Showing papers on "Symmetric group published in 2000"

Finite group theory

Abstract: 1. Preliminary results 2. Permutation representations 3. Representation of groups on groups 4. Linear representations 5. Permutation groups 6. Extensions of groups and modules 7. Spaces with forms 8. p-groups 9. Change of field of a linear representation 10. Presentation of groups 11. The generalized Fitting subgroup 12. Linear representation of finite groups 13. Transfer and fusion 14. The geometry of groups of Lie type 15. Signalizer functors 16. Finite simple groups References List of symbols Index.

Asymptotics of Plancherel measures for symmetric groups

Abstract: 1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G∧ of irreducible representations of G which assigns to a representation π ∈ G∧ the weight (dim π)/|G|. For the symmetric group S(n), the set S(n)∧ is the set of partitions λ of the number n, which we shall identify with Young diagrams with n squares throughout this paper. The Plancherel measure on partitions λ arises naturally in representation– theoretic, combinatorial, and probabilistic problems. For example, the Plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation [31]. We denote the Plancherel measure on partitions of n by Mn,

Hyperelliptic jacobians without complex multiplication

Abstract: has only trivial endomorphisms over an algebraic closure of the ground field K if the Galois group Gal(f) of the polynomial f ∈ K[x] is “very big”. More precisely, if f is a polynomial of degree n ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then End(J(C)) = Z. Notice that it easily follows that the ring of K-endomorphisms of J(C) coincides with Z and the real problem is how to prove that every endomorphism of J(C) is defined over K. There are some results of this type in the literature. Previously Mori [8], [9] has constructed explicit examples (in all characteristics) of hyperelliptic jacobians without nontrivial endomorphisms. In particular, he provided examples over Q with semistable Cf and big (doubly transitive) Gal(f) [9]. The semistability of Cf implies the semistability of J(Cf ) and, thanks to a theorem of Ribet [14], all endomorphisms of J(Cf ) are defined over Q. (Applying to Cf/Q the Shafarevich conjecture [17] (proven by Fontaine [3] and independently by Abrashkin [1], [2]) and using Lemma 4.4.3 and arguments on p. 42 of [16], one may prove that the Galois group Gal(f) of the polynomial f involved is S2g+1 where deg(f) = 2g + 1.) Andre ([7], pp. 294-295) observed that results of Katz ([5], [6]) give rise to examples of hyperelliptic jacobians J(Cf ) over the field of rational function C(z) with End(J(Cf )) = Z. Namely, one may take f(x) = h(x)−z where h(x) ∈ C[x] is a Morse function. In particular, this explains Mori’s example [8]

The partition algebra and the Potts model transfer matrix spectrum in high dimensions

Abstract: We construct generalizations Pmn(Q) of the partition algebra Pn(Q) (Martin P P 1996 J. Algebra 183 319), facilitating a representation theoretic approach to the n-site transfer matrix spectrum of a high-dimensional Q-state Potts model with magnetic field and source terms (and to corresponding dichromatic polynomials). For each Q we describe the irreducible representation theory of the sequence of algebras P*(Q) = {Pn(Q) ⊂ Pn1(Q) ⊂ Pn + 1(Q)| n = 0,1,2,...} approaching the large-n limit. For each positive integer Q we extend the Potts model representation ρn of Pn(Q) to a representation of P1n(Q). We show how these Potts representations embed in the representation theory of the partition algebras. These results together provide a tool with which to examine the nature of physical correlation functions. For large n the irreducible content of the Potts representations can be summarized by SQ EndPn(Q)({VQ⊗n}) and SQ-1 EndPn1(Q)({VQ⊗n}), where SQ is the symmetric group, and VQ is the space of states of a Potts spin. We show how the partition algebra formalism matches up the correlation functions of the Potts model and the corresponding absolute spectrum degeneracies of its transfer matrix.

Littlewood–Richardson Coefficients and Kazhdan–Lusztig Polynomials

Abstract: We show that the Littlewood–Richardson coefficients are values at 1 of certain parabolic Kazhdan–Lusztig polynomials for affine symmetric groups. These $q$-analogues of Littlewood–Richardson multiplicities coincide with those previously introduced in [21] in terms of ribbon tableaux.

Symmetric groups and the cup product on the cohomology of Hilbert schemes

Abstract: The integral cohomology ring of the Hilbert scheme of n-tuples on the affine plane is shown to be isomorphic to the graded ring associated to a filtration of the ring of integral class functions on the symmetric group.

Symmetric products, permutation orbifolds and discrete torsion

Abstract: Symmetric product orbifolds, i.e. permutation orbifolds of the full symmetric group S n are considered by applying the general techniques of permutation orbifolds. Generating functions for various quantities, e.g. the torus partition functions and the Klein-bottle amplitudes are presented, as well as a simple expression for the discrete torsion coefficients.

Riemann-Hilbert problem and the discrete Bessel Kernel

Abstract: We use discrete analogs of Riemann-Hilbert problem's methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann-Hilbert problem and of an integrable integral operator and show that computing the resolvent of a discrete integrable operator can be reduced to solving a corresponding discrete Riemann-Hilbert problem. We also give an example, explicitly solvable in terms of classical special functions, when a discrete Riemann-Hilbert problem converges in a certain scaling limit to a conventional one; the example originates from the representation theory of the infinite symmetric group.

Ribbon tile invariants

Abstract: Let T be a finite set of tiles, and B a set of regions F tileable by T. We introduce a tile counting group G(T, B) as a group of all linear relations for the number of times each tile T C T can occur in a tiling of a region r C B. We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group. The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group. The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.

Minimal Resolutions and the Homology of Matching and Chessboard Complexes

Abstract: We generalize work of Lascoux and Jozefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 2 minors of generic matrices and generic symmetric matrices, respectively Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras Our motivation is two-fold: • We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Bjorner, Lovasz, Vrecica and Zivaljevic This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes • The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers

Actions by the classical Banach spaces

Abstract: The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that the Borel cardinalities of the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.One of the first things to be addressed concerns the complexity of these equivalence relations. This question for appears in [1].

Tree actions of automorphism groups

Abstract: We introduce conditions on a group action on a tree that are sufficient for the action to extend to the automorphism group. We apply this to two different classes of one-relator groups: certain BaumslagSolitar groups and one-relator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.

Sporadic groups and automorphisms of linear spaces

Abstract: This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let be a finite linear space on which G acts as a line-transitive automorphism group. Then the socle of G is not a sporadic group. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 353–362, 2000

On Cayley Graphs on the Symmetric Group Generated by Tranpositions

Abstract: , we denote by \(\) its smallest non-zero Laplacian eigenvalue. In this paper we show that among all sets of n-1 transpositions which generate the symmetric group, \(\), the set whose associated Cayley graph has the highest \(\) is the set {(1, n), (2, n), ..., (n-1, n)} (or the same with n and i exchanged for any i

Plane Partitions and Characters of the Symmetric Group

Abstract: In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product χ^λ ⊗ χ^μ of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand χ^ν equals the number of pairs of Littlewood-Richardson multitableaux of shape (λ, μ), conjugate content and type ν. We also give lower and upper bounds for these numbers.

The Automorphism Tower of a Free Group

Abstract: It is proved that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof is that the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F).

Representations of the symmetric group are reducible over simply transitive subgroups

Abstract: If the characteristic of F is zero, this problem has been solved by Saxl [16]. An important feature of Saxl’s result is that a groupG as in the problem above is either 2-transitive or fixes a point, i.e. is contained in some Σn−1. The caseG ≤ Σn−1 can then be settled using the branching rule and induction. On the other hand, an explicit list of k-transitive groups (for k ≥ 2) is available, which can be used to complete the proof in characteristic zero, see [16] for more details. From now on we assume that p > 0. In this case the problem is important for determining maximal subgroups of finite classical groups [1],[13]. However, the situation is nowmore complicated. For example, to determine the pairs (G,D) as above withG = An one needs the Mullineux conjecture [7], [2]. IfG is intransitive then, up to a conjugation, it is contained in a standard Young subgroup of the form Σn−k ×Σk. The irreducible restrictions from Σn toΣn−1 have been described in [14], see also [11], [6]. In [11] it is also

Automorphisms and enumeration of switching classes of tournaments

Abstract: Two tournaments T1 and T2 on the same vertex set X are said to be switching equivalent if X has a subset Y such that T2 arises from T1 by switching all arcs between Y and its complement XnY . The main result of this paper is a characterisation of the abstract nite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral. Moreover, if G is such a group, then there is a switching class C ,w ith Aut(C) = G, such that every subgroup ofG of odd order is the full automorphism group of some tournament in C. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. The proof follows as a special case of a result on the full automorphism group of randomG-invariant digraphs selected from a certain class of probability distributions. We also show that a permutation group G, acting on a set X ,i s contained in the automorphism group of some switching class of tournaments with vertex set X if and only if the Sylow 2-subgroups of

Fundamental groups of moduli and the Grothendieck-Teichmüller group

Abstract: Let Mo,, denote the moduli space of Riemann spheres with n ordered marked points In this article we define the group Out# of quasispecial symmetric outer automorphisms of the algebraic fundamental group 1r1(Mo,n) for all n > 4 to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of Irl(M4,n) and commuting with the group of outer automorphisms of rl (MO,n) obtained by permuting the marked points Our main result states that Outd is isomorphic to the Grothendieck-Teichmiiller group GT for all n > 5

Quantum Vertex Representations via Finite Groups¶and the McKay Correspondence

Abstract: We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ×ℂ× and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU 2, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of ADE type.

Immanants and finite point processes

Abstract: Given a Hermitian, non-negative definite kernel K and a character ? of the symmetric group on n letters, define the corresponding immanant function K?x1, ?, xn]????(?)?ni=1K(xi, x?(i)), where the sum is over all permutations ? of {1, ?, n}. When ? is the sign character (resp. the trivial character), then K? is a determinant (resp. permanent). The function K? is symmetric and non-negative, and, under suitable conditions, is also non-trivial and integrable with respect to the product measure ??n for a given measure ?. In this case, K? can be normalised to be a symmetric probability density. The determinantal and permanental cases or this construction correspond to the fermion and boson point processes which have been studied extensively in the literature. The case where K gives rise to an orthogonal projection of L2(?) onto a finite-dimensional subspace is studied here in detail. The determinantal instance of this special case has a substantial literature because of its role in several problems in mathematical physics, particularly as the distribution of eigenvalues for various models of random matrices. The representation theory of the symmetric group is used to compute the normalisation constant and identify the kth-order marginal densities for 1?k?n as linear combinations of analogously defined immanantal densities. Connections with inequalities for immanants, particularly the permanental dominance conjecture of Lieb, are considered, and asymptotics when the dimension of the subspace goes to infinity are presented.