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Representation Theory of the Symmetric Group
Abstract
No group is of greater importance than the symmetric group. After all, any group can be embedded as a subgroup of a symmetric group. In this chapter, we construct the irreducible representations of the symmetric group S
n
. The character theory of the symmetric group is a rich and important theory filled with important connections to combinatorics. One can find whole books dedicated to this subject, cf. [16, 17, 19, 12]. Moreover, there are important applications to such diverse areas as voting and card shuffling [7, 8, 3].
... As the n × n permutation matrices over C are a faithful representation of the symmetric group of all permutations of n letters [7], there will be n! such matrices. In the case of a system of N qubits, the time evolution operator is a 2 N × 2 N matrix, and thus we consider all permutation matrices of the same size as potential goal operators. ...
We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Certain time bounds are known, but these are not tight bounds. For computation, we also need to consider bounds in the presence of control functions. Here, we use a numerical search approach to discover specific optimal control schemes. We present results for two coupled spins controlled in two scenarios: (i) a single control field influencing each spin separately; (ii) two orthogonal control fields influencing each spin.
The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.
The present section is central for the development of the theory of composite nucleon systems. We start in section 6.2 from an interacting system of n nucleons and introduce a partition of n to specify j occupation numbers (n1 n2 ... nj). Using these occupation numbers we pass from the coordinates x1 x2 ... xn of the single particles to j center coordinates z and n - j internal coordinates y. Next we specify the orbital state with respect to the internal coordinates, establish the orbital partition through the application of an orbital Young operator and couple the orbital state with the spin-isospin state to an antisymmetric state. In section 6.3 we consider linear superpositions of configurations as described above. From the stationary variational principle we develop by a variation restricted to the state u(z) describing the relative motion of the composite particles, the integral equations which govern this state. These integral equations are characterized by kernels of interaction and normalization operators which depend on the choice of the internal states, on the choice of the orbital partition and on the nucleon-nucleon interaction. The kernels of these operators are expressible as matrix elements between particular orbital states involving, with respect to center coordinates z, the coherent states described in section 4.3. In section 6.4 we develop the concept of a distribution of n particles into j shell configurations which are then subject to translations in phase space.
In the present section, we extend the study of composite nucleon systems to more complex internal states. In section 8.2, we discuss algebraic and analytic properties of the overlap matrix between single-particle states. By a modification of the single-particle basis, we arrive at a new biorthogonal basis which leads to a simpler form of the matrix ε. This modification is implemented through the generalized Weyl operators introduced in section 4.4. For three important configurations we determine the matrix ε explicitly. In sections 8.3 and 8.4, we examine the interaction of a simple composite particle with a composite particle having an internal closed oscillator shell. The corresponding configuration covers the interaction of a single nucleon, a deuteron, a 3H, 3He or a 4He nucleus with the closed-shell nuclei 4He, 16O and 40Ca. We obtain the normalization kernel, examine its exchange decomposition and compute its eigenfunctions and eigenvalues. The eigenvalues reflect the action of the Pauli principle on the composite particle interaction as a function of the relative excitation and the mass numbers of the fragments. In section 8.5, we study the configuration s4p12 + s4p12 which covers the interaction of 16O + 16O. Particular emphasis is given to the exchange properties of the normalization operator and to the accessibility of compound states in 32S. The interaction of a simple composite particle with a composite particle having an open oscillator shell is examined in section 8.6. It is shown that the necessary modifications of the scheme involve shell model concepts related to the introduction of angular momentum for shell configurations. The configurations are arranged in the order of increasing complexity.
In principle it is possible to analyse the permutational symmetry of orbital states entirely in terms of the symmetric group, and for the nuclear states this approach has been adopted in [KR 69, 69a, 72]. The use of the unitary and general linear groups for the same purpose can be justified both on physical and mathematical grounds.
Simple composite nucleon systems were characterized in section 6.2 by the assumption that the internal orbital state of each composite system be stable under internal permutations. The occupation numbers of these composite particles then form a weight w = (w1w2 ... wj) and the stability group is the group of the weight S(w). The orbital partition f is established through a Gelfand pattern q. To this assumption on the permutational structure of the orbital states we add the specification of any internal state as an unexcited oscillator state. With this choice we obtain in sections 7.2 and 7.3 closed analytic expressions for the normalization and interaction kernels. In section 7.4 we choose j = 3 and pass from the kernels of the operators to the oscillator representation. The oscillator representation will be used in section 10 to study states of the lightest nuclei.
One of the fundamental properties of nuclear systems is that they are composed out of fermions. The variety of nuclear phenomena related to this property can hardly be overestimated. In the present section we elaborate some tools for a more detailed analysis of this property.
At some places in the preceding sections full use has been made of harmonic oscillator state wave functions. But no attention was directed to the internal width of the different composite particle systems. The frequencies of the oscillator states in different degrees of freedom were all chosen to be equal. It was possible therefore to transform from one set of coordinates to another, for example from cluster coordinates to single-particle coordinates, by orthogonal transformations. Even in Bargmann space these could be performed by substitution without explicitly using integral transforms (see section 5.1).
For n particles moving in three-dimensional space, each coordinate should carry an index i = 1, 2,3 denoting the vector component and an index l = 1, 2,..., n denoting the particle number. Among the linear point transformations acting on these coordinates there are two commuting subgroups of transformations. The elements of the first one are given by product matrices M3 × In where the 3 × 3 matrix M3 acts on the vector components. The elements of the second one are of the form I3 × Mn where the n × n matrix Mn acts on the vectors for the different particles. To the first subgroup there belong the rotations giving rise to the rotation group SO (3, IR). The second subgroup contains permutations and transformations to various types of relative coordinates. In the present section we shall deal with this second subgroup and hence shall drop the reference to the three vector components. Moreover we shall need at present only orthogonal n × n transformation matrices.
The character table of a finite group G is constructed
by computing the eigenvectors of matrix equations determined
by the centre of the group algebra. The numerical character
values are expressed in algebraic form. A variant using a
certain sub-algebra of the centre of the group algebra is
used to ease problems associated with determining the
conjugacy classes of elements of G. The simple group of
order 50,232,960 and its subgroups PSL(2,17) and PSL(2,19)
are constructed using general techniques.
A combination of hand and machine calculation gives the
character tables of the known simple groups of order < 106
excepting Sp(4,4) and PSL(2,q). The characters of the non-
Abelian 2-groups of order < 2 6 are computed.
Miscellaneous computations involving the symmetric
group Sn are given.
In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!m ⁿ is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn } which commute with
Obviously, S (1, m ) is a cyclic group of order m , while S ( n , 1) is the symmetric group of order n! . If c i = ( i, n + i , …, ( m – 1) n + i ) and
then { c 1 , c 2 , …, c n } generate a normal subgroup Q(n) of order m ⁿ and { s 1 , s 2 , …, s n …1 } generate a subgroup S(n) isomorphic to S ( n , 1).
Let be a decomposable symmetrized tensor corresponding to the subgroup G of Sm and the irreducible character λ. Various conditions on the vectors are shown either to be necessary or sufficient for . Applications to generalized matrix functions are given.
This paper is the result of our attempt to extend the classical theory of polynomial representations of the general linear group GL,, (~;) and of complex representations of the symmetric group S r to the more general case in which the field of complex numbers is replaced by an infinite field of arbitrary characteristic. The characteristic zero situation is a well-established theory created at the beginning of this century by Young, Schur and Frobenius (see Weyl [15]). The characteristic p case has been investigated by several authors: Brauer, Nesbitt, Thrall, Littlewood, Robinson, Kerber, and others, but the question is far from being completely understood. Let A be an infinite field, and let V be the n-dimensional A-vector space of which GL,,(A) is the group of automorphisms. Let ~| be the r-fold tensor product of V over A. V~r is both a GL,,(A) and an A [S Jmodule 1 (St acts on V| by permuting the factors, and this action clearly commutes with that of GL n(A)). Let 2 be a partition of r into at most n parts. In 3.2 and 3.6 we define subspaces ~ (resp. ~) of V| which are GL,,(A) (resp. A[Sr]) submodules of V| The definitions involve only relations with integral coefficients and we prove that Vx, ~ have dimensions independent of the field A; in fact we prove that the V~, ~ corresponding to a field A of characteristic p >0 can be regarded as the reductions modulo p of the corresponding modules over ~. If A = iI~, ~ can be regarded as the image of the appropriate Young symmetrizer (primitive idempotent in II; [S,]) but this does not hold if char A > 0. We also note that ~ is isomorphic to the classical Specht-module involving Vandermonde determinants. It is however more convenient for us to regard ~ as a subspace of V| Actually we can define ~U~ as the set of tensors in V| which are U-invariant of weight 2. (Here U denotes the group of upper unitriangular matrices in GL,,(A) and we t Given a ring A and a group G we denote by A [G] the group algebra of G over A.
The structure of the current algebra representation in the state space of fermions in an external Yang-Mills field in 3+1 space-time dimensions is analyzed; the topology of the vector space is determined by a countable family of semi-definite inner products. We show that there is no hermitian non-trivial Hilbert space representation such that the energy is bounded from below. The structure of the Hilbert space for the quantized coupled Dirac-Yang-Mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite-dimensional Grassmannians.
We present a fast algorithm for computing the global crystal basis of the basic-module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an
th root of 1.
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