Irreducible invariant 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... 

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. 

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. ... 

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

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. 

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. 

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... 

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. 

The representation is called unitary if p(g) is a unitary operator for all g. The representation p g) is called irreducible if a non-trivial invariant subspace V does not exist4. We are now ready to introduce the Heisenberg group, which is the main object of this section. 

After the factorization, one can derive SALCs within each invariant subspace via the action of the projection operator. The projection operator for irreducible representation j is defined as... 

Although Fq describes a set of equivalent elements, it is not degenerate, since it can be further reduced into invariant subspaces. For the case of a triangle this representation gives rise to two irreducible representations (irreps) of Csv... 

It is then demonstrated that tire irreducible representations of the symmetry group G decompose the electronic state space of the CNT into invariant subspaces in which die eigenfunctions of H act as bases. 

Weyl answered the first point with the insight that all irreducible representations of the special linear group can be made as invariant subspaces of tensor powers of the underlying standard representation. They were conceived as operations of the linear group transformations with a determinant on a geometrical coordinate space. Any representation of the linear group can be characterized with a tensor product of the coordinate space by a symmetry property. 

The Wigner matrices multiply just like the rotations themselves. There is a one-to-one correspondence between the Wigner matrices of index l and the rotations R. These matrices form a representation of the rotation group. In fact, since the 2/ + 1 spherical harmonics of order / form an invariant subspace of Hilbert space with respect to all rotations, it follows that the matrices D ,m(R) form a (21 + 1) dimensional irreducible representation of the rotation R. Explicit formulas for these matrices can be found in books on angular momentum (notably Edmunds, 1957). 

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-... 

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. 

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... 

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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