Irreducible subspace

Definition 6.2 Suppose (G, V, p) is a representation and (G. VE, pyy) is a subrepresentation. Suppose that (G, VE, pw) is an irreducible representation. Then we call VE an irreducible subspace or an irreducible invariant subspace... 

Next, we consider the symmetry operations of the system. The free energy is expanded as a function of the strains (as defined above) and the corresponding harmonic polynomials A (a,-). The resulting expression must be invariant under the symmetry transformations. If the symmetry is low enough, one can reduce further the vector space(s) introduced above, by choosing a suitable basis. The resulting irreducible subspaces are indicated... 

Extra considerations are required to construct suitable sets of polynomials, which provide basis functions for the irreducible subspaces of the cubic and icosahedral point groups. Clearly, such a set of central functions is invariant under the point group G. Eor such a function, f, then (fgi, fg2,. .., fgn> is a subspace of the central functions invariant under G. But it is not, in general, an irreducible subspace, i.e. it may contain further subspaces that transform according to different irreducible representations. 

In the case of the icosahedral point groups, Ih and I, Table 3.10, the analysis is more complicated and there is a need to identify the combinations of the spherical harmonics, which will generate higher dimensional irreducibile subspaces. For example, at level 3, there are 7 harmonics, but the irreducible subspaces in icosahedral symmetry are four-fold [Gu] and three-fold [T2u]. It is found that three of the original functions can be carried over to provide basis functions in icosahedral, symmetry but that four distinct linear combinations of... 

In this Appendix, Htickel theory calculations are used to demonstrate that the polynomials of Appendix 1 can be applied as basis functions for the irreducible subspaces of a Hamiltonian invariant under icosahedral point symmetry, while extended Htickel theory calculations on cubium cages of cubic point symmetry are used to demonstrate the same result for the kubic harmonics, since single bond-length regular orbits are not possible in all cases. 

From the above result, it could be inferred exactly that such irreducible subspaces of the state space establish the proper mathematical domain of the classical physical field quantities. In fact, the demonstration was undertaken by using a relativistic electromagnetic field tensor, F,y, and its antisymmetric property ... 

The dimensional space that is spanned by the hi, independent products can be reduced into irreducible subspaces a, 0,, ... 

An obvious feature of a reducible distortion space, is that it can in fact be decomposed into irreducible subspaces. The conserved subgroups of these irreducible subspaces may immediately be determined from Tables 1 and 2. Since all these resulting subgroups are conserved in a part of the total distortion space, they will all be epikernels of the total space. Symmetry reduction thus facilitates the search for epikernels in reducible spaces. 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

In this section we will use the idea of invariant subspaces of a representation (see Definition 5.1) to define irreducible representations. Then we will prove Schur s lemma, which tells us that irreducible representations are indeed good building blocks. 

Definition 6.1 A representation (G, V, p) is irreducible z/to only invariant subspaces are V itself and the trivial subspace 0. Representations that are not irreducible are called reducible. 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

We saw in Section 4.5 that a quantum mechanical system with symmetry determines a unitary representation of the symmetry group. It is natural then to ask about the physical meaning of representation-theoretic concepts. In this section, we consider the meaning of invariant subspaces and irreducible representations. 

We know from numerous experiments that every quantum system has elementary states. An elementary state of a quantum system should be observer-independent. In other words, any observer should be able (in theory) to recognize that state experimentally, and the observations should all agree. Second, an elementary state should be indivisible. That is. one should not be able to think of the elementary state as a superposition of two or more more elementary states. If we accept the model that every recognizable state corresponds to a vector subspace of the state space of the system, then we can conclude that elementary states correspond to irreducible representations. The independence of the choice of observer compels the subspace to be invariant under the representahon. The indivisible nature of the subspace requires the subspace to be irreducible. So elementary states correspond to irreducible representations. More specifically, if a vector w represents an elementary state, then w should lie in an irreducible invariant subspace W, that is, a subspace whose only invariant subspaces are itself and 0. In fact, every vector in W represents a state indistinguishable from w, as a consequence of Exercise 6.6. 

Proposition 6.11 implies that irreducible representations are the identifiable basic building blocks of all finite-dimensional representations of compact groups. These results can be generalized to infinite-dimensional representations of compact groups. The main difficulty is not with the representation theory, but rather with linear operators on infinite-dimensional vector spaces. Readers interested in the mathematical details ( dense subspaces and so on) should consult a book on functional analysis, such as Reed and Simon [RS],... 

Next we must check that Qn is irreducible. Suppose W is a subspace of invariant under Qn. Then W must be invariant under 7 , since for any g e SU (2) and w e W we have, by the definition of the pushforward repre-... 

Since Rn is irreducible, it follows that W is either the zero subspace or is all of P . Hence Qn is irreducible. ... 

Proof. Let w denote a weight vector of weight n. Let W denote the smallest invariant subspace containing w. Since w 0 by the definition of a weight vector, we have W 7 0. Let W be a nontrivial irreducible invariant subspace of W and note that w = 7 0, because otherwise W- - would contain w... 

We know that is an irreducible invariant subspace of P by Proposition 7.2. By Proposition 6.5 and Proposition 7.1 we know that is not isomorphic to any subrepresentation of the Cartesian sum... 

Proposition 7.6 Suppose that V is a nontrivial irreducible invariant subspace of the natural representation of S 0(f) on Lf(S f Then there is a nonnegative integer such that V = y. ... 

From Proposition 5.1 we know that y is an invariant subspace. Since the natural representation of S<9(3) on L (W ) is unitary. Proposition 5.4 implies that is a homomorphism of representations. Since V and y are irreducible, it follows from Schur s Lemma and the nontriviality of n [V] that fit gives an isomorphism of representations from V to y. ... 

Proposition 7.7 Suppose f e I is nonzero and f is a nonnegative integer. Let F denote the one-dimensional subspace ofF spanned by f. Then F is an invariant, irreducible, nontrivial subspace of Furthermore, every invariant, irreducible, nontrivial subspace of has this... 

Hence w is not trivial. Because W is an irreducible invariant subspace, Proposition 7.6 implies that there is a nonnegative integer such that... 

We can define invariant subspaces, subrepresentafions and irreducible representations exactly as we did for groups. 

All of the results of Section 6.1 apply, mutatis mutandis, to irreducible Lie algebra representations. For example, if T is a homomorphism of Lie algebra representations, then the kernel of T and the image of T are both invariant subspaces. This leads to Schur s Lemma for Lie algebra representations. 

Definition 10.8 Suppose G is a group, V is a complex scalar product space and p. G PU (V) is a projective unitary representation. We say that p is irreducible if the only subspace W of V such that [VT] is invariant under p is V itself. 

Thus the eigenstates of HSF are labeled by the partitions [ASP] associated with the irreducible representations of S%v. These [Asp] labels are called permutation quantum numbers. The spin-free Hilbert space Fsp of the Hamiltonian may be decomposed into subspaces FSF([ASF]) invariant to 5 p... 

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