Showing papers on "Symmetric group published in 2002"

Automorphisms of Free Groups and Outer Space

Abstract: This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.

Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams

Abstract: Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure M n. That is, the weight M n(λ) of a diagram λ equals dim2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan-Shepp 1977, Vershik-Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999

New Approaches to Designing Public Key Cryptosystems Using One-Way Functions and Trapdoors in Finite Groups

Abstract: A symmetric key cryptosystem based on logarithmic signature s for finite permutation groups was described by the first author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message space a large finite group G , using logarithmic signature s and their generalizations. The first approach relies on the fact that permutations of the message space G induced by transversal logarithmic signature s almost always generate the full symmetric group SG on the message space. The second approach could potentially lead to new ElGamal-like systems based on trapdoor, one-way functions induced by logarithmic signature -like objects we call meshes , which are uniform covers for G .

Order Structure on the Algebra of Permutations and of Planar Binary Trees

Abstract: Let i>Xn be either the symmetric group on i>n letters, the set of planar binary i>n-trees or the set of vertices of the (i>n − 1)-dimensional cube In each case there exists a graded associative product on ⊕i>n≥0i>K[i>Xn] We prove that it can be described explicitly by using the weak Bruhat order on i>Sn, the left-to-right order on planar trees, the lexicographic order in the cube case

Quantum automorphism groups of finite graphs

Abstract: A quantum analogue of the automorphism group of a finite graph is introduced. These are quantum subgroups of the quantum permutation groups defined by Wang. The quantum automorphism group is a stronger invariant for finite graphs than the usual automorphism group. We get a quantum dihedral group D 4 .

Cohomology of buildings and of their automorphism groups

Abstract: Buildings were created by Tits. They serve as a vehicle for understanding real semi-simple groups and their p-adic analogs. These are undoubtedly the two most interesting cases, but existing constructions provide many more interesting examples, some of which are remarkably symmetric. While classical (spherical or Euclidean) buildings correspond to spherical or Euclidean reflection groups, we are mainly concerned with buildings related to other (infinite) reflection groups. The class of examples which this paper addresses is that of Kac–Moody buildings, as described by Tits [Ti]. It is a very large family, and it is precisely the great symmetry of these buildings that we are using. Our initial interest in the cohomology of buildings was that we suspected that buildings provided a rich source of Kazhdan groups. Recall that a locally compact group G is a Kazhdan group (or has Property (T)) if the trivial representation of G is isolated in the space of all irreducible unitary representations of G. Equivalently, G is a Kazhdan group if H1 ct(G, ρ) = 0: i.e., its first continuous cohomology group with coefficients in any unitary representation vanishes. In this paper we use the latter definition (for their equivalence, cf. [HV]). It turns out that not only can we give an almost definitive statement about when the automorphism group of a Kac–Moody

Transitive Permutation Groups Without Semiregular Subgroups

Abstract: A transitive nite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and ane group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups. Part of the motivation for studying this class of groups was a conjecture due to Maru si c, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

Projective representations of symmetric groups via Sergeev duality

Abstract: In this article, we determine the irreducible projective representations of the symmetric group Sd and the alternating group Ad over an algebraically closed field of characteristic p 6= 2. These matters are well understood in the case p = 0, thanks to the fundamental work of Schur [24] in 1911, as well as the much more recent work of Nazarov [19, 20], Sergeev [25, 26] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set RPp(d) of restricted p-strict partitions of d. Such partitions arose recently in work of Kashiwara, Miwa, Peterson and Yung [11] and Leclerc and Thibon [14] on the q-deformed Fock space of the affine Kac-Moody algebra of type A p−1. Leclerc and Thibon proposed that RPp(d) should label the irreducible projective representations in some natural way, and we show here how this can be done. Note that for p = 3, 5, the labelling problem was solved in [1, 3], while if p = 2 all projective representations of Sd and Ad are linear so do not need to be considered further here. To be more precise, recall that λ is a partition of d if λ = (λ1, λ2, . . . ) is a non-increasing sequence of non-negative integers summing to d. Call λ p-strict if in addition

A new proof of the Mullineux conjecture

Abstract: Let S_d be the symmetric group on d letters and let k be a field of characteristic p>2. Tensoring an irreducible S_d module with the sign representation defines an involution on the p-regular partitions of d. It is suprisingly difficult to describe this involution combinatorially. Mullineux conjectured an algorithmic description in 1979. Kleshchev gave an entirely new algorithm describing the involution in 1996 and proved with Ford that it agrees with Mullineux's. Using the modular representation theory of the supergroup GL(m|n) we provide the first new proof of the Mullineux conjecture which is independent of Kleshchev's approach. Similar techniques allow us to classify the irreducible polynomial representations of GL(m|n), completing recent partial results by Donkin.

The geometry of point particles

Abstract: There is a very natural map from the configuration space of n distinct points in Euclidean 3space into the flag manifold U(n)/U(1)n, which is compatible with the action of the symmetric group. The ...

A geometric and algebraic description of annular braid groups

Abstract: We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph An - 1. This provides a new example of an infinite type Artin group which injects into a finite type Artin group. In fact, we show that the affine braid group with Coxeter graph An - 1 injects into the braid group on n + 1 stings. Recently it has been shown that the braid groups are linear, see [3]. Therefore, this shows that the affine braid groups are also linear.

Transitive Permutation Groups of Prime-Squared Degree

Abstract: We explicitly determine all of the transitive groups of degree i>p2, i>p a prime, whose Sylow i>p-subgroup is not isomorphic to the wreath product {\Bbb Z}_p\wr{\Bbb Z}_p. Furthermore, we provide a general description of the transitive groups of degree i>p2 whose Sylow i>p-subgroup is isomorphic to {\Bbb Z}_p\wr{\Bbb Z}_p, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order i>p2, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order i>p2, and find all nonnormal Cayley graphs of order i>p2.

A note on self-dual group codes

Abstract: We classify group algebras over Galois rings containing self-dual ideals; i.e., ideals C which satisfy C = C/sup /spl perp// with respect to the natural nondegenerate bilinear form given on group algebras.

CONSTRUCTING AUTOMORPHISM GROUPS OF p-GROUPS

Abstract: We present an algorithm to construct the automorphism group of a finite p-group. The method works down the lower exponent-p central series of the group. The central difficulty in each inductive step is a stabiliser computation; we introduce various approaches designed to simplify this computation.

Representation theory of symmetric groups and their double covers

Abstract: In this article we will give an overview of the new Lie theoretic approach to the p-modular representation theory of the symmetric groups and their double covers that has emerged in the last few years. There are in fact two parallel theories here: one for the symmetric groups Sn involving the affine Kac-Moody algebra of type A p−1, and one for their double covers Ŝn involving the twisted algebra of type A p−1. In the case of Sn itself, the theory has been developed especially by Kleshchev [19], Lascoux-Leclerc-Thibon [21], Ariki [1] and Grojnowski [9], while the double covers are treated for the first time in [4] along the lines of [9], after the important progress made over C by Sergeev [35, 36] and Nazarov [30, 31]. One of the most striking results at the heart of both of the theories is the explicit description of the modular branching graphs in terms of Kashiwara’s crystal graph for the basic module of the corresponding affine Lie algebra. Note that the results described are just a part of a larger picture: there are analogous results for the cyclotomic and affine Hecke algebras, and their twisted analogues, the cyclotomic and affine Hecke-Clifford superalgebras. However we will try here to bring out only those parts of the theory that apply to the symmetric group, since that is the most applicable to finite group theory.

Crossingless matchings and the cohomology of (n,n) Springer varieties

Abstract: The sequence of rings $H^n, n\ge 0,$ introduced in math.QA/0103190, controls categorification of the quantum sl(2) invariant of tangles. We prove that the center of $H^n$ is isomorphic to the cohomology ring of the (n,n) Springer variety and show that the braid group action in the derived category of $H^n$-modules descends to the Springer action of the symmetric group.

Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations

Abstract: Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

The rational function analogue of a question of Schur and exceptionality of permutation representations

Abstract: In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of linear and Dickson polynomials, was proved by M. Fried in 1970. Here we investigate the analogous question for rational functions, also we allow the base field to be any number field. As a result, there are many more rational functions for which the analogous property holds. Some infinite series come from rational isogenies of elliptic curves and deformations. There are several sporadic examples which do not fit in any of the series we obtain. First we translate the arithmetic property to a question about finite permutation groups, and classify those groups which fulfill the necessary conditions. The proofs depend on the classification of the finite simple groups. Then we use arithmetic arguments to either rule out many cases, or to prove that the remaining cases indeed give rise to examples. This part is based on Mazur's classical results about rational points on the modular curves X_0(p) and X_1(p), results about Galois images in GL_2(p) coming from action of the absolute Galois group of Q on p-torsion points of elliptic curves, the theory of complex multiplication, and the techniques used in the inverse regular Galois problem.

New refinements of the McKay conjecture for arbitrary finite groups

Abstract: Let G be an arbitrary finite group and fix a prime number p. The McKay conjecture asserts that G and the normalizer in G of a Sylow p-subgroup have equal numbers of irreducible characters with degrees not divisible by p. The Alperin-McKay conjecture is version of this as applied to individual Brauer p-blocks of G. We offer evidence that perhaps much stronger forms of both of these conjectures are true.

Symmetric groups, wreath products, Morita equivalences, and Broue's abelian defect group conjecture

Abstract: It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group Sp [rmoust ] Sw. Combined with work of J. Rickard, this proves that Broue's abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5.

On the automorphism groups of cayley graphs of finite simple groups

Abstract: Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.

Kazhdan-Lusztig and R-polynomials, Young's lattice, and Dyck partitions

Abstract: We give explicit combinatorial product formulas for the maximal parabolic Kazhdan-Lusztig and R-polynomials of the symmetric group. These formulas imply that these polynomials are combinatorial invariants, and that the Kazhdan-Lusztig ones are nonnegative. The combinatorial formulas are most naturally stated in terms of Young's lattice, and the one for the Kazhdan-Lusztig polynomials depends on a new class of skew partitions which are closely related to Dyck paths.

Combinatorial Laplacian of the Matching Complex

Abstract: A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral.

Non-crossing cumulants of type B

Abstract: We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of {\em non-crossing cumulant of type B;} the inspiration for its definition is found by looking at an operation of ``restricted convolution of multiplicative functions'', studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B). The non-crossing cumulants of type B live in an appropriate framework of ``non-commutative probability space of type B'', and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of ``vanishing of mixed cumulants of type B'', we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.