Showing papers on "Symmetric group published in 2004"

Derived equivalences for symmetric groups and sl\_2-categorification

Abstract: We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Brou\'e's abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category O of gl\_n(C) and for rational representations of general linear groups over an algebraically closed field of characteristic p, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard.

Random Walks on Finite Groups

Abstract: Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great variety of different fields — Probability, Algebra, Representation Theory, Functional Analysis, Geometry, Combinatorics — have been used to attack special instances of this problem. This article gives a general overview of this area of research.

Quantum marginal problem and representations of the symmetric group

Abstract: We discuss existence of mixed stateAB of two (or multy-) com- ponent system HAB = HA HB with reduced density matricesA, �B and given spectraAB,�A,�B. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.

Cuntz-pimsner algebras of group actions

Abstract: . We associate a ∗ -bimodule over the group algebra to every self-similar group action on the space of one-sided sequences. Completions ofthe group algebra, which agree with the bimodule are investigated. Thisgives new examples of Hilbert bimodules and the associated Cuntz-Pimsneralgebras. A direct proof of simplicity of these algebras is given. We showalso a relation between the Cuntz algebras and the Higman-Thompson groupsand define an analog of the Higman-Thompson group for the Cuntz-Pimsneralgebra of a self-similar group action.Keywords: Self-similar group actions, one-sided shift, bimodules, Cuntz al-gebras, Cuntz-Pimsner algebras, Higman-Thompson groups,Grigorchuk group. MSC (2000): 46L89, 16D20, 20E08. 1. INTRODUCTION The notion of a self-similar group action (fractal group or state-closed group)naturally appears in the theory of groups acting on rooted trees and groups definedby finite transducers. Study of such actions is motivated by examples of exoticgroups which are easily defined through their self-similar action.An example of such a group is the Grigorchuk 2-group. In [12] this group isintroduced as a group of measure-preserving transformations of the unit interval.Probably it is the most simple example of a finitely generated infinite torsion group.The action of the Grigorchuk group on the interval has the self-similarity property:the restrictions of the action of any its element on the halfs of the interval is againan element of the group. This is used in a very short proof of the fact that it is aninfinite torsion group. Later the same self-similarity was used to prove that thegroup has intermediate growth (see [13], [14]). It has other interesting propertiessuch as just-infiniteness and finite width (see [15]).Examples of self-similar groups of this sort include the Gupta-Sidkigroup ([18]), the just-nonsolvable torsion free group from [5] and other (see [15],[2] and [16] for more examples).

Higher Spin Symmetry and N = 4 SYM

Abstract: We assemble the spectrum of single-trace operators in free N = 4 SU(N) SYM theory into irreducible representations of the Higher Spin symmetry algebra hs(2,2|4). Higher Spin representations or YT-pletons are associated to Young tableaux (YT) corresponding to representations of the symmetric group compatible with the cyclicity of color traces. After turning on interactions gYM 6 0, YT-pletons decompose into infinite towers of representations of the superconformal algebra psu(2,2|4) and anomalous dimensions are generated. We work out the decompositions of tripletons with respect to the N = 4 superconformal algebra psu(2,2|4) and compute their anomalous dimensions to lowest non-trivial order in g 2 YMN at large N. We then focus on operators/states sitting in semishort multiplets of psu(2,2|4). By passing them through a semishort-sieve that removes superdescendants, we derive compact expressions for the partition function of semishort primaries.

Free Wreath Product by the Quantum Permutation Group

Abstract: Let A be a compact quantum group, let n∈N * and let A aut(X n ) be the quantum permutation group on n letters. A free wreath product construction A*w A aut(X n ) is introduced. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.

Harmonic analysis on the infinite symmetric group

Abstract: The infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space l2(S(∞)) generates a type II1 von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification \(\mathfrak{S}\supset{S(\infty)}\), which we call the space of virtual permutations. Although \(\mathfrak{S}\) is no longer a group, it still admits a natural two–sided action of S(∞). Thus, \(\mathfrak{S}\) is a G–space, where G stands for the product of two copies of S(∞). On \(\mathfrak{S}\), there exists a unique G-invariant probability measure μ1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ1 into a family {μt: t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {Tz: z∈ℂ} of unitary representations of G, called generalized regular representations (each representation Tz with z≠=0 can be realized in the Hilbert space \(L^2(\mathfrak{S}, \mu_t)\), where t=|z|2). As |z|→∞, the generalized regular representations Tz approach, in a suitable sense, the “naive” two–sided regular representation of the group G in the space l2(S(∞)). In contrast with the latter representation, the generalized regular representations Tz are highly reducible and have a rich structure. We prove that any Tz admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z1, z2, the spectral types of the representations \(T_{z_1}\) and \(T_{z_2}\) are shown to be disjoint. In the case z∈ℤ, a complete description of the spectral type is obtained. Further work on the case z∈ℂ∖ℤ reveals a remarkable link with stochastic point processes and random matrix theory.

The classification of p-local finite groups over the extraspecial group of order p 3 and exponent p

Abstract: In this paper we classify the p-local finite groups over p1+2+, the extraspecial group of order p3 and exponent p for odd p. This study reduces to the classification of the saturated fusion systems over p1+2+, which will be characterized by the outer automorphism group, the number of Open image in new window-radical subgroups and the automorphism group of each nontrivial Open image in new window-radical subgroup. As part of this classification, we obtain three new exotic 7-local finite groups.

The virtual and universal braids

Abstract: We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi-direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi-direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group. Recently some generalizations of classical knots and links were de- fined and studied: singular links (20, 5), virtual links (15, 12) and welded links (10). One of the ways to study classical links is to study the braid group. Singular braids (1, 5), virtual braids (15, 21), welded braids (10) were defined similar to the classical braid group. A theorem of A. A. Markov (4, Ch. 2.2) reduces the problem of classification of links to some alge- braic problems of the theory of braid groups. These problems include the word problem and the conjugacy problem. There are generaliza- tions of Markov's theorem for singular links (11), virtual links, and welded links (14). There are some dierent ways to solve the word problem for the singular braid monoid and singular braid group (8, 7, 22). The solution of the word problem for the welded braid group follows from the fact that this group is a subgroup of the automorphism group of the free group (10). A normal form of words in the welded braid group was constructed in (13). In this paper we study the structure of the virtual braid group V Bn. This is similar to the classical braid group Bn and welded braid group

CROSSINGLESS MATCHINGS AND THE COHOMOLOGY OF (n, n) SPRINGER VARIETIES

Abstract: In an earlier paper, the author introduced a collection of rings that control a categorification of the quantum sl(2) invariant of tangles. We prove that centers of these rings are isomorphics to the cohomology rings of the (n, n) Springer varieties and show that the braid group action in the derived category of modules over these rings descends to the Springer action of the symmetric group.

On Monomial Characters and Central Idempotents of Rational Group Algebras

Abstract: We give a method to obtain the primitive central idempotent of the rational group algebra ℚG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-by-supersolvable groups this method takes a particularly easy form that can be used to compute the Wedderburn decomposition of ℚG.

The Bruhat Order on the Involutions of the Symmetric Group

Abstract: In this paper we study the partially ordered set of the involutions of the symmetric group Sn with the order induced by the Bruhat order of Sn. We prove that this is a graded poset, with rank function given by the average of the number of inversions and the number of excedances, and that it is lexicographically shellable, hence Cohen-Macaulay, and Eulerian.

The Spectra of Density Operators 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.

Vertex Operators and the Class Algebras of Symmetric Groups

Abstract: We exhibit a vertex operator that implements the multiplication by power sums of Jucys–Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of $$\mathcal{W}_{1 + \infty } $$ , to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski. Bibliography: 18 titles.

Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions

Abstract: The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis $\{h_\la\}_{\la\in\CY^k}$ to $\{s_\la^{(k)}\}_{\la\in\CY^k}$ may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.

Applied Abstract Algebra

Abstract: With the advent of computers that can handle symbolic manipulations, abstract algebra can now be applied. In this book David Joyner, Richard Kreminski, and Joann Turisco introduce a wide range of abstract algebra with relevant and interesting applications, from error-correcting codes to cryptography to the group theory of Rubik's cube. They cover basic topics such as the Euclidean algorithm, encryption, and permutations. Hamming codes and Reed-Solomon codes used on today's CDs are also discussed. The authors present examples as diverse as "Rotation," available on the Nokia 7160 cell phone, bell ringing, and the game of NIM. In place of the standard treatment of group theory, which emphasizes the classification of groups, the authors highlight examples and computations. Cyclic groups, the general linear group GL( n), and the symmetric groups are emphasized. With its clear writing style and wealth of examples, Applied Abstract Algebra will be welcomed by mathematicians, computer scientists, and students alike. Each chapter includes exercises in GAP (a free computer algebra system) and MAGMA (a noncommercial computer algebra system), which are especially helpful in giving students a grasp of practical examples.

Dimension and randomness in groups acting on rooted trees

Abstract: We explore the structure of the p-adic automorphism group of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of generated by three random elements are full-dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.

Specht Filtrations for Hecke Algebras of Type A

Abstract: Let Hq (d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive lth root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l 4 the multiplicities in a Specht or dual Specht module filtration of an Hq (d)-module are well defined. A cohomological criterion is given for when an Hq (d)-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p 5. These results are false in general for p = 2 or 3.

Stein’s method and Plancherel measure of the symmetric group

Abstract: We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.

Structure of the Loday-Ronco Hopf algebra of trees

Abstract: Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto-Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. We also obtain a transparent proof of its isomorphism with the non-commutative Connes-Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes-Kreimer Hopf algebra is related to symmetric functions.

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. Our results show that recently discovered analogy between random partitions arising in representation theory and spectra of random matrices extends to the associated time-dependent models.

On the first zassenhaus conjecture for integral group rings

Abstract: It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I. S. Luthar and I. B. S. Passi which sometimes per- mits an answer to this conjecture. We illustrate the method on certain explicit examples. We prove with additional arguments that the conjecture is valid for any 3-dimensional crystallographic point group. Finally we apply the method to generic character tables and establish a p-variation of the conjecture for the simple groups PSL(2,p).

Invariant algebras and major indices for classical Weyl groups

Abstract: Given a classical Weyl group $W$, that is, a Weyl group of type $A$, $B$ or $D$, one can associate with it a polynomial with integral coefficients $Z_W$ given by the ratio of the Hilbert series of the invariant algebras of the natural action of $W$ and $W^t$ on the ring of polynomials ${\bf C}[x_1, \ldots , x_n]^{\otimes t}$. We introduce and study several statistics on the classical Weyl groups of type $B$ and $D$ and show that they can be used to give an explicit formula for $Z_{D_n}$. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, $Dmaj$ and $ned$ on $D_n$. The statistic $Dmaj$, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for $B_n$; namely, it allows us to find an explicit formula for $Z_{D_n}$. Our proof is based on the theory of $t$-partite partitions introduced by Gordon and further studied by Garsia and Gessel. Using similar ideas, we define the Mahonian statistic $ned$ also on $B_n$ and we find a new and simpler proof of the Adin?Roichman formula for $Z_{B_n}$. Finally, we define a new descent number $Ddes$ on $D_n$ so that the pair $(Ddes,Dmaj)$ gives a generalization to $D_n$ of the Carlitz identity on the Eulerian?Mahonian distribution of descent number and major index on the symmetric group.

Liouville property for groups and manifolds

Abstract: We introduce a method to estimate the entropy of random walks on groups. We apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not Liouville. We also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non-trivial Poisson boundary if and only if this group is not virtually nilpotent. This, in particular, shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient Lie group. As another application we get a series of examples of amenable groups such that any finite entropy non-degenerate measure on them has non-trivial Poisson boundary. Since the groups in question are amenable, they do admit measures such that the corresponding random walks have trivial boundary; the above shows that such measures on these groups have infinite entropy.

Extensions of modules over Schur algebras, symmetric groups and Hecke algebras

Abstract: We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.

Morita equivalence of Cherednik algebras

Abstract: We classify the rational Cherednik algebras H_c(W) (and their spherical subalgebras) up to isomorphism and Morita equivalence in case when W is the symmetric group and `c' is a generic parameter value.

Braids and Permutations

Abstract: E. Artin described all irreducible representations of the braid group B_k to the symmetric group S(k). We strengthen some of his results and, moreover, exhibit a complete picture of homomorphisms of B_k to S(n) for n 6 there exist, up to conjugation, exactly 3 irreducible representations of B_k into S(2k) with non-cyclic images but they all are imprimitive. We use these results to prove that for n 4 the intersection PB_k\cap B'_k of PB_k with the commutator subgroup B'_k=[B_k,B_k] is a completely characteristic subgroup of B'_k.