Induced representations

It follows that there are two semi-direct products involving these groups. The rich-ness of the semi-direct product construction is evidenced by looking at the order sixteen groups, where all but one of the nine non-Abelian groups can be expressed either as a direct or semi-direct product

Q8, D8(= Z8−1Z2), Q × Z2, D4× Z2, Z83Z2, Z85Z2, Z43Z4, (Z4× Z2)  Z2, (Z4× Z2) ^Z2.

The last two groups correspond to the action of the actor group on the first and second group we just discussed.

3.11 Induced representations

This is a method which constructs representations of a group using the representa-tions of its subgroups. It is particularly well-suited for groups which are semi-direct products. LetH be a subgroup of G of index N

G ⊃ H. (3.134)

Suppose we know a d-dimensional representation r ofH, that is for every h ∈ H, h : | i  → | i^ = M(h)^[r]_ij| j , (3.135) acting on the d-dimensional Hilbert spaceH, spanned by | i , i = 1, 2, . . . , d. We want to show how to construct a representation ofG, starting from the known rep-resentation onH. It is called the induced representation of G by the r representation ofH.

We begin by considering the coset made up of N points,

H ⊕ g1H ⊕ g2H ⊕ · · · ⊕ gN−1H, (3.136) where gkare elements ofG not in H. Consider the set of elements

G × H : ( g, | i  ); g∈ G, | i  ∈ H. (3.137) In order to establish a one-to-one correspondence with the coset, we assume that whenever theG group element is of the form

g= gkh_a, (3.138)

we make the identification

g_kh_a, | i 

=

g_k, M(ha)^[r]i j | j 

, (3.139)

so that both g and gh are equivalent in the sense that they differ only by reshuffling H. In this way, we obtain N copies of H, the Hilbert space of the r representation ofH, one at each point of the coset, which we take to be

( e, H ), ( g1, H ), ( g2, H ), · · · , ( gN−1, H ). (3.140) The action ofG on this set is simply group multiplication

g: ( gk, | i  ) → ( ggk, | i  ), g ∈ G, k = 1, 2, · · · , N. (3.141) Suppose that g= ha, the coset decomposition tells us that

h_ag_k = glh_b, (3.142)

where gl and hbare uniquely determined. Hence, h_a :  We have shown that these N copies of the vector space are linearly mapped into one another under the action ofG. The action of G is thus represented by a (d N × d N) matrix. It may or may not be irreducible.

In the special case whenH is Abelian, its irreducible representations are one-dimensional, and the induced representation onG by an irrep of H is in terms of (N × N) matrices. This suggest that we think of the induced representation as matrices in block form,(N × N) matrix, whose elements are themselves (d × d) matrices.

For the more mathematically minded, we define N mapping (called sections) which single out N vectors inH:

f_l : ( gk, | i  ) −→

This defines a basis, and we can arrange these N vectors into a column, which is being acted on by the induced representation.

3.11 Induced representations 63 As an example, consider the group of order 3n²

(3n²) = (Zn× Zn)  Z3, (3.147) generated by a ∈ Z3, c∈ Znand d ∈ Zn, with the presentation

< a, c, d | a³= cⁿ= dⁿ= e, cd = dc, aca⁻¹= c⁻¹d⁻¹, ada⁻¹= c > . (3.148) Let us construct the representation induced by the one-dimensional representation ofH = Zn× Zn, for which

c= η^k; d = η^l, ηⁿ = 1, (3.149)

acting on complex numbers z. The coset contains just three points, and we consider the action ofG on

( e, z ), ( a, z ), ( a², z ). (3.150) Then

a( e, z ) = ( a, z ), a( a, z ) = ( a², z ), a( a², z ) = ( e, z ), (3.151) so that a is represented by the(3 × 3) permutation matrix

a=

⎛

⎝0 1 0 0 0 1 1 0 0

⎞

⎠ . (3.152)

Next, we consider the action of c

c( e, z ) = ( c, z ) = ( e, cz ) = ( e, η^kz) = η^k( e, z ). (3.153) Using

ca= ad; ca²= a²c⁻¹d⁻¹, (3.154) derived from the presentation, we find

c( a, z ) = ( ca, z ) = ( ad, z ) = ( a, dz ) = ( e, η^lz) = η^l( e, z ), (3.155) as well as

c( a², z ) = ( ca², z ) = ( a²c⁻¹d⁻¹, z ) = ( a², c⁻¹d⁻¹z)

= ( e, ¯η^k¯η^lz) = η^−k−l( a², z ). (3.156) Similarly, the action of d can be derived, using

da= c⁻¹d⁻¹; da²= a²c. (3.157)

The above show that c and d are represented by the diagonal matrices

completing the induced(3 × 3) representation.

In general, the induced representation is reducible, but when one builds it out of a subgroup that has a low index, one has a better chance of obtaining an irrep from this procedure. This is the case for this example.

Alas, it is not so easy to work out the character tables for more complicated groups. One runs out of tricks, and must rely on numerical methods. Fortunately for us, character tables and other useful group properties have been tabulated in the Atlas of Simple Groups for the simple groups, in book form (Oxford University Press) and also on the internet: http://brauer.maths.qmul.ac.uk/Atlas.

A word of caution about the ATLAS’s dot notation. By A4· 2, they mean the group S4 ⊃ A4, and derive specific matching rules for their irrepsA5 → S4: 1→ 1 + 11, 11+ ¯11→ 2, and 3 → 31+ 32.

On the other hand, the group 2· A4 is the binary extension or double cover of A4, to be discussed in detail later, but it does not containA4as a subgroup; rather A4is the quotient group of the binary tetrahedral group withZ2.

It is useful to tabulate group properties in a convenient form that general-izes the tables of Thomas and Wood. In Appendix 1, we gather in one place important information (presentation, classes, character table, Kronecker products, embeddings, representations) for groups which appear ubiquitously in physical applications.

3.12 Invariants

In physics, invariance manifests itself when the Hamiltonian (or action in field theory) can be expressed in terms of group invariants. Typically, the Hamilto-nian is a function of the dynamical variables which transform as some repre-sentation of the group. Mathematically, this amounts to constructing invariants out of polynomials built out of group representations. Invariants appear when-ever the (multiple) Kronecker products of representations contains 1, the singlet irrep.

For instance, the A4 Kronecker products show two quadratic invariants in the product of two irreps

[3 × 3]singlet, 11× ¯11. (3.159)

3.12 Invariants 65 There are more ways to form cubic invariants. We can form cubic invariants in four

ways 

where the representation subscript denotes the projection of the product on that representation: 3 appears in both the symmetric and antisymmetric product of two triplets. The second cubic invariant requires at least two triplets.

The same reasoning leads to four quartic invariants built out of triplets alone:

(3 × 3)^sym₃ × (3 × 3)^sym₃  but they are not all independent. This procedure, while very useful to identify invariants in the product of representations, produces too many redundancies.

How do we find out how many of those are independent? Furthermore, their actual construction requires detailed knowledge encoded in the Clebsch–Gordan decompositions.

The identification and construction of finite group invariants is clearly a formidable task, but it is tractable. The determination of invariants constructed out of several irreducible representations, can be somewhat alleviated by the fact that all irreps of most groups can be generated by the Kronecker products of its fun-damental irrep. There are exceptions to this rule: in some cases, these products generate different irreps always in the same combination, making it impossible to distinguish them by this construction, but in practice a good first step is to find all invariants made out of the fundamental irrep.

Consider a finite groupG of order N, with n irreducible representations ra, with r1 = 1 the singlet. We express the k-fold symmetric Kronecker product of any irrep as a sum of irreps as

(ra× ra× · · · × ra)s

The integer coefficientN^[k](rb; ra) denotes the number of rb irreps in the k-fold product of rairreps. Consider Molien’s remarkable generating function

M(rb; ra; λ) = 1

where i labels the class Ci of G, with ni elements. A^[r_i ^a] is any group element

yields the desired coefficients. When we set rb to be the singlet irrep, the Molien function yields the number of possible invariants constructed out of one irreducible representation

whereN^[k] denotes the number of invariants of order k. What makes the Molien function particularly useful is that it can always be written in the form

M(1; r; λ) = (1 +# d_kλ^k)

(1 − λ^a1)ⁿ¹(1 − λ^a2)ⁿ²· · ·, (3.164) where the numerator is a finite polynomial in λ, and the dk, ak and nk are pos-itive integers. The numerator yields dk invariants of order k, while expansion of each factor in the denominator generates nkinvariants of order ak, etc. This infinite number of invariants can be expressed as products of a finite set of basic invariants, but the Molien expansion does not single them out. However, the subset of powers in the denominators ak which are smaller than those in the numerator determine invariants of lesser order, and those must form the bulk of the basic invariants, but whenever they are comparable, the distinction becomes blurred. To make matters worse, invariants satisfy non-linear relations among themselves called syzygies.

In some cases, it can even happen that the invariants of order a1, a2, etc., from the denominator of the Molien function, do not satisfy syzygies among themselves.

They are called free invariants. The powers in the numerator then refer to con-strained invariants which satisfy syzygies with the free invariants. Unfortunately, this neat distinction among invariants is of limited validity.

We apply this approach to the tetrahedral groupA4, with four irreps, 1, 11, ¯11, and 3. Using its character table, it is an easy matter to compute the Molien function for each. For the singlet irrep, we have of course

M(1; 1; λ) = 1

1− λ, (3.165)

3.13 Coverings 67 corresponding to the one-dimensional trivial invariant. Applied to 11, the other one-dimensional representations, we find

M(1; 11; λ) = M(1; ¯11; λ) = 1

1− λ³, (3.166)

yielding one invariant of cubic order with the 11. Calling its one component z, it means that z³is invariant, as one can easily check.

For the triplet, a straightforward computation of the Molien function yields M(1; 3; λ) = 1+ λ⁶

(1 − λ²)(1 − λ³)(1 − λ⁴). (3.167) We infer three free invariants, of order 2, 3, and 4, and one sixth-order constrained invariant. There is no easy way to construct these invariants, so we simply state their expression in terms of the three real coordinates x_i, i = 1, 2, 3 spanning the triplet irrep. The quadratic invariant is the length of the vector, as in S O(3)

2 = 

x₁²+ x2²+ x3²

. (3.168)

The free cubic and quartic invariants are found to be

3 = x1x₂x₃, 4 = x1⁴+ x2⁴+ x3⁴, (3.169) and the constrained sixth-order invariant is

6 =

x₁²− x2²

 x₂²− x3²

 x₃²− x1²

. (3.170)

Its square is related by one syzygy, to sums of polynomials in the free invariants:

46²−24³+1083⁴+2⁶+3643²2−203²2³+54²2²−442⁴= 0, (3.171) which gives an idea of the level of complication.

3.13 Coverings

We have noted the close similarity between the dihedral and dicyclic groups, indeed their presentations are almost the same

Dn: < a, b | aⁿ= b²= e ; bab⁻¹= a⁻¹>, Q2n : < a, b | a²ⁿ = e ; aⁿ= b²; bab⁻¹= a⁻¹> .

In both, aⁿ and b²are equal to one another, but in the dicyclic group, they are of order two. HenceQ2n contains aZ2 subgroup generated by b², called the Schur multiplier. In symbols (following the ATLAS), we write

Q2n = 2 · Dn, and we say thatQ2nis the double cover ofDn.

This is a special case of a general construction. Let G be a finite group gen-erated by a set of elements ai (letters), with presentation defined by a bunch of relationswα(a1, a2, . . . ) = e (words). We define a new group G^where the words are no longer equal to the unit element, and commute with the letters, that is, wα(a1, a2, . . . ) = e, but such that they commute with the generators of G, that is,wαai = aiwα. Thewα generate a subgroup C of G^, which commutes with G.

G^is called the covering group of G by a central subgroup C with G= G^/C. It is clearly not simple, and the Schur multiplier is the intersection of C with its com-mutator subgroup. In the dicyclic group, there is only onew of order two. Dicyclic groups have spinor representations while the dihedral groups does not. For physi-cists, this is a familiar situation: the spinor representations are obtained by going from S O(3) to SU(2) where SU(2) is the cover of SO(3).

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