Gelfand transforms of polyradial Schwartz functions on the Heisenberg group

We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.     

Stable decompositions for some symmetric group characters arising in braid group cohomology

We prove that certain permutation characters for the symmetric group Σn decompose in a manner that is independent of n for large n. This result is a key ingredient in the recent work of T. Church and B. Farb, who obtain a “representation stability” theorem for the character of Σn acting on the cohomology Hp(Pn,C) of the pure braid group Pn.     

Partial sum of matrix entries of representations of the symmetric group and its asymptotics

Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we extend a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose each of these objects into a main term and a reminder, and for each such a decomposition we prove a central limit theorem for the main term. We apply these results to prove a law of large numbers for the partial sum. Our main tool is the expansion of symmetric functions evaluated on Jucys–Murphy elements.     

Compact weakly symmetric spaces and spherical pairs

Let be a spherical pair and assume that is a connected compact simple Lie group and a closed subgroup of . We prove in this paper that the homogeneous manifold is weakly symmetric with respect to and possibly an additional fixed isometry . It follows that M. Krämer's classification list of such spherical pairs also becomes a classification list of compact weakly symmetric spaces. In fact, our proof involves a case-by-case study of the isotropy representations of all spherical pairs on Krämer's list.

     

Projective characters of the infinite generalized symmetric group

We consider the infinite generalized symmetric group S(∞)?? _m ^∞ , introduce its covering $\tilde B_m $ , and describe all indecomposable characters on the group $\tilde B_m $ .     

Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of ²n−1 permutations that pairwise generate the symmetric group Sn. There is no set of ²n−1+1 permutations having this property. For all sufficiently large integers n with n≡2mod4, there exists a set of ²n−2 even permutations that pairwise generate the alternating group An. There is no set of ²n−2+1 permutations having this property.     

Stanley's character polynomials and coloured factorisations in the symmetric group

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.     

Supercuspidal Gelfand pairs

If G is a totally disconnected group and H is a closed subgroup then, according to the Gelfand-Kazhdan Lemma, if the double coset space H?G/H is preserved by an antiautomorphism of G of order two then (G,H) must be a Gelfand pair in the sense that Hom_H(π,1) has dimension at most one for each irreducible, admissible representation π of G. Under certain rather general restrictions, we show that if the symmetry property holds only for almost all double cosets, then (G,H) is a supercuspidal Gelfand pair in the sense that for all irreducible, supercuspidal representations π of G. There exist examples of supercuspidal Gelfand pairs which are not Gelfand pairs.     

A theorem of the Wiener—Tauberian type forL 1(H n)

The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hⁿ and the unitary group U(n), acts on Hⁿ in a natural way. Here we prove a Wiener-Tauberian theorem for L¹ (Hⁿ) with this HM(n)-action on Hⁿ i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L¹ (Hⁿ) in order that the linear span of^gf : g∈HM(n) is dense in L¹(Hⁿ), where^gf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hⁿ.     

On the decomposition map for symmetric groups

Let R ^d be the ℤ-module generated by the irreducible characters of the symmetric group . We determine bases for the kernel of the decomposition map. It is known that R ^d ⊗ _ℤ F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I _br ^d is the kernel of the decomposition map for prime p then R ^d /I _br ^d ⊗ _ℤ F is isomorphic to the radical quotient of the p-modular Solomon descent algebra. To the memory of Manfred Schocker.     

Equality of multiplicity free skew characters

In this paper we show that two skew diagrams λ/μ and α/β can represent the same multiplicity free skew character [λ/μ]=[α/β] only in the the trivial cases when λ/μ and α/β are the same up to translation or rotation or if λ=α is a staircase partition λ=(l,l−1,…,2,1) and λ/μ and α/β are conjugate of each other.     

Minimizing symmetric submodular functions

We describe a purely combinatorial algorithm which, given a submodular set functionf on a finite setV, finds a nontrivial subsetA ofV minimizingf[A] + f[V A]. This algorithm, an extension of the Nagamochi—Ibaraki minimum cut algorithm as simplified by Stoer and Wagner [M. Stoer, F. Wagner, A simple min cut algorithm, Proceedings of the European Symposium on Algorithms ESA '94, LNCS 855, Springer, Berlin, 1994, pp. 141–147] and by Frank [A. Frank, On the edge-connectivity algorithm of Nagamochi and Ibaraki, Laboratoire Artémis, IMAG, Université J. Fourier, Grenbole, 1994], minimizes any symmetric submodular function using O(|V|³) calls to a function value oracle. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.A preliminary version of this paper was presented at the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) in January 1995. This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.     

On Schur's Q-functions and the Primitive Idempotents of a Commutative Hecke Algebra

Let B _n denote the centralizer of a fixed-point free involution in the symmetric group S _2n . Each of the four one-dimensional representations of B _n induces a multiplicity-free representation of S _2n , and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of S _n.     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

     

Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups Hi, i=1,…,k of the symmetric group Sn (or the alternating group An) such that each element in the group Sn (respectively An) lies in some conjugate of one of the Hi. We prove that this number lies between a?(n) and bn for certain constants a,b, where ?(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+?(n)/2, and we determine the exact value for Sn when n is odd and for An when n is even.     

Asymptotic value distribution of additive functions defined on the symmetric group

We examine the asymptotic value distribution of additive functions defined via the multiplicities of lengths of the cycles comprising a random permutation taken from the symmetric group with equal probability. We establish necessary and sufficient conditions for the weak law of large numbers and for the relative compactness of the sequence of distributions. Considering particular cases, we demonstrate that long cycles play an exceptional role and that, sometimes, in order to obtain a Poisson limit law, their influence must be negligible. The proofs are based on the ideas going back to the seminal papers of I.Z. Ruzsa on the classical additive arithmetic functions.