Showing papers on "Symmetric group published in 1978"

The Probability of Generating the Symmetric Group

Abstract: In [2], Dixon considered the following question: \" Suppose two permutations are chosen at random from the symmetric group Sn of degree n. What is the probability that they will generate Sn? \" Actually, Netto conjectured last century that almost all pairs of elements from Sn will generate Sn or An. Dixon showed that this is true in the following sense: The proportion of ordered pairs (x, y) (x, y e Sn) which generate either An or Sn is greater than 1 — 2/(log log n) 2 for all sufficiently large n. In fact this bound is only a very rough result, as Dixon himself suggests. In this paper we shall prove

Minimal degree for a permutation representation of a classical group

Abstract: The minimal degree for a permutation representation of the finite linear groups, and finite classical groups is determined.

Some combinatorial results involving Young diagrams

Abstract: In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

On the transfer in the homology of symmetric groups

Abstract: The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphismin group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

On a conjecture of Carter concerning irreducible Specht modules

Abstract: We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

Calculation of permutation matrices using graphical methods of spin algebras: Explicit expressions for the Serber-coupling case

Abstract: We present a general method, based on the diagrammatic techniques of spin algebras, for the calculation of permutation matrices of two-rowed (two-columned) irreducible representations of the symmetric group ${\mathcal{S}}_{N}$ relative to a basis (or bases) adapted to the subgroup(s) ${{\mathcal{S}}_{N}}_{A}\ensuremath{\bigotimes}{\mathcal{S}}_{N}\ensuremath{\subset}{\mathcal{S}}_{N}$ or to (a) subgroup chain(s) which may be generated by a recursive application of the above-given chain. These matrix elements are needed in spin-adapted configuration interaction calculations. This general technique is applied to the Serber coupling scheme, and general and explicit closed formulas are obtained for the matrix elements of an arbitrary transposition. An extension of this formalism to cases with an arbitrarily large frozen core is also outlined using the same technique. A computer program based on these derivations was written and its effectiveness compared with that of other approaches is discussed. An extension of these applications to more complex permutations than transpositions, as well as to other coupling schemes, is also briefly discussed.

Chern classes of certain representations of symmetric groups

Abstract: A formula is derived for the Chern classes of the representation id / £: P f H -» Up" where P is cyclic of order P and i: H -» U" is a fintie dimensional unitary representation of the group //. The formula is applied to the problem of calculating the Chern classes of the "natural" representa- tions iry. Sj -» Uj of symmetric groups by permutation matrices. 1. Introduction. In (Ch), one of us derived "formulas" for the Chern classes of an induced representation in the case where the inducing representation of the subgroup is 1-dimensional. The formula for the /th Chern class involved a leading term expressed in terms of the multiplicative generalization of transfer in (N) plus additional terms arising from Chern classes Cj(irk) of "natural representations" by permutation matrices. The situation was somewhat un- satisfactory since little or nothing was known about the classes Cj(irk). Recently, through access to an interesting paper of C. B. Thomas (Th), our attention was drawn to this question again. Thomas makes use of estimates of Grothendieck (G) on the orders of Chern classes of representations of discrete groups. Grothendieck's results (G, Corollary 4.11, p. 263) imply that, for a rational representation p: G->GL(rt, Q), the p-primary component of the order of e,(p) is bounded by p° if / 5E 0 mod/? - 1, ^i+,"(,) if/ = 0mod/? - 1,

The cohomology of the symmetric groups

Abstract: Let Sn be the symmetric group on n letters and SG the limit of the sets of degree + 1 homotopy equivalences of the n 1 sphere. Letp be an odd prime. The main results of this paper are the calculations of H*(Sn, Z/p) and H*(SG, Z/p) as algebras, determination of the action of the Steenrod algebra, C?(p), on H*(Sn, Z/p) and H*(SG, Z/p) and integral analysis of H*(Sn, Z, p) and H*(SG, Z, p). 0. Introduction. Let K and L be discrete groups with L abelian. The groups H"(K, L) have been of interest for years. [12] and [11] first considered these cohomology groups algebraically and their relation with topological problems. The algebraic groups H" (K, L) are isomorphic to H" (BK, L) where BK is the topological classifying space for the group K. Suppose K is Sn, the symmetric group on n letters. Then H*(Sn, L) is especially important. In the 1950's, work on cohomology operations, [29] and [30], showed the necessity for knowledge of H*(Spi, Z/p). The construction of the mod p Steenrod operations depends on properties of Sp. Furthermore the Adem relations were derived using the structure of H*(Sp2, Z/p). If L is a ring then H*(K, L) is a graded ring. The homology of symmetric products, [9], [17], [20], [21], and [28], computed the groups H'(Sn, Z/p) as Z/p vector spaces. The graded ring structure, which was not analyzed, becomes important in later problems. There is an interesting link that ties Sn to SG. Recall Q (S) = dir lim 52nSn is the space of "infinite loops of Soo" and SG = dir lim SGn where SGn is the space of degree + 1 homotopy equivalences of S" -'. SG is homotopy equivalent to the + 1 component of Q (SO). THEOREM. (1) There is a canonical map co: BS = dir lim BSQ (S )o inducing integral and modp homology isomorphisms. (2) The inclusions Sn X Sm S> n+m give H*(500) the structure of an algebra. o* is an algebra isomorphism and a Hopf algebra isomorphism mod p where H*(Q (S %) is an algebra under the loop sum product. Received by the editors October 21, 1975 and, in revised form, March 29, 1977. AMS (MOS) subject classifications (1970). Primary 18H10; Secondary 55F40.

On doubly transitive permutation groups

Abstract: Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the stabilizer G α of αhas a set σ = { B 1 , …, B t } of nontrivial blocks of imprimitivity in Ω – {α}. If G α is 3-transitive on σ it is shown that either G is a collineation group of a desarguesian projective or affine plane or no nonidentity element of G α fixes B pointwise.

Recoupling coefficients of the symmetric group involving outer plethysms

Abstract: In a series of papers we have examined the properties of certain double coset matrix elements (DCME) in the representations of the symmetric group SN that act as recoupling coefficients for outer products carried out via alternate subgroup sequences. In this paper we examine these same properties using symmetrized outer products in SN, which are also known as outer plethysms. The notions of double coset representative, symbol, and matrix element are extended to this case using the theory of semidirect products and little groups. The recoupling coefficients between bases symmetry adapted with respect to the usual outer product and the outer plethysm are examined in detail. Because of the Weyl–Schur construction of irreducible tensors, the recoupling theory of SN is central to a unified recoupling theory of the general linear group and its subgroups.

Recoupling coefficients of the general linear group in bases adapted to shell theories

Abstract: Recoupling coefficients for tensor representations of the general linear group Gl(n) are identified with analogous quantities in representations of the symmetric group SN. Two basis labeling schemes in Gl(n) are considered: (a) uses weights and outer product labels from SN, and (b) uses outer plethysms in SN and labels with respect to some elementary subgroup, usually SU(2). Scheme (a) corresponds to a generalized Gel’fand–Tsetlin basis and is the one usually adopted in elementary particle theories. Scheme (b) corresponds to the basis usually adopted in nuclear and atomic shell theory. The transformation between the two equivalent bases is identified with certain weighted double coset matrix elements (WDCME) of SN. Racah factors are generalized isoscalar factors in scheme (a) and have previously been identified with certain WDCME in that basis. In scheme (b) Racah factors determine the coefficients of fractional parentage (CFP) and are here identified with certain double coset matrix elements (DCME) of SN...

Injective modules for group algebras of locally finite groups

Abstract: Two recent results relate the existence of injective modules for group algebras which are ‘small’ in some sense to the structure of the group.(1) The trivial kG-module is injective if and only if G is a locally finite group with no elements of order p = char k (9).(2) If (G) is a countable group, then every irreducible kG-module is injective if and only if G is a locally finite p′ group which is abelian-by-finite (9) and (11)

The use of the symmetric group in the construction of multispinor Lagrangians

Abstract: The construction of Lagrange functions for nth rank multispinor (Bargmann–Wigner) fields is developed using the symmetric group Sn. Restrictions on the number of fields present in the Lagrangian and their couplings to one another are obtained by constructing differential operators which transform irreducibly under Sn. The technique is illustrated by a brief discussion of the second and third rank cases and a more detailed fourth rank example. The Lagrangians thus obtained are precisely those specified by Lorentz invariance, but the method used greatly facilitates their construction.

Some simple homeomorphism groups having nonsolvable outer automorphism groups

Abstract: Let B(ℝ) be the group of homeomcrphisms of the real line having bounded support. Higman has shown that (ℝ) is simple. Schreier conjectured in 1928 that for any simple group, the outer automorphism groiftp is solvable. But it turns out that not only is Outer(ℝ) non-sol table, but its derived series stabilizes after one step. Thus, although the counterexample is infinite, solvability fails as it must in finite groups (rather than because the derived series descends ad infinitum). For another (new) simple group G, Outer(G) is its own derived group. A countable counterexample is also given. Most important among the new simple groups is the group of diffeomorphisms (differentiable, but not necessarily C1) of the real line having bounded support.

Color-charge algebras in Adler's chromodynamics

Abstract: We show that the color-charge algebra in the three-quark sector generated by the matrices of the fundamental representation of U(n) does not have the trace properties required in Adler's extension of chromodynamics. We also discuss a diagrammatic representation of algebras generated by quark and antiquark charges in general, and an embedding of the N-quark algebra in the symmetric group S/sub N/+1.