Showing papers on "Symmetric group published in 1998"
TL;DR: In this paper, the quantum automorphism groups of finite spaces were determined, i.e., compact matrix quantum groups in the sense of Woronowicz, and the quantum groups were defined.
Abstract: We determine the quantum automorphism groups of finite spaces. These are compact matrix quantum groups in the sense of Woronowicz.
422 citations
TL;DR: Brou et al. as mentioned in this paper introduced monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others.
Abstract: Presentations a la Coxeter are given for all irreducible nite com plex re ection groups They provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite Coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra Hecke algebras We introduce monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others Summary Introduction Complex re ection groups and their presentations A Background from complex re ection groups B Presentations Braid groups and their diagrams A Generalities about hyperplane complements B Generalities about the braid groups C The braid diagrams Proofs of the main theorems for the braid groups B de e r A Notation and prerequisites B Computation of B de e r and of its center for d C Computation of B e e r and of its center Hecke algebras A Background from di erential equations and monodromy B A family of monodromy representations of the braid group C Hecke algebras Diagrams and tables Appendix Generators of the monodromy around a divisor Appendix tables to Mathematics Subject Classi cation Primary G We thank Jean Michel Peter Orlik Pierre Vogel for useful conversations and the Isaac Newton Institute for its hospitality while the last version of this manuscript was written up The second named author gratefully acknowledges nancial support by the Fondation Alexander von Humboldt for his stays in Paris Michel Brou e Gunter Malle Raphael Rouquier
367 citations
TL;DR: Based on the theory of Dunkl operators, the authors presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ N fixme.
Abstract: Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ
N
. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S
N
, our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem
involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.
286 citations
TL;DR: In this article, the authors consider representations of symmetric groups and give the asymptotic behaviour of the characters when the corresponding Young diagrams, rescaled by a factorq−1/2, converge to some prescribed shape.
215 citations
01 Jan 1998
TL;DR: In this article, the authors consider hyperplane arrangements that interpolate between two well-known arrangements: (1) the set B n of hyperplanes xi = xj, for 1 ≤ i < j ≤ n and m ∈ Ω(n) ∈ n, and (2) the sets B n (xi, xj) = m.
Abstract: A (real) hyperplane arrangement is a discrete set of hyperplanes in ℝn. We will be concerned with hyperplane arrangements that “interpolate” between two well-known arrangements: (1) the set B n of hyperplanes xi = xj, for 1 ≤ i < j ≤ n, and (2) the set B n of hyperplanes xi = xj = m, for 1 ≤ i < j ≤ n and m ∈ ℤ. The arrangement B n is known as the braid arrangement or the reflection arrangement of type An−1 (i. e., the set of reflecting hyperplanes of the symmetric group
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which is the Coxeter group of type An−1). Similarly, B n is the affine braid arrangement or reflection arrangement of type б n i. e., the set of reflecting hyperplanes of the affine Weyl group
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of type A n .
106 citations
TL;DR: In this article, the authors prove the quantum version of Kleshchev's modular branching rule for symmetric groups for Hecke algebras at the roots of unity.
Abstract: We prove the quantum version - for Hecke algebras at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This
result describes the socle of the restriction of an irreducible -module to the subalgebra
. As a consequence, we describe the quantum version of the Mullineux involution describing the
irreducible module obtained on twisting an irreducible module with the sign representation.1991
Mathematics Subject Classification: 20C05, 20G05.
97 citations
TL;DR: In this article, the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes is illuminated.
Abstract: We illuminate the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes. Equivalently, the structure constants for the ring of polynomials in variables $x_1,x_2,...$ in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young's lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.
87 citations
TL;DR: In this paper, the authors compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups.
81 citations
TL;DR: In this paper, a Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators, i.e., w is a fully commutable element whose reduced expression can be expressed by any generator.
Abstract: A Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree n, the number of fully commutative elements is the nth Catalan number. The Coxeter groups with finitely many fully commutative elements can be arranged into seven infinite families A_n, B_n, D_n, E_n, F_n, H_n and I_2(m). For each family, we provide explicit generating functions for the number of fully commutative elements and the number of fully commutative involutions; in each case, the generating function is algebraic.
80 citations
TL;DR: In this article, the structure of a semigroup S generated by a permutation group G of units and an idempotent ϵ is studied and it is shown that S is a regular semigroup.
Abstract: It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n − 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n − 1 while the analogous result is true for the semigroup of all n × n matrices over a field. In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ϵ . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow. In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ϵ is an idempotent of rank n − 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup. The fourth section deals with a special ca...
76 citations
TL;DR: For a semigroup S and a set B, the relative rank of S modulo A is the minimal cardinality of a setB such that S can be generated.
Abstract: For a semigroup S and a set the relative rank of S modulo A is the minimal cardinality of a setB such that generates S. We show that the relative rank of an infinite full transformation semigroup modulo the symmetric group, and also modulo the set of all idempotent mappings, is equal to 2. We also characterise all pairs of mappings which, together with the symmetric group or the set of all idempotents, generate the full transformation semigroup.
TL;DR: This work considers Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank, and relates the descriptive complexity of the equivalence relation to the nature of its complete invariants.
TL;DR: This paper replaces the matrix multiplications in Clausen's algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and writes the result in a form similar to Horner's rule.
Abstract: This paper introduces new techniques for the efficient computation of Fourier transforms on symmetric groups and their homogeneous spaces. We replace the matrix multiplications in Clausen's algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and write the result in a form similar to Horner's rule. The algorithm we obtain computes the Fourier transform of a function on S n in no more than 3/4n(n-1)|S n | multiplications and the same number of additions. Analysis of our algorithm leads to several combinatorial problems that generalize path counting. We prove corresponding results for inverse transforms and transforms on homogeneous spaces.
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TL;DR: In this paper, the authors compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups.
Abstract: We compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups.
TL;DR: In this article, a base for SU(3) irreps is constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions, represented as the symmetric group of permutations of the particle coordinates of these space.
Abstract: Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these space. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.
TL;DR: In this paper, the authors define the vexillary elements in the hyperoctahedral group to be those for which the Stanley 1 symmetric function is a single Schur Q-function.
Abstract: In analogy with the symmetric group, we define the vexillary elements in the hyperoctahedral group to be those for which the Stanley 1 symmetric function is a single Schur Q-function. We show that the vexillary elements can be again determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Q -functions with a superfied Schur function and a method for determining the shape of a vexillary signed permutation using jeu de taquin.
TL;DR: In this paper, the authors give some structure to the Brown-Peterson cohomology of a wide class of spaces and show that it is even dimensional (concentrated in even degrees) and flat as a BP ∗-module for the category of finitely presented BP ∆(BP )-modules.
Abstract: We give some structure to the Brown-Peterson cohomology (or its p-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even dimensional. We can say that the Brown-Peterson cohomology is even dimensional (concentrated in even degrees) and is flat as a BP ∗-module for the category of finitely presented BP ∗(BP )-modules. At first glance this would seem to be a very restricted class of spaces, but the world abounds with naturally occurring examples: Eilenberg-Mac Lane spaces, loops of finite Postnikov systems, classifying spaces of all finite groups whose Morava K-theory is known (including the symmetric groups), QS2n, BO(n), MO(n), BO, ImJ , etc. We finish with an explicit algebraic construction of the Brown-Peterson cohomology of a product of Eilenberg-Mac Lane spaces. ∗Partially supported by the National Science Foundation
TL;DR: In this article, the correlation of adjacency operators on the infinite symmetric group which are parametrized by the Young diagrams is studied and the correlation function under suitable normalization and through the infinite volume limit is computed.
Abstract: An adjacency operator on a group is a formal sum of (left) regular representations over a conjugacy class. For such adjacency operators on the infinite symmetric group which are parametrized by the Young diagrams, we discuss the correlation of their powers with respect to the vacuum vector state. We compute exactly the correlation function under suitable normalization and through the infinite volume limit. This approach is viewed as a central limit theorem in quantum probability, where the operators are interpreted as random variables via spectral decomposition. In [K], Kerov showed the corresponding result for one-row Young diagrams. Our formula provides an extension of Kerov's theorem to the case of arbitrary Young diagrams.
TL;DR: It is shown that the cyclic Eulerian elements linearly span a commutative semisimple algebra of QSn, which is naturally isomorphic to the algebra of the classical Euleria elements.
Abstract: LetSnbe the symmetric group on {1,?,n} and QSn its group algebra over the rational field; we assumen?2. ??Snis said a descent ini, 1?i?n-1, if ?(i)? (i+1); moreover, ? is said to have a cyclic descent if ?(n)?(1). We define the cyclic Eulerian elements as the sums of all elements inSnhaving a fixed global number of descents, possibly including the cyclic one. We show that the cyclic Eulerian elements linearly span a commutative semisimple algebra of QSn, which is naturally isomorphic to the algebra of the classical Eulerian elements. Moreover, we give a complete set of orthogonal idempotents for such algebra, which are strictly related to the usual Eulerian idempotents.
01 Jan 1998
TL;DR: In the recent years, the theory of deformations, mainly in the spirit of quantum groups and quantum algebras, has been the subject of considerable interest in statistical physics as discussed by the authors.
Abstract: In the recent years, the theory of deformations, mainly in the spirit of quantum groups and quantum algebras, has been the subject of considerable interest in statistical physics. More precisely, deformed oscillator algebras have proved to be useful in parasiatistics(connected to irreducible representations, of dimensions greater than 1, of the symmetric group), in anyonic statistics(connected to the braid group) that concerns only particles in (one or) two dimensions, and in q-deformed statisticsthat may concern particles in arbitrary dimensions. In particular, the q-deformed statistics deal with:
(i)
q-bosons (which are bosons obeying a q-deformed Bose-Einstein distribution),
(ii)
q-fermions (which are fermions obeying a q-deformed Fermi-Dirac distribution), and
(iii)
quons (with qsuch that q k = 1, where k∈ ℕ \ {0,1}) which are objects, refered to as k-fermions in this work, interpolating between fermions (corresponding to k= 2) and bosons (corresponding to k→ ∞).
TL;DR: An asymptotic normality theorem for the number of descents and major index in conjugacy classes with large cycles is proved.
TL;DR: In this article, the degree and point stabilizer of finite simple exceptional twisted groups are studied, and it is shown that these parameters are known for all finite simple groups, modulo the classification of simple groups.
Abstract: A minimal permutation representation of a group is its faithful permutation representation of least degree. Here the minimal permutation representations of finite simple exceptional twisted groups are studied: their degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, are found. We can thus assert that, modulo the classification of finite simple groups, the aforesaid parameters are known for all finite simple groups.
[...]
TL;DR: Symplectic matroids as mentioned in this paper are a special case of Coxeter matroid, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube.
Abstract: A symplectic matroid is a collection {\cal B} of k-element subsets of J e l1, 2, …, n, 1a, 2a, … nar, each of which contains not both of i and ia for every i ≤ n, and which has the additional property that for any linear ordering p of J such that i p j implies ja p ia and i p ja implies j p ia for all i, j ≤ n, {\cal B} has a member which dominates element-wise every other member of {\cal B}. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k e n, and which are equivalent to Bouchet‘s symmetric matroids or 2-matroids.
TL;DR: Barbasch and Vogan gave a beautiful rule for restricting and inducing Kazhdan-Lusztig representations of Weyl groups as discussed by the authors, and this rule implies and generalizes the Littlewood-Richardson rule for decomposing outer products of representations of the symmetric groups.
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TL;DR: Olshanski and Borodin this article considered the tail point processes which describe the limit behavior of the Thoma parameters with large numbers and showed that these processes have determinantal form with a kernel which generalizes the well-known sine kernel arising in random matrix theory.
Abstract: In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin, math.RT/9804087) we developed an approach to certain probability distributions on the Thoma simplex. The latter has infinite dimension and is a kind of dual object for the infinite symmetric group. Our approach is based on studying the correlation functions of certain related point stochastic processes.
In the present paper we consider the so-called tail point processes which describe the limit behavior of the Thoma parameters (coordinates on the Thoma simplex) with large numbers. The tail processes turn out to be stationary processes on the real line. Their correlation functions have determinantal form with a kernel which generalizes the well-known sine kernel arising in random matrix theory. Our second result is a law of large numbers for the Thoma parameters. We also produce Sturm-Liouville operators commuting with the Whittaker kernel introduced in Part II and with the generalized sine kernel.
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TL;DR: The symmetric group S_n possesses a nontrivial central extension, whose irreducible representations coincide with the representations of a certain algebra A_n and Young symmetrizers.
Abstract: The symmetric group S_n possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S_n itself, coincide with the irreducible representations of a certain algebra A_n. Recently M.~Nazarov realized irreducible representations of A_n and Young symmetrizers by means of the Howe duality between the Lie superalgebra q(n) and the Hecke algebra H_n, the semidirect product of S_n with the Clifford algebra C_n on n indeterminates.
Here I construct one more analog of Young symmetrizers in H_n as well as the analogs of Specht modules for A_n and H_n.
TL;DR: In this paper, the W -module structure of a large number of cells of Harish-Chandra modules for a classical group G, where W is the complex Weyl group, is computed.
Abstract: We compute the W -module structure of a large number of cells of Harish-Chandra modules for a classical group G , where W is the complex Weyl group.
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TL;DR: Olshanski and Borodin this paper studied the correlation functions for point stochastic processes in terms of multivariate hypergeometric functions and showed that the lifted correlation functions are given by a determinantal formula involving a kernel.
Abstract: We continue the study of the correlation functions for the point stochastic processes introduced in Part I (G.Olshanski, math.RT/9804086). We find an integral representation of all the correlation functions and their explicit expression in terms of multivariate hypergeometric functions. Then we define a modification (``lifting'') of the processes which results in a substantial simplification of the structure of the correlation functions. It turns out that the ``lifted'' correlation functions are given by a determinantal formula involving a kernel. The latter has the form (A(x)B(y)-B(x)A(y))/(x-y), where A and B are certain Whittaker functions. Such a form for correlation functions is well known in the random matrix theory and mathematical physics. Finally, we get some asymptotic formulas for the correlation functions which are employed in Part III (A.Borodin and G.Olshanski, math.RT/9804088).
TL;DR: In this article, the Murnaghan-Nakayama type for mulas for computing the irreducible characters of these algebras was given. But this was not the case for the characters of the Weyl groups.
Abstract: Iwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike (AK), Broue and Malle (BM), and Ariki (Ari). In this paper we give Murnaghan-Nakayama type for- mulas for computing the irreducible characters of these algebras. Our method is a gen- eralization of that in our earlier paper (HR) in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene (Gre), who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike (AK) and Ariki (Ari). The finite irreducible complex reflection groups come in three infinite families: the symmetric groups Sn on n letters; the wreath product groups r Sn ,w here r denotes the cyclic group of order r; and a series of index-p subgroups G(r, p, n )o f r S n for each positive integer p that divides r. In the classification of finite irreducible reflection groups, besides these infinite families Sn, r ,a nd G ( r , p , n), there exist only 34 excep- tional irreducible reflection groups, see (ST). A formula for the irreducible characters of the Iwahori-Hecke algebras for Sn is known (Ram), (KW), (vdJ). This formula is a q-analogue of the classical Murnaghan- Nakayama formula for computing the irreducible characters of Sn. Similar formulas for the characters of the groups G(r, p, n) are classically known, see (Mac), (Ste), (AK), (Osi) and the references there. Formulas of this type are also known for the Iwahori-Hecke al- gebras of Weyl groups of types B and D (HR), (Pfe1), (Pfe2). Recently, Iwahori-Hecke algebras have been constructed for the groups r Sn and G(r, p, n) (AK), (BM), (Ari). In this paper we derive Murnaghan-Nakayama type formulas for computing the irreducible characters of the Iwahori-Hecke algebras that correspond to r Sn and G(r, p, n). Hoefsmit (Hfs) has given explicit analogues of Young's seminormal representations for the Iwahori-Hecke algebras of types An 1, Bn ,a nd D n. Ariki and Koike, (AK) and (Ari), have constructed "Hoefsmit-analogues" of Young's seminormal representations
TL;DR: In this paper, it was shown that B is the matrix with rows and columns indexed by permutations with σ, τ entry equal to qi(στ −1) where i(π) is the number of inversions of π.
Abstract: Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system An−1. Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matrix with rows and columns indexed by permutations with σ, τ entry equal to qi(στ −1) where i(π) is the number of inversions of π. Equivalently B is the matrix for left multiplication on CSn by Γn(q) = ∑ π∈Sn qπ. Clearly B commutes with the right-regular action of Sn on CSn. A general theorem of Varchenko applied in this special case shows that B is singular exactly when q is a j(j − 1)st root of 1 for some j between 2 and n. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the Sn-module structure of the nullspace of B in the case that B is singular. Our first result is that ker(B) = indn Sn−1(Lien−1)/Lien in the case that q = e2πi/n(n−1) where Lien denotes the multilinear part of the free Lie algebra with n generators. Our second result gives an elegant formula for the determinant of B restricted to the virtual Sn-module P η with characteristic the power sum symmetric function pη(x).