Fusion Procedure for Wreath Products of Finite Groups by the Symmetric Group

Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.     

The symmetric genus of alternating and symmetric groups

The symmetric genus of a finite group G has been defined by Thomas W. Tucker as the smallest genus of all surfaces on which G acts faithfully as a group of automorphisms (some of which may reverse the orientation of the surface). This note announces the symmetric genus of all finite alternating and symmetric groups.     

The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.     

On the exponentiality of affine symmetric spaces

An affine symmetric space G/H is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in G/H. For the group space (G × G)/G _diag of a Lie group G, these properties are equivalent to the exponentiality and weak exponentiality of G, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.     

Rational permutation groups containing a full cycle

A finite group whose irreducible complex characters are rational valued is called a rational group. Thus, G is a rational group if and only if N _G (〈x〉)/C _G (〈x〉) ≌ Aut(〈x〉) for every x ∈ G. For example, all symmetric groups and their Sylow 2-subgroups are rational groups. Structure of rational groups have been studied extensively, but the general classification of rational groups has not been able to be done up to now. In this paper, we show that a full symmetric group of prime degree does not have any rational transitive proper subgroup and that a rational doubly transitive permutation group containing a full cycle is the full symmetric group. We also obtain several results related to the study of rational groups.     

Relative symmetric polynomials

In this article, we introduce the notion of a relative symmetric polynomial with respect to a permutation group and an irreducible character and we give answers for some natural questions about their structures. In order to study symmetric polynomials with respect to linear characters, we define the concept of relative Vandermonde polynomial. Finally, we present some interesting research problems concerning relative symmetric polynomials.     

Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

Embeddable transversal designs

The embeddability of certain (group) divisible designs in symmetric 2-designs is investigated. These designs are symmetric resolvable transversal designs. It is proved that all such transversal designs with v = 2k are embeddable and some necessary and sufficient conditions for other cases are given.     

On the converse of singer's theorem

A construction is given in which the nonzero elements of a planar difference set give rise to a totally symmetric quasi-group. Examples are provided which suggest that the quasi-group is essentially the additive group of the field. The evidence supports the conjecture that the converse of Singer's theorem holds. The Multiplier Theorem is used to characterize when the totally symmetric quasi-groups are totally symmetric loops. The results extend to Abelian group difference sets (λ = 1).     

Square integrability of representations on p-adic symmetric spaces

A symmetric space analogue of Casselman's criterion for square integrability of representations of a p-adic group is established. It is described in terms of exponents of Jacquet modules along parabolic subgroups associated to the symmetric space.     

Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks

We develop a number of statistical aspects of symmetric groups (mostly dealing with the distribution of cycles in various subsets of Sn), asymptotic properties of (ordinary) characters of symmetric groups, and estimates for the multiplicities of root number functions of these groups. As main applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, a finiteness result for the number of Fuchsian presentations of such a group (resolving a long-standing problem of Roger Lyndon), as well as a proof of a well-known conjecture of Roichman concerning the mixing time of random walks on symmetric groups.     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

On the combinatorial invariance of Kazhdan-Lusztig polynomials

In this paper, we solve the conjecture about the combinatorial invariance of Kazhdan-Lusztig polynomials for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. We also obtain explicit formulas for the R-polynomials and for the Kazhdan-Lusztig polynomials associated with any interval of length 5 in any Coxeter group, showing in particular what they look like in the symmetric group.     

Symmetric sign patterns with maximal inertias

The inertia of an n by n symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order n. In this note we classify all the maximal inertias for symmetric sign patterns of order n, and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.     

The Calogero–Moser partition and Rouquier families for complex reflection groups

Let W be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify part of the conjecture in the case that W is a wreath product of a symmetric group with a cyclic group of order l.     

Monomorphism categories associated with symmetric groups and parity in finite groups

Monomorphism categories of the symmetric and alternating groups are studied via Cayley’s Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley’s theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.