Showing papers on "Symmetric group published in 2006"

The Theory of Group Characters and Matrix Representations of Groups

Abstract: Matrices Algebras Groups The Frobenius algebra The symmetric group Immanants and $S$-functions $S$-functions of special series The calculation of the characters of the symmetric group Group characters and the structure of groups Continuous matrix groups and invariant matrices Groups of unitary matrices Appendix Bibliography Supplementary bibliography Index.

The 192 solutions of the Heun equation

Abstract: A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D n . Each of the 192 expressions is labeled by an element of D 4 . Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S 4 . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

Factor and normal subgroup theorems for lattices in products of groups

Abstract: A central result in the theory of semisimple groups and their lattices is Margulis’ normal subgroup theorem: any normal subgroup of an irreducible lattice in a center free, higher rank semisimple group, has finite index [Mar79,Mar91]. In the present paper we establish a Margulis-type theorem for a large family of lattices, including all Kac-Moody groups over (sufficiently large) finite fields. As in Margulis’ strategy, we establish along the way a “factor theorem” for measurable quotients of boundaries, which is of independent interest. Its proof introduces new ideas relying heavily on Furstenberg’s boundary theory (pertaining to harmonic functions, random walks and stationary measures for group actions). An adelic extension of the latter, and factor theorems in which the boundary is not a homogeneous space, follow as well. Recall that a group is called just infinite, if every non-trivial normal subgroup of it has finite index. The elementary observation that every finitely generated infinite group admits an infinite just infinite quotient, is one motivation for studying this property (see [Wil00] for more on the general structure of such groups). Extending this notion, we shall call a topological group G just non-compact, if every non-trivial closed normal subgroup N G is co-compact, and topologically just infinite, if every such N has finite index. Of course, for an abstract group G, all the three notions agree when it is viewed as a topological group with discrete topology.

The Spectra of Quantum States and the Kronecker Coefficients of the Symmetric Group

Abstract: Determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics. In this paper we consider the spectra of a bipartite quantum state and its two marginal states. To each spectrum we can associate a representation of the symmetric group defined by a Young diagram whose normalised row lengths approximate the spectrum. We show that, for allowed spectra, the representation of the composite system is contained in the tensor product of the representations of the two subsystems. This gives a new physical meaning to representations of the symmetric group. It also introduces a new way of using the machinery of group theory in quantum informational problems, which we illustrate by two simple examples.

Generating infinite symmetric groups

Abstract: Let S = Sym(Ω) be the group of all permutations of an infinite set Ω. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, then there exists a positive integer n such that every element of S may be written as a group word of length at most n in the elements of U . Likewise, if U is a generating set for S as a monoid, then there exists a positive integer n such that every element of S may be written as a monoid word of length at most n in the elements of U . Some related questions and recent results are noted, and a brief proof is given of a result of Ore’s on commutators, which is used in the proof of the above result. 1. Introduction, notation, and some lemmas on full moieties In [14, Theorem 1.1], Macpherson and Neumann show that if Ω is an infinite set, then the group S = Sym(Ω) is not the union of a chain of |Ω| or fewer proper subgroups. We will repeat the beautiful proof of that result, with modifications that will allow us to obtain, along with it, the result stated in the abstract. The present section is devoted to obtaining strengthened versions of the lemmas used in that proof. Following the notation of [14], for Ω an infinite set, Sym(Ω), generally abbreviated to S, will denote the group of all permutations of Ω, and such permutations will be written to the right of their arguments. For subsets Σ ⊆ Ωa ndU ⊆ S ,t he symbol U(Σ) will denote the set of elements of U that stabilize Σ pointwise, and U{Σ} is the set {f ∈ U :Σ f =Σ } .( In [14] this notation is used only for U a subgroup.) A subset Σ ⊆ Ω will be called full with respect to U ⊆ S if the set of permutations of Σ induced by members of U{Σ} is all of Sym(Σ). The cardinality of a set X will be written |X|, and a subset Σ ⊆ Ω will be called a moiety if |Σ| = |Ω| = |Ω − Σ|. Suppose that Σ1 and Σ2 are moieties of Ω whose intersection is also a moiety, and whose union is all of Ω. Then [14, Lemma 2.3] says that if G is a subgroup of S = Sym(Ω) such that Σ1 and Σ2 are both full with respect to G, then G = S. To strengthen this result, we will consider subsets U, V ⊆ S, closed under inverses, such that Σ1 is full with respect to U and Σ2 with respect to V . By the lemma cited, � U ∪ V � = S; our version of this result will bound the number of factors from U and V needed to obtain the general element of S. Our proof will use the following fact, first proved by Ore [18]. Much stronger results have been proved since. In § 4 we will give a self-contained proof of a statement of intermediate strength.

Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products

Abstract: I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack Sg+1(X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the Gromov- Witten invariants of a point.

Partial augmentations and brauer character values of torsion units in group rings

Abstract: For a torsion unit u of the integral group ring ZG of a nite group G, and a prime p which does not divide the order of u (but the order of G), a relation between the partial augmentations of u on the p-regular classes of G and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable G, consequences concerning rational conjugacy of u to a group element are discussed, considering as examples the symmetric group S5 and the groups PSL(2; p f ).

Affine insertion and Pieri rules for the affine Grassmannian

Abstract: We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. 2) A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of the pairing between homology and cohomology of the affine Grassmannian. These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously. Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.

All p-local finite groups of rank two for odd prime p

Abstract: In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3.

Gaussian fluctuations of characters of symmetric groups and of Young diagrams

Abstract: We study asymptotics of reducible representations of the symmetric groups Sq for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, the left-regular representation (Plancherel measure), irreducible representations and tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.

Markov processes on partitions

Abstract: We introduce and study a family of Markov processes on partitions. The processes preserve the so-called z-measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels. We also compute the scaling limits of the kernels in two different regimes. The limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random Young diagrams, and the behavior of the Young diagrams near the diagonal.

Pattern Avoidance and the Bruhat Order

Abstract: The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length.

q-Eulerian Polynomials: Excedance Number and Major index

Abstract: In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Bj\"orner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.

A survey on Zariski pairs

Abstract: As Zariski pointed out in the Introduction of [130], this question was first considered by Enriques and the problem is reduced to finding the fundamental group of the complement of the given curve (the word complement is understood and often omitted for short). Zariski considered some explicit cases and proved important results. Here we detail some of the most relevant: (Z1) If two curves lie in a connected family of equisingular curves, then they have isomorphic fundamental groups. (Z2) If a continuous family {Ct}t∈[0,1] is equisingular for t ∈ (0, 1] and C0 is reduced, then there is a natural epimorphism π1(P\C0, p0) π1(P\Ct, pt), where the base point pt (t ∈ [0, 1]) depends on t continuously. (Z3) The fundamental group of an irreducible curve of order n, possessing ordinary double points only, is cyclic of order n ([130, Theorem 7]), see Remark 1. (Z4) Consider the projection from the general cubic surface in P onto P, centered at a general point outside the surface. Its branch locus is a sextic C6 with six cusps whose fundamental group is isomorphic to Z/2Z ∗ Z/3Z. (Z5) He noted that the six cusps of any sextic described in (Z4) satisfy the extra condition of lying on a conic –without decreasing the dimension of their family. Moreover, if C6 is a sextic with six cusps and its fundamental group has a representation onto the symmetric group of three letters, then C6 is the branch curve of a cubic surface and its six cusps lie on a conic. In particular if a sextic C′ 6 with six cusps not on a conic exists, then π1(P \ C6, po) 6∼= π1(P \ C′ 6, po).

Young modules and filtration multiplicities for Brauer algebras

Abstract: We define permutation modules and Young modules for the Brauer algebra B k (r,δ), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups.

Kazhdan¿Lusztig cells and the Murphy basis

Abstract: Let $H$ be the Iwahori?Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan?Lusztig basis and the Murphy basis. We establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan?Lusztig basis and Lusztig's results on the $a$-function.

Quantum permutation groups: a survey

Abstract: This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments.

Random measurement bases, quantum state distinction and applications to the hidden subgroup problem

Abstract: We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group (Moore at al., 2005). We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel'fand pair, e.g., Abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large (Grigni et al., 2004), which happens for example in the HSP in the symmetric group. The drawback of random POVMs is that they are not efficient to implement, since measuring in a random basis takes exponential time as can be seen by a counting argument. This leads us to the open question of efficiently implementable pseudorandom measurement bases.

Zassenhaus conjecture for central extensions of $S_{5}$

Abstract: We confirm a conjecture of Zassenhaus about rational conjugacy of torsion units in integral group rings for a covering group of the symmetric group $S_{5}$ and for the general linear group $\text{GL}(2,5)$.

Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

Abstract: This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to related background material. Before defining our generalization $NC^{(k)}(W)$, we develop from scratch the theory of algebraic noncrossing partitions $NC(W)$. This involves studying a finite Coxeter group $W$ with respect to its generating set $T$ of {\em all} reflections, instead of the usual Coxeter generating set $S$. This is the first time that this material has appeared in one place. Finally, it turns out that our poset $NC^{(k)}(W)$ shares many enumerative features in common with the ``generalized nonnesting partitions'' of Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In particular, there is a generalized ``Fuss-Catalan number'', with a nice closed formula in terms of the invariant degrees of $W$, that plays an important role in each case. We give a basic introduction to these topics, and we describe several conjectures relating these three families of ``Fuss-Catalan objects''.

A remark on rational Cherednik algebras and differential operators on the cyclic quiver

Abstract: We show that the spherical subalgebra Uk,c of the rational Cherednik algebra associated to Sn 2 Cl, the wreath product of the symmetric group and the cyclic group of order l, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size l. This confirms a version of [5, Conjecture 11.22] in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov [12] on the deformed Harish–Chandra homomorphism, and of Crawley–Boevey, [3] and [4], and Gan and Ginzburg [7] on preprojective algebras.

Presentation of the singular part of the Brauer monoid

Abstract: We obtain a presentation for the singular part of the Brauer monoid with respect to an irreducible system of generators, consisting of idempotents. As an application of this result we get a new construction of the symmetric group via connected sequences of subsets. Another application describes the lengths of elements in the singular part of the Brauer monoid with respect to the system of generators, mentioned above.

Meixner polynomials and random partitions

Abstract: The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov's Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional lattice. In the particular case of z-measures, the correlation kernel of this process, called the discrete hypergeometric kernel, has especially nice properties. The aim of the paper is to derive the discrete hypergeometric kernel by a new method, based on a relationship between the z-measures and the Meixner orthogonal polynomial ensemble. The present paper can be viewed as an introduction to another our paper where the same approach is applied to studying a dynamical model related to the z-measures (Markov processes on partitions, Prob. Theory Rel. Fields 135 (2006), 84-152; arXiv: math-ph/0409075).

The distinguishing number of the direct product and wreath product action

Abstract: Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted D G(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ?Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S m × S n on [m] × [n].

Peer-to-peer group management framework and methodology

Abstract: A group management framework and methodology for managing symmetric groups using peer-to-peer network communications. Group management is distributed throughout the group members (602, 604, 606) of the peer-to-peer network (608), and group information (334) is mirrored in each of the members (602, 604, 606) of the group (600). The framework facilitates management of group member lists (336), provides group communication capabilities for external applications (204, 350), and security for the group communication without the need for intermediary servers to provide services.

Character formulas for the operad of two compatible brackets and for the bihamiltonian operad

Abstract: We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.