Unitary Subgroup Research Articles

MacMahon's Master Theorem, Double Tableau Polynomials, and Representations of Groups

It is shown that MacMahon's master theorem gives the diagonal elements of a class of irreducible representations of the general linear group,Gl(n,C), and hence the trace of these representations, or the group characters. These representations are unitary under restriction to the unitary subgroup and constitute the so-called totally symmetric representations. A generalization of MacMahon's master theorem that generates the elements of the representation matrix themselves is given. These relations for groups are, in fact, more general in that they apply to arbitrary matrices of indeterminates. These results are proved by using the properties of a class of Young–Weyl tableau polynomials defined overn2indeterminates. Forn=2, two sets of basic polynomials are defined in terms of Young–Weyl tableaux. The first set corresponds to the unitary irreducible representations of U(2), as defined in the physics literature and whose properties comprise a substantive part of what is known as the quantum theory of angular momentum; the second set is Rota's double standard tableau polynomials. These two sets of polynomials span the same space, and a comprehensive set of relations between them is given. A further generalization of MacMahon's master theorem pertaining to the general polynomial representations ofGL(n,C), as enumerated by partitions, is conjectured and is shown to contain the classical Schur function generating function as a special case.

A basis for the unitary subgroup of the group of units in a finite commutative group algebra

A basis for the unitary subgroup of the group of units in a finite commutative group algebra

Unitary subgroup of integral group rings

Let A be finite abelian group and G = A x abn, b2 = 1, ab = a-1, a I A. We find generators up to finite index of the unitary subgroup of ZG. In fact, the generators are the bicyclic units. For an arbitrary group G, let B2(ZG) denote the group generated by the bicyclic units. We classify groups G such that B2(ZG) is unitary.

Unitary subgroup of integral group rings

Unitary subgroup of integral group rings

Ordinary and modular characters of SU(3, p)

This article deals with the relationship between ordinary and p-modular characters of the group SU(3, p), parallel to the treatment of SL(3, p) in J. E. Humphreys, Ordinary and modular characters of SL(3, p), J. Algebra72 (1981), 8–16. Here we denote by SU(3, q) the special unitary subgroup of SL(3, q2), where q is a power of the prime p; the notation SU(3, q2) is perhaps more standard, but we want to emphasize the comparison with SL(3, q). Our main conclusion is that the decomposition behavior is essentially the “same” for both groups, in a sense to be made more precise below.

Unitary subgroup of the multiplicative group of a modular group algebra of a finite abelian p-group

Unitary subgroup of the multiplicative group of a modular group algebra of a finite abelian p-group

Unitary subgroup of the multiplicative group of the integral group ring of a cyclic group

Unitary subgroup of the multiplicative group of the integral group ring of a cyclic group

The evaluation of irreducible polynomial representations of the general linear groups and of the unitary groups over fields of characteristic 0

We describe an efficient method for the computer evaluation of the ordinary irreducible polynomial representations of general linear groups using an integral form of the ordinary irreducible representations of symmetric groups. In order to do this, we first give an algebraic explanation of D. E. Littlewood's modification of I. Schur's construction. Then we derive a formula for the entries of the representing matrices which is much more concise and adapted to the effective use of computer calculations. Finally, we describe how one obtains — using this time an orthogonal form of the ordinary irreducible representations of symmetric groups — a version which yields a unitary representation when it is restricted to the unitary subgroup. In this way we adapt D. B. Hunter's results which heavily rely on Littlewood's methods, and boson polynomials come into the play so that we also meet the needs of applications to physics.

A COMPLETE CLASSIFICATION OF THE REPRESENTATIONS OF $ \mathrm{GL}(\infty)$ CONTAINING THE IDENTITY REPRESENTATION OF THE UNITARY SUBGROUP

It is proved that a previously known list of indecomposable spherical functions of the group that are bilaterally invariant with respect to the unitary subgroup is complete.Bibliography: 9 titles.

Isomorphisms of unitary groups over associative rings

Automorphisms of unitary linear groups u (n,Λ,q) over the ring Λ are found to be standard on an elementary unitary subgroup in the case when the hyperbolic rank of the form q is strictly greater than one and n⩾5 (for a commutative ring, n⩾4).

Phase conventions consistent with duality between SUn and S L

The Schur–Weyl construction of irreducible tensors by symmetrized Lth rank exterior products of a defining n-dimensional vector space establishes a duality between the coupling algebra of the unitary group Un and the symmetric group SL . The coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in Un and association in SL . The unitary group is factored by its unitary unimodular subgroup SUn≂Un/U1 so it is usual to identify the coupling algebras of Un and SUn by choosing the trivial (phaseless) coupling for U1. This establishes a set of equivalence relations since all pseudoscalar irreducible representations (irreps) of Un subduce onto the scalar irrep of SUn. The question remains can this trivial identification with the coupling algebra of SUn be made consistent with the symmetries which follow from duality? Specifically the symmetries which follow from duality must be consistent with the equivalence relations. We argue such consistency may be obtained. The resulting symmetries place significant restrictions on the resolution of multiplicities and the choice of phase conventions for transformations which in the absence of duality are normally considered independently. We review the usual formulation of a coupling algebra and the identifications which follow from duality using a double coset decomposition of SL . Particular attention is given to the phase transformation introduced by a transposition in order of the two component irreps being coupled. Phase transformations needed for association and complex conjugation are examined to determine the symmetries they must exhibit to be consistent with duality. Two phase transformations needed for complex conjugation, the Derome–Sharp matrix and the 1jm factor, are shown to be related by association. A transposition phase convention and an association phase convention are proposed and shown to be consistent with all the symmetries required by duality.

Description of representations of a group of invertible operators of a Hilbert space that contain the identity representation of the unitary subgroup

Description of representations of a group of invertible operators of a Hilbert space that contain the identity representation of the unitary subgroup

Integrity bases for corepresentations of magnetic crystal point groups

A relation between integrity bases of invariants of an unitary-antiunitary group G( H) and integrity bases of invariants of its unitary subgroup is described. Images of corepresentations of magnetic crystal point groups are listed and integrity bases are given for them.

Antiunitary symmetry operators in quantum mechanics

A criterion to decide that some symmetries of a quantum system must be realized as antiunitary operators is given. It is based on some mathematical theorems about the second cohomology group of the symmetry group when expressed in terms of those of a normal subgroup and the corresponding factor group. It is also shown that this criterion implies that the only possibility for the unitary subgroup in the Galilean case is that generated by the space reflection and the connected component containing the identity; otherwise only massless systems would arise.

Selection rules for P n3′n

Clebsch–Gordan coefficients for the type II Shubnikov space group Pn3′n are calculated in terms of such coefficients for the unitary subgroup Pn3n.

Selection rules for type II Shubnikov space groups

Clebsch–Gordan coefficients for type II Shubnikov space groups, containing the inversion as point group operation are expressed by simple formulas in terms of convenient Clebsch–Gordan coefficients for the unitary subgroup.

The reduction of symmetrized powers of corepresentations of magnetic groups

The problem of reducing the symmetrized powers of the corepresentations of a magnetic group is solved by relating it to the analogous problem for representations of the unitary subgroup. Use is made of a formula given by Littlewood (1950) for the symmetrization of the direct sum of two matrices. A procedure is also given for the complete reduction of a reducible corepresentation which assumes that the unitary part has already been completely reduced to canonical form.

Clebsch-Gordan coefficients for nonunitary groups

It is shown that calculating Clebsch-Gordan coefficients of a nonunitary group can be reduced to formulas containing only representations of the unitary subgroup and additional conditions due to the antiunitary symmetry. This is another example demonstrating that, in applications involving corepresentations of nonunitary groups, the analysis can be made mainly in terms of representations of its unitary part.

Time-Reversal Degeneracies in the Band Structure of a Ferromagnetic Metal

Symmetry of a single domain of a ferromagnetic metal should be described by the appropriate magnetic (Shubnikov) space group M. Previous workers have used only the unitary subgroup of M. It is shown here that the inclusion of the antiunitary elements of M may lead to some degeneracies, beyond those noted previously, in the electronic band structure for a hexagonal close-packed metal, but not for a cubic metal.

Magnetic Groups and Their Corepresentations

A review is given of the theory of magnetic groups and of their unitary corepresentations. Particular application is made to magnetic space groups, this part of the work being set in the framework of little-group theory. The symmetry problems in physics which lead to magnetic groups are analyzed and various applications of the theory to such problems are pointed out. Finally a method is given for obtaining the Kronecker products of corepresentations of magnetic groups, and an example is presented in which the unitary subgroup of the magnetic group is the point group ${\mathrm{C}}_{2v}$.