# Some notes on group theory.

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1 Some notes on group theory. Eef van Beveren Departamento de Física Faculdade de Ciências e Tecnologia P- COIMBRA (Portugal) February

2 Contents Some intuitive notions of groups.. What is a group? The cyclic notation for permutation groups Partitions and Young diagrams Subgroups and cosets Equivalence classes Representations of finite groups.. What is a matrix representation of a group? The word-representation D (w) of S The regular representation The symmetry group of the triangle One-dimensional and trivial representations Reducible representations Irreducible representations (Irreps) Unitary representations The standard irreps of S n. 7. Standard Young tableaux and their associated Yamanouchi symbols.. 7. Hooklength, hookproduct and axial distance The dimension of irreps for S n Young s orthogonal form for the irreps of S n The standard irreps of S The standard irreps of S The classification of irreps The character table of a representation The first lemma of Schur The second lemma of Schur The orthogonality theorem The orthogonality of characters for irreps Irreps and equivalence classes Abelian groups Series of matrices and direct products Det(Exp)=Exp(Trace) The Baker-Campbell-Hausdorff formula i

3 5. Orthogonal transformations Unitary transformations The direct product of two matrices The special orthogonal group in two dimensions The group SO() Irreps of SO() Active rotations in two dimensions Passive rotations in two dimensions The standard irreps of SO() The generator of SO() A differential operator for the generator The generator for physicists The Lie-algebra The orthogonal group in two dimensions Representations and the reduction procedure The group O() Irreps of O() The special orthogonal group in three dimensions The rotation group SO() The Euler angles The generators The rotation axis The Lie-algebra of SO() The Casimir operator of SO() The (l+)-dimensional irrep d (l) The three dimensional irrep for l = The standard irreps of SO() The spherical harmonics Weight diagrams Equivalence classes and characters The special unitary group in two dimensions Unitary transformations in two dimensions The generators of SU() The relation between SU() and SO() The subgroup U() The (j +)-dimensional irrep {j +} The two-dimensional irrep {} The three-dimensional irrep {} The four-dimensional irrep {4} The product space {} {} Clebsch-Gordan or Wigner coefficients The Lie-algebra of SU() The product space {} {} {} Tensors for SU() ii

4 9.4 Irreducible tensors for SU() Rotations in the generator space Real representations The special unitary group in three dimensions. 9. The generators The representations of SU() Weight diagrams for SU() The Lie-algebra of SU() The three-dimensional irrep {} of SU() The conjugate representation { } Tensors for SU() Traceless tensors Symmetric tensors Irreducible tensors D (p,q) The matrix elements of the step operators The six-dimensional irrep D (,) The eight-dimensional irrep D (,) The classification of semi-simple Lie groups. 55. Invariant subgoups and subalgebras The general structure of the Lie algebra The step operators The root vectors Dynkin diagrams The Dynkin diagram for SU() The orthogonal group in four dimensions. 69. The generators of SO(N) The generators of SO(4) The group SU() SU() The relation between SO(4) and SU() SU() Irreps for SO(4) Weight diagrams for SO(4) Invariant subgroups of SO(4) The Hydrogen atom iii

5 iv

6 Chapter Some intuitive notions of groups. In this chapter we will discuss the definition and the properties of a group. As an example we will study the symmetric groups Sn of permutations of n objects.. What is a group? For our purpose it is sufficient to consider a group G as a set of operations, i.e.: G = {g,g,,gp}. (.) Each element g i (i =,,p) of the group G (.) represents an operation. Groups can be finite as well as infinite. The number of elements of a finite group is called the order p of the group. As an example let us study the group S of permutations of three objects. Let us define as objects the letters A, B and C. With those objects one can form threeletter words, like ABC and CAB (six possibilities). We define the operation of an element of S on a three-letter word, by a reshuffling of the order of the letters in a word, i.e.: g i (word) = another (or the same) word. (.) Let us denote the elements of S in such a way that from the notation it is immediatly clear how it operates on a three-letter word, i.e. by: g = [] = I, g = [], g = [], g 4 = [], g 5 = [], and g 6 = []. Table.: The elements of the permutation group S. Then, we might define the action of the elements of the group S by: [](ABC) = the st letter remains in the st position the nd letter remains in the nd position the rd letter remains in the rd position = ABC

7 [](ABC) = [](ABC) = [](ABC) = [](ABC) = [](ABC) = the nd letter moves to the st position the rd letter moves to the nd position the st letter moves to the rd position the rd letter moves to the st position the st letter moves to the nd position the nd letter moves to the rd position the st letter remains in the st position the rd letter moves to the nd position the nd letter moves to the rd position the nd letter moves to the st position the st letter moves to the nd position the rd letter remains in the rd position the rd letter moves to the st position the nd letter remains in the nd position the st letter moves to the rd position = BCA = CAB = ACB = BAC = CBA (.) In equations (.) a definition is given for the operation of the group elements of table (.) at page of the permutation group S. But a group is not only a set of operations. A group must also be endowed with a suitable definition of a product. In the case of the example of S such a product might be chosen as follows: First one observes that the repeated operation of two group elements on a threeletter word results again in a three-letter word, for example: []([](ABC)) = [](CAB) = ACB. (.4) Then one notices that such repeated operation is equal to one of the other operations of the group; in the case of the above example (.4) one finds: []([](ABC)) = ACB = [](ABC). (.5) In agreement with the definition of the product function in Analysis for repeated actions ofoperations onthe variable ABC, we define here forthe product of group operations the following: ([] [])(ABC) = []([](ABC)). (.6) From this definition and the relation (.5) one might deduce for the product of [] and [] the result: [] [] = []. (.7) Notice that, in the final notation of formula (.7) for the product of two group elements, we dropped the symbol, which is used in formula (.6). The product of group elements must, in general, be defined such, that the product of two elements of the group is itself also an element of the group.

8 The products between all elements of S are collected in table (.) below. Such table is called the multiplication table of the group. It contains all information on the structure of the group; in the case of table (.), of the group S. S b I [] [] [] [] [] a ab I I [] [] [] [] [] [] [] [] I [] [] [] [] [] I [] [] [] [] [] [] [] [] I [] [] [] [] [] [] [] I [] [] [] [] [] [] [] I Table.: The multiplication table of the group S. The product which is defined for a group must have the following properties:. The product must be associative, i.e.: ( ) ( ) gi g j gk = g i gj g k. (.8). There must exist an identity operation, I. For example the identity operation for S is defined by: I = [] i.e. : I(ABC) = ABC. (.9). Each element g of the group G must have its inverse operation g, such that: gg = g g = I. (.) From the multiplication table we find for S the following results for the inverses of its elements: I = I, [] = [], [] = [], [] = [], [] = [], [] = []. Notice that some group elements are their own inverse. Notice also that the group product is in general not commutative. For example: [][] = [] [] = [][]. A group for which the group product is commutative is called an Abelian group. Notice moreover from table (.) that in each row and in each column appear once all elements of the group. This is a quite reasonable result, since: if ab = ac, then consequently b = a ac = c.

9 . The cyclic notation for permutation groups. A different way of denoting the elements of a permutation group Sn is by means of cycles (e.g. (ijk l)(m)). In a cycle each object is followed by its image, and the image of the last object of a cycle is given by the first object, i.e.: (ijk l)(m) = i j j k k... l l i m m for the numbers i,j,k,...l all different. For example, the cyclic notation of the elements of S is given by: [] = [] = [] = [] = [] = = ()()() = () = () = () = () = () = () = ()() = () = () = ()() = () = () (.) [] = = ()() = () = () (.) As the length of a cycle one might define the amount of numbers which appear in between its brackets. Cycles of length, like (), () and (), are called transpositions. A cycle of length l is called an l-cycle. Noticethatcycles oflengthmaybeomitted, since itisunderstoodthat positions in the n-letter words, which are not mentioned in the cyclic notation of an element of Sn, are not touched by the operation of that element on those words. This leads, however, to some ambiguity in responding to the questions as to which group a certain group element belongs. For example, in S 6 the meaning of the group element () is explicitly given by: () = ()()(4)(5)(6). 4

10 In terms of the cyclic notation, the multiplication table of S is given in table (.) below. S b I () () () () () a ab I I () () () () () () () () I () () () () () I () () () () () () () () I () () () () () () () I () () () () () () () I Table.: The multiplication table of S in the cyclic notation. As the order q of an element of a finite group one defines the number of times it has to be multiplied with itself in order to obtain the identity operator. In table (.4) is collected the order of each element of S. One might notice from that table that the order of an element of the permutation group S (and of any other permutation group) equals the length of its cyclic representation. element (g) order (q) g q I I () ()()() = ()() = I () ()()() = ()() = I () ()() = I () ()() = I () ()() = I Table.4: The order of the elements of S. Let us study the product of elements of permutation groups in the cyclic notation, in a bit more detail. First we notice for elements of S the following: ()() = ()() = = () = () (.) i.e. In multiplying a cycle by a transposition, one can add or take out a number from the cycle, depending on whether it is multiplied from the left or from the right. In the first line of (.) the number is added to the cycle () by multiplying it by () from the left. In the second line the number is taken out of the cycle () by multiplying it from the right by (). 5

11 These results for S can be generalized to any permutation group Sn, as follows: For the numbers i, i,..., i k all different: (i i )(i i k ) = (i i i )(i i k ) = and : (i i k )(i k i ) = (i i k )(i k i i ) = i i i i i i i i 4 i 4. i k i i i i i i i i i i 4 i 4 = (i i i k ) = (i i i k ). i k i i etc (.4) i i k i i i i i i i 4. i k i i i i k i i i i i i i 4 = (i i k ) = (i i k ). i k i i etc (.5) From the above procedures (.4) and (.5) one might conclude that starting with only its transpositions, i.e. (), (),..., (n), (),..., (n),..., (n,n), it is possible, using group multiplication, to construct all other group elements of the group Sn, by adding numbers to the cycles, removing and interchanging them. However, the same can be achieved with a more restricted set of transpositions, i.e.: {(),(),(4),...,(n,n)}. (.6) For example, one can construct the other elements of S starting with the transpositions () and (), as follows: I = ()(), () = ()(), () = ()() and () = ()()(). This way one obtains all elements of S starting out with two transpositions and using the product operation which is defined on the group. 6

12 For S 4 we may restrict ourselves to demonstrating the construction of the remaining transpositions, starting from the set {(), (), (4)}, because all other group elements can readily be obtained using repeatingly the procedures (.4) and (.5). The result is shown below: () = ()()(), (4) = ()()()(4)()()() and (4) = ()()(4)()(). For other permutation groups the procedure is similar as here outlined for S and S 4. There exist, however, more interesting properties of cycles: The product of all elements of the set of transpositions shown in (.6), yields the following group element of Sn: ()()(4) (n,n) = ( n). It can be shown that the complete permutation group Sn can be generated by the following two elements: () and ( n). (.7) We will show this below for S 4. Now, since we know already that it is possible to construct this permutation group starting with the set {(), (), (4)}, we only must demonstrate the construction of the group elements () and (4) using the generator set {(), (4)}, i.e.: () = ()(4)()(4)(4)()(4)(4)() and (4) = ()(4)(4)()(4)(4)(). Similarly, one can achieve comparable results for any symmetric group Sn. 7

13 . Partitions and Young diagrams The cyclic structure of a group element of Sn can be represented by a partition of n. A partition of n is a set of positive integer numbers: with the following properties: [n, n,, n k ] (.8) n n n k, and n +n + +n k = n. Such partitions are very often visualized by means of Young diagrams as shown below. A way of representing a partition [n, n,, n k ], is by means of a Young diagram, which is a figure of n boxes arranged in horizontal rows: n boxes in the upper row, n boxes in the second row,, n k boxes in the k-th and last row, as shown in the figure.. n k n n Let us assume that a group element of Sn consists of a n -cycle, followed by a n -cycle, followed by..., followed by a n k -cycle. And let us moreover assume that those cycles are ordered according to their lengths, such that the largest cycle (n ) comes first and the smallest cycle (n k ) last. In that case is the cyclic structure of this particular group element represented by the partition (.8) or, alternatively, by the above Young diagram. In table (.5) are represented the partitions and Young diagrams for the several cyclic structures of S. Cyclic notation partition Young diagram ()()() [ ]=[] () () [] ()() ()() ()() [] Table.5: The representation by partitions and Young diagrams of the cyclic structures of the permutation group S. 8

14 .4 Subgroups and cosets It might be noticed from the multiplication table (.) at page 5 of S, that if one restricts oneself to the first three rows and columns of the table, then one finds the group A of symmetric permutations of three objects. In table (.6) those lines and columns are collected. A b I () () a ab I I () () () () () I () () I () Table.6: The multiplication table of the group A of symmetric permutations of three objects. Notice that A = {I, () ()} satisfies all conditions for a group: existence of a product, existence of an identity operator, associativity and the existence of the inverse of each group element. A is called a subgroup of S. Other subgroups of S are the following sets of operations: {I, ()}, {I, ()}, and {I, ()}. (.9) A coset of a group G with respect to a subgroup S is a set of elements of G which is obtained by multiplying one element g of G with all elements of the subgroup S. There are left- and right cosets, given by: gs represents a left coset and Sg a right coset. (.) In order to find the right cosets of S with respect to the subgroup A, we only have to study the first three lines of the multiplication table (.) at page 5 of S. Those lines are for our convenience collected in table (.7) below. A b I () () () () () a ab I I () () () () () () () () I () () () () () I () () () () Table.7: The right cosets of S with respect to its subgroup A. Each column of the table forms a right coset A g of S. We find from table (.7) two different types of cosets, i.e.: {I,(),()} and {(),(),()}. Notice that no group element is contained in both types of cosets and also that each group element of S appears at least in one of the cosets of table (.7). This is valid for any group: 9

15 . Either Sg i = Sg j, or no group element in Sg i coincides with a group element of Sg j. This may be understood as follows: Let the subgroup S of the group G be given by the set of elements {g, g,..., g k }. Then we know that for a group element gα of the subgroup S the operator products g gα, g gα,..., g k gα are elements of S too. Moreover we know that in the set {g gα, g gα,..., g k gα} each element of S appears once and only once. So, the subgroup S might alternatively be given by this latter set of elements. Now, if for two group elements gα and g β of the subgroup S and two elements g i and g j of the group G one has gαg i = g β g j, then also g gαg i = g g β g j, g gαg i = g g β g j,... g k gαg i = g k g β g j. Consequently, the two cosets are equal, i.e. {g gαg i, g gαg i,..., g k gαg i } = {g g β g i, g g β g i,..., g k g β g i }, or equivalently Sg i = Sg j.. Each group element g appears at least in one coset. This is not surprising, since I belongs always to the subgroup S and thus comes g = Ig at least in the coset Sg.. The number of elements in each coset is equal to the order of the subgroup S. As a consequence of the above properties for cosets, one might deduce that the number of elements of a subgroup S is always an integer fraction of the number of elements of the group G, i.e.: order of G = positive integer. (.) order of S A subgroup S is called normal or invariant, if for all elements g in the group G the following holds: One might check that A is an invariant subgroup of S. g Sg = S. (.)

16 .5 Equivalence classes Two elements a and b of a group G are said to be equivalent, when there exists a third element g of G such that: This definition of equivalence has the following properties: b = g a g. (.). Each group element g of G is equivalent to itself, according to: g = Ig = g g g. (.4). If a is equivalent to b and a is also equivalent to c, then b and c are equivalent, because according to the definition of equivalence (.) there must exist group elements g and h of G such that: c = h ah = h g bgh = (gh) b(gh), (.5) and because the product gh must represent an element of the group G. Because of the above properties (.4) and (.5), a group G can be subdivided into sets of equivalent group elements, the so-called equivalence classes. S b I () () () () () a b ab I I I I I I I () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () Table.8: The equivalence operation for all elements of the group S. In table (.8) we have constructed all possible group elements g ag which are equivalent to group element a of the group S. Every horizontal line in that table contains the group elements of one equivalence class of S. Notice from table (.8) that group elements which have the same cyclic structure are in the same equivalence class of S. Whereas, group elements with different cyclic structures are in different equivalence classes of S. In table (.5) at page 8 are separated the group elements of S which belong to different equivalence classes, as well as the representation of their cyclic structure by partitions and Young diagrams. From that table it might be clear that for S and similarly for the other permutation groups, the equivalence classes can be represented by partitions and Young diagrams.

17 As an example of an other permutation group, we collect in table (.9) the equivalence classes and their representations by partitions and Young diagrams of the group S 4. Equivalence class partition Young diagram ()()()(4) [] ()()(4) ()()(4) ()(4)() ()()(4) ()()(4) (4)()() [] ()(4) ()(4) (4)() [] ()(4) ()(4) ()(4) ()(4) (4)() (4)() (4)() (4)() (4) (4) (4) (4) (4) (4) [] [4] Table.9: The equivalence classes and their representation by partitions and Young diagrams of the permutation group S 4.

18 Chapter Representations of finite groups. In this chapter we will discuss the definition and the properties of matrix representations of finite groups.. What is a matrix representation of a group? An n-dimensional matrix representation of a group element g of the group G, is given by a transformation D(g) of an n-dimensional (complex) vector space Vn into itself, i.e.: D(g) : Vn Vn. (.) The matrix for D(g) is known, once the transformation of the basis vectors ê i (i =,,n) of Vn is specified.. The word-representation D (w) of S. Instead of the three-letter words introduced in section (.), we represent those words here by vectors of a three-dimensional vector space, i.e.: ABC A B C = A +B +C. (.) Moreover, the operations (.) are translated into vector transformations. For example, the operation of () on a three-letter word, given in (.), is here translated into the transformation: D (w) (()) A B C = B A C. (.) Then, assuming arbitrary (complex) values for A, B and C, the transformations D (w) (g) can be represented by matrices. In the above example one finds:

19 D (w) (()) =. (.4) Similarly, one can represent the other operations of S by matrices D (w) (g). In table (.) below, the result is summarized. g D (w) (g) g D (w) (g) g D (w) (g) I () () () () () Table.: The word-representation D (w) of S. The structure of a matrix representation D of a group G reflects the structure of the group, i.e. the group product is preserved by the matrix representation. Consequently, wehaveforanytwogroupelementsaandbofgthefollowingproperty of their matrix representations D(a) and D(b): D(a)D(b) = D(ab). (.5) For example, for D (w) of S, using table (.) for the word-representation and multiplication table (.) at page 5, we find for the group elements () and (): D (w) (())D (w) (()) = = = D (w) (()) = D (w) (()()). (.6) Noticefromtable(.)that all groupelements ofs arerepresented by adifferent matrix. A representation with that property is called a faithful representation. Notice also from table (.) that I is represented by the unit matrix. This is always the case, for any representation. 4

20 . The regular representation. In the regular representation D (r) of a finite group G, the matrix which represents a group element g is defined in close analogy with the product operation of g on all elements of the group. The dimension of the vector space is in the case of the regular representation equal to the order p of the group G. The elements ê i of the orthonormal basis {ê,,êp} of the vector space are somehow related to the group elements g i of G. The operation of the matrix which represents the transformation D (r) (g k ) on a basis vector ê j is defined as follows: D (r) p (g k ) ê j = δ(g k g j,g i )ê i, (.7) i = where the Kronecker delta function is here defined by: if g k g j = g i δ(g k g j,g i ) =. if g k g j g i Let us study the regular representation for the group S. Its dimension is equal to the order of S, i.e. equal to 6. So, each element g of S is represented by a 6 6 matrix D (r) (g). For the numbering of the group elements we take the one given in table (.) at page. As an explicit example, let us construct the matrix D (r) (g = ()). From the multiplication table (.) at page we have the following results: g g = g, g g = g, g g = g, g g 4 = g 5, g g 5 = g 6, g g 6 = g 4. Using these results and formula (.7), we find for the orthonormal basis vectors ê i of the six-dimensional vector space, the set of transformations: D (r) (g )ê = ê D (r) (g )ê = ê D (r) (g )ê = ê D (r) (g )ê 4 = ê 5 D (r) (g )ê 5 = ê 6 D (r) (g )ê 6 = ê 4. (.8) Consequently, the matrix which belongs to this set of transformations, is given by: 5

21 D (r) (g ) =. (.9) The other matrices can be determined in a similar way. The resulting regular representation D (r) of S is shown in table (.) below. g D (r) (g) g D (r) (g) I () () () () () Table.: The regular representation D (r) of S. Notice from table (.) that, as in the previous case, each group element of S is represented by a different matrix. The regular representation is thus a faithful representation. From formula(.7) one might deduce the following general equation for the matrix elements of the regular representation of a group: [ D (r) ] (g k ) ij = δ(g k g j,g i ). (.) 6

22 .4 The symmetry group of the triangle. In this section we study those transformations of the -dimensional plane into itself, which leave invariant the equilateral triangle shown in figure (.). y x and ( ) y ê ( / / ) ê x ( ) / / (a) (b) Figure.: The equilateral triangle in the plane. The centre of the triangle (a) coincides with the origin of the coordinate system. In (b) are indicated the coordinates of the corners of the triangle and the unit vectors in the plane. A rotation, R, of of the plane maps the triangle into itself, but maps the unit vectors ê and ê into ê and ê as is shown in figure (.). From the coordinates of the transformed unit vectors, one can deduce the matrix for this rotation, which is given by: R = / / / /. (.) Twice this rotation (i.e. R ) also maps the triangle into itself. The matrix for that operation is given by: R = / / / /. (.) Three times (i.e. R ) leads to the identity operation, I. Reflection, P, around the x-axis also maps the triangle into itself. In that case the unit vector ê maps into ê and ê maps into ê. So, the matrix for reflection around the x-axis is given by: P = 7. (.)

23 y x and ê ê y x (a) (b) Figure.: The effect of a rotation of of the plane: In (a) the transformed triangle is shown. In (b) are indicated the transformed unit vectors. The other possible invariance operations for the triangle of figure (.) at page 7 are given by: PR = / / / / and PR = / / / / This way we obtain the six invariance operations for the triangle, i.e.:. (.4) { I,R,R,P,PR,PR }. (.5) Now, let us indicate the corners of the triangle of figure (.) at page 7 by the letters A, B and C, as indicated in figure (.) below. A C B Figure.: The position of the corners A, B and C of the triangle. Then, the invariance operations (.5) might be considered as rearrangements of the three letters A, B and C. 8

24 For example, with respect to the positions of the corners, the rotation R defined in formula (.) is represented by the rearrangement of the three letters A, B and C which is shown in figure (.4). A C B R B A C ABC BCA Figure.4: The position of the corners A, B and C of the triangle after a rotation of. We notice, that with respect to the order of the letters A, B and C, a rotation of hasthesameeffectastheoperation()ofthepermutationgroups (compare formula.). Consequently, one may view the matrix R of equation (.) as a two-dimensional representation of the group element () of S. Similarly, one can compare the other group elements of S with the above transformations (.5). The resulting two-dimensional representation D () of the permutation group S is shown in table (.) below. g D () (g) g D () (g) I () / / / / () / / / / () () / / / / () / / / / Table.: The two-dimensional representation D () of S. Notice that D () is a faithful representation of S. 9

25 .5 One-dimensional and trivial representations. Every group G has an one-dimensional trivial representation D () for which each group element g is represented by, i.e.: D () (g)ê = ê. (.6) This representation satisfies all requirements for a (one-dimensional) matrix representation. For example: if g i g j = g k, then D () (g i )D () (g j ) = = = D () (g k ) = D () (g i g j ), in agreement with the requirement formulated in equation (.5). For the permutation groups Sn exists yet another one-dimensional representation D ( ), given by: D ( ) (g) = { for even permutations for odd permutations. (.7) Other groups might have other (complex) one-dimensional representations. For example the subgroup A of even permutations of S can be represented by the following complex numbers: D(I) =, D(()) = e πi/ and D(()) = e πi/, (.8) or also by the non-equivalent representation, given by: D(I) =, D(()) = e πi/ and D(()) = e πi/. (.9)

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