r ~ r 3 }. Pick r. I claim that { r ~ r 3 }. r 3 rr -3 = r r 3 rr -3 = r r 2 rr -2 = r r 2 rr -2 = r hrh -1 = r 3 hrh -1 = r 3 vrv -1 = r 3 vrv -1 = r 3 mrm -1 = r 3 mrm -1 = r 3 nrn -1 = r 3 nrn -1 = r 3 Thus { r, r 3 } is one conjugacy class. Thus { r, r 3 } is one conjugacy class. A similar procedure gives us the remainder of the conjugacy classes. A similar procedure gives us the remainder of the conjugacy classes.

52. Impact on Crystallography Crystallographers study the geometric structure of crystals in many materials in order to to determine a material’s physical properties, which benefits society as a whole. Crystallographers study the geometric structure of crystals in many materials in order to to determine a material’s physical properties, which benefits society as a whole. Knowing these conjugacy classes (i.e. knowing that certain transformations are equivalent) can speed up their computer analyses of the material. Knowing these conjugacy classes (i.e. knowing that certain transformations are equivalent) can speed up their computer analyses of the material.

53. References “Modern Geometries, 5 th Ed.” by James R. Smart, Brooks/Cole Publishing Company 1998 “Modern Geometries, 5 th Ed.” by James R. Smart, Brooks/Cole Publishing Company 1998 “Symmetry Groups and their Applications” by Willard Miller Jr., Academic Press 1972 “Symmetry Groups and their Applications” by Willard Miller Jr., Academic Press 1972 “General Chemistry” by Linus Pauling, Dover 1970 “General Chemistry” by Linus Pauling, Dover 1970 “The Fenyman Lectures on Physics”, Fenyman, et al, Addison-Wesley 1963 “The Fenyman Lectures on Physics”, Fenyman, et al, Addison-Wesley 1963

54. Questions?