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Showing papers on "Symmetric group published in 1984"


MonographDOI
01 Jan 1984
TL;DR: In this paper, the authors propose a representation theory of symmetric groups and their young subgroups, which is based on the notion of irreducible matrix representations of groups.
Abstract: 1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetric groups 4. Representations of wreath products 5. Applications to combinatories and representation theory 6. Modular representations 7. Representation theory of Sn over an arbitrary field 8. Representations of general linear groups Appendices Index.

2,445 citations


Journal ArticleDOI
TL;DR: Using the theory of symmetric functions, a formula is found for r(w) when W is the symmetric group Sn and for the element w0 ∈ Sn of longest length and certain other w ∉ Sn the formula is particularly simple.

415 citations


Book
28 Dec 1984
TL;DR: The Representation Theory of the Symmetric Group as mentioned in this paper provides an account of both the ordinary and modular representation theory of the symmetric groups, and its applications are vast, varying from theoretical physics through combinatories to the study of polynomial identity algebras.
Abstract: The Representation Theory of the Symmetric Group provides an account of both the ordinary and modular representation theory of the symmetric groups. The range of applications of this theory is vast, varying from theoretical physics, through combinatories to the study of polynomial identity algebras; and new uses are still being found.

282 citations


Proceedings ArticleDOI
24 Oct 1984
TL;DR: A theory of black box groups is built, and it is proved that for such subgroups, membership and divisor of the order are in NPB, and under a plausible mathematical hypothesis on short presentations of finite simple groups, nom membership and exaact order will also be inNPB.
Abstract: We build a theory of black box groups, and apply it to matrix groups over finite fields. Elements of a black box group are encoded by strings of uniform length and group operations are performd by an oracle. Subgroups are given by a list of generators. We prove that for such subgroups, membership and divisor of the order are in NPB. (B is the group box oracle.) Under a plausible mathematical hypothesis on short presentations of finite simple groups, nom membership and exaact order will also be in NPB and thus in NPB ∩ NPB.

227 citations


Journal ArticleDOI
TL;DR: In this paper, a combinatorial method for the construction of free bases for the symmetric polynomials over the subset lattice has been proposed, where the action of a symmetric group on the Stanley-Reisner ring is studied.

129 citations


Journal ArticleDOI
TL;DR: In this paper, a complete characterization of compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)⊗Mn of continuous functions from X to M n can be continuously diagonalized.

122 citations


MonographDOI
24 May 1984
TL;DR: In this article, the authors examined the representation theory of the general linear groups and revealed that there is a close analogy with that of the symmetric groups, and they suggested many lines for further investigation.
Abstract: The most important examples of finite groups are the group of permutations of a set of n objects, known as the symmetric group, and the group of non-singular n-by-n matrices over a finite field, which is called the general linear group. This book examines the representation theory of the general linear groups, and reveals that there is a close analogy with that of the symmetric groups. It consists of an essay which was joint winner of the Cambridge University Adams Prize 1981-2, and is intended to be accessible to mathematicians with no previous specialist knowledge of the topics involved.Many people have studied the representations of general linear groups over fields of the natural characteristic, but this volume explores new territory by considering the case where the characteristic of the ground field is not the natural one. Not only are the results in the book elegant and interesting in their own right, but they suggest many lines for further investigation.

95 citations


Journal ArticleDOI
TL;DR: In this article, the (k, l, m) triangle group has a presentation δ(k,l, m), where m is the number of vertices in the triangle group.
Abstract: Given positive integers k, l, m, the (k, l, m) triangle group has presentation δ(k, l, m) = . This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

48 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups and obtain asymptotic formulas for the number of blocks of the symmetric group Sn over a field of characteristic p for n → ∞.
Abstract: We give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups. Using this we can obtain asymptotic formulas for the number of blocks of the symmetric group Sn over a field of characteristic p for n → ∞. For p < 7 we give simple explicit formulas for the number of blocks of defect zero. The study of the modular forms leads to interesting identities involving the Dedekind η-function.

36 citations


Journal ArticleDOI
TL;DR: Assuming that the classification theorem for finite simple groups is complete, a conjecture that a Steiner triple system with a doubly transitive automorphism group is a projective or affine geometry, is verified.

28 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that for the case f(B) = 1, the possible defect groups are either cyclic or a Klein four group, and this result can be seen as a generalization of Brauer's result.

Journal ArticleDOI
TL;DR: In this article, an extension of the Schur-Weyl duality connecting the representations of the symmetric and unitary groups is given, and three factorisation lemmas are derived.
Abstract: An extension of the Schur-Weyl duality connecting the representations of the symmetric and unitary groups is given. The Schur-Weyl basis is constructed using annihilation and creation operators. Three factorisation lemmas are derived. Their importance lies in the fact that they relate the phase freedoms within the Racah-Wigner algebra of the symmetric groups and the unitary groups. Extensions of the Regge symmetries are also given. These are expressed in five duality relations.

Journal ArticleDOI
TL;DR: A complete solution is obtained for the problem of determining the invariance groups admitted by the various configurations enumerated, by the use of the conjugate sets of the subgroups of the frame group.
Abstract: Wider extension is here given to the theory developed in an earlier paper, in which the enumeration of certain configurations or combinatorial entities was made to depend on two or more groups (range groups). The present paper recognizes for the first time the essential part played by an additional group, the frame group, which includes all the range groups as subgroups, and which was implicit in the earlier theory as the symmetric group of the degree of the range groups. The use of frame groups which are not symmetric greatly extends the range of application of the theory, and allows its development in terms of abstract groups. By the use of the conjugate sets of the subgroups (both cyclic and noncyclic) of the frame group, instead of merely the conjugate sets of its operations, a complete solution is obtained for the problem (solved only for special cases in the earlier paper) of determining the invariance groups admitted by the various configurations enumerated.



Dissertation
01 Jan 1984
TL;DR: In this article, it was shown that the left, right and two-sided cells in the affine Weyl group A n of type A n - 1 > 2 are T-invariant.
Abstract: In [1], Kazhdan and Lusztig introduce the concept of a W-graph for a Coxeter group W. In particular, they define left, right and two-sided cells. These W-graphs play an important role in the representation theory. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated. These cells have been worked out only in a very few cases. In the present thesis, we shall find all the left, right and two-sided cells in the affine Weyl group A n of type A n - 1 > 2. Our main results show that each left (resp. right) cell of A n determines a partition, say λ of n and, is characterized by a λ-tabloid and also by its generalized right (resp. left ) T-invariant. There exists a one-to-one correspondence between the set of two-sided cells of A n and the set A n of partitions of n. The number of left (resp. right) cells corresponding to a given partition λ ϵ A n is equal to n l / m п j = 1 u j l , where {u 1 > … > um} is the dual partition of λ. Each two- sided cell in A n is also an RL-equivalence class of A n and is a connected set. Each left (resp. right) cell in A n is a maximal left (resp. right) connected component in the two-sided cell of A n containing it. Let P be any proper standard parabolic subgroup of A n isomorphic to the symmetric group S n - then the intersection of P with each two-sided cell of A n is non-empty and is just a two-sided cell of P. The intersection of P with each left (rasp, right) cell of A n is either empty or a left (resp. right) cell of A n. Most of these results were conjectured by Lusstig [2], [3].

Book
01 Dec 1984
TL;DR: An introduction to the complex representations of the symmetric group and general linear Lie algebra by P. Hanlon and G. Billera is given in this paper, where Chen, Chen, A. M. Garsia, and J. C. Greene constructions on rim hook tableaux by D. E. Chen.
Abstract: An introduction to the complex representations of the symmetric group and general linear Lie algebra by P. Hanlon The cyclotomic identity by N. Metropolis and G. C. Rota Arrangements of hyperplanes and differential forms by P. Orlik, L. Solomon, and H. Terao Double centralizing theorems for wreath product by A. Regev On the construction of the maximal weight vectors in the tensor algebra of $\mathrm{gl}_n(\mathbb{C})$ by P. Hanlon The q-Dyson conjecture, generalized exponents, and the internal product of Schur functions by R. P. Stanley Spherical designs and group representations by E. Bannai Algorithms for plethysm by Y. M. Chen, A. M. Garsia, and J. Remmel Combinatorial correspondences for Young tableaux, and maximal chains in the weak Bruhat order of $S_n$ by P. Edelman and C. Greene Constructions on rim hook tableaux by D. E. White Orderings of Coxeter groups by A. Bjorner Modularly complemented lattices and shellability by D. Stanton and M. Wachs Counting faces and chains in polytopes and posets by M. M. Bayer and L. J. Billera The combinatorics of $(k,l)$-hook Schur functions by J. B. Remmel Multipartite P-partitions and inner products of skew Schur functions by I. M. Gessel Problem session.

Journal ArticleDOI
TL;DR: In this paper, it was shown that G can be embedded in the group Aut(A) of the automorphisms of a permutation group, provided that the characteristic of K (denoted by char(K) henceforth) is zero or greater than n + 1.

Journal ArticleDOI
TL;DR: In this article, the decomposition rules of tensor products of the representations of the classical Weyl groups were studied and three conditions were established: condition 1, condition 2, condition 3.

Journal ArticleDOI
TL;DR: In this article, the stabilizer of a point is the Ree group F = 2 F 4 (2) and the subdegrees are 1755 and 2304, and the conjugacy classes of elements of a rank 3 permutation group are determined.

Journal ArticleDOI
TL;DR: In this paper, bounds and asymptotic formulas for the size of the irreducible representations of the symmetric groups of a PI-algebra were given for rings with no right (or left) total annihilators.
Abstract: Bounds and asymptotic formulas are given for the size of the irreducible representations of the symmetric groups. These are applied to obtain information on the identities and codimension sequencec n(R) of a PI-algebraR, of a PI-algebraR of characteristic zero, e.g., the “ultimate” width of the hook in which the diagrams of the cocharacters ofR lies is <=(lim c n (R)1/n ) 2 , and lim cn(R)1/n≦ 2(lim cn(R)1/n)2 for rings with no right (or left) total annihilators.

Journal ArticleDOI
TL;DR: This paper defines operations ψ H and λ H in a β- ring such that whenever the β-ring is a λ-ring, and H a Young subgroup S π of S n , then the β H, ψH and ε H reduce to β π, π and δ H, respectively.

Journal ArticleDOI
TL;DR: In this article, it was shown that the symmetric group S of degree three cannot be embedded subnormally in a finite perfect group, that is, a group which is equal to its own derived subgroup.

Journal ArticleDOI
TL;DR: In this paper, a simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups U(n) and SU(n), including the case of mixed tensor representations.
Abstract: A simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups U(n) and SU(n); including the case of mixed tensor representations. We observe that simultaneity is incompatible with the conventional permutation symmetries for U(n) with n>3. The relevance of these results to the Schur–Weyl duality of U(n) with the symmetric groups Sl is discussed.




Journal ArticleDOI
TL;DR: In this paper, an algorithm for computing d(A) when X corresponds to the partition (2, n-2) was presented. But it is not known whether the algorithm can be used to compute a generalized matrix function.
Abstract: Let X be an irreducible character of the symmetric group S,. For A = (a,J) an n-by-n matrix, define the immanant of A corresponding to X by d(A) = E X(gf)Ha[lt uGSn t=1 The article contains an algorithm for computing d(A) when X corresponds to the partition (2, n-2). Introduction. Denote by Xk the (irreducible, characteristic zero) character of the symmetric group Sn, corresponding to the partition (k, In-k), for k = 1, 2,.. ,n. If A = (ai1) is an n-by-n matrix, define n dk(A) = , Xk(Uf) H at,(t) a(S, t=l Then, for example, dl(A) = det(A) and dn(A) = per(A), the permanent of A. In general, dk is known as an immanant or a generalized matrix function. (An immanant is a generalized matrix function based on Sn.) Suppose G is a (simple) graph on n vertices. Denote by L(G) the Laplacian matrix corresponding to some labeling of the vertices of G, i.e., L(G) is an n-by-n matrix, the (i, j) entry of which is the degree of vertex i when i = j, -1 if i 7 j but vertex i is adjacent to vertex j, and zero otherwise. It is shown in [5] that the number of Hamiltonian circuits in G is given by the formula I n k (1) ~h(G)= (-1) dk(L(G)). 2nk= 2 While there is an immense literature on generalized matrix functions, Eq. (1) is already sufficient motivation to seek "fast" algorithms for their actual computation. The main result of this note is an algorithm for computing d2. (See the next section.) It seems that d2 may be especially appropriate for the study of Laplacian matrices for the following reason: If G is a graph on n vertices, then L(G) is positive semidefinite symmetric and singular. Moreover, G is connected if and only if rank L(G) = n 1. For arbitrary positive semidefinite symmetric matrices without a zero row, it was established in [3, Corollaries 5 and 6] that d2(A) > 0 with equality if and only if rank(A) < n 1. Received August 22, 1983. 1980 Mathematics Subject Classification. Primary 65F30, 15A15, 05C50. *Work of the second author was supported by the National Science Foundation under Grant No. MCS-8300097. ?01984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page

Book ChapterDOI
01 Jan 1984
TL;DR: In this article, it was shown that there are two nonconjugate units of order 3 in U1 ℤ S3, where S3 is the symmetric group on three elements.
Abstract: Let U ℤ G be the group of units of ℤ G, the integral group ring of a finite group G. Obviously, ± g ∈ U ℤ G for g ∈ G; these are called trivial units. The augmentation of a unit has to be ± 1 . Thus we have, U ℤ G = ± U1 ℤ G with U1ℤG denoting the group of units of augmentation one. It is a classical result of G. Higman [4] that every torsion unit of a commutative integral group ring is trivial. Hughes and Pearson [5] showed that there are two nonconjugate units of order 3 in U1 ℤ S3 , where S3 is the symmetric group on three elements. These units cannot be conjugate to trivial units. Accordingly, Zassenhaus made the following conjectures.