Showing papers on "Symmetric group published in 2005"

A combinatorial formula for the character of the diagonal coinvariants

Abstract: Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doubly graded S-module can be expressed using the Frobenius characteristic map as nabla en, where en is the n-th elementary symmetric function and nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture. Finally, we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en.

Handbook of Computational Group Theory

Abstract: History of Computational Group Theory BACKGROUND MATERIALS Fundamentals Group Actions Series Presentation of Groups Presentation of Subgroups Abelian Group Presentations Representation Theory, Modules, Extension, Derivations, and Complements Field Theory REPRESENTING GROUPS ON A COMPUTER Representing Groups on Computers The Use of Random Methods in CGT Some Structural Calculators Computing with Homorphisms COMPUTATION IN FINITE PERMUTATION GROUPS The Calculation of Orbits and Stabilizers Testing for Alt (W) and Sym (W) Finding Block Systems Bases and Strong Generating Sets Homomorphisms from Permutation Groups Backtrack Searches Sylow Subgroups, P-cores, and the Solvable Radical Applications COSET ENUMERATION The Basic Procedure Strategies for Coset Enumeration Presentations of Subgroups Finding All Subgroups Finding All Subgroups Up to a Given Index Applications PRESENTATION OF GIVEN GROUPS Finding a Presentation of a Given Group Finding a Presentation of a Strong Generating Set The Sims 'Verify' Algorithm REPRESENTATIONS, COHOMOLOGY, AND CHARACTERS Computation in Finite Fields Elemetary Computational Linear Algebra Factorizing Polynomials Over Finite Fields Testing KG-Models for Irreducibility - The Meataxe Related Computations Cohomology Computing Character Tables Structural Investigation of Matrix Groups COMPUTATION WITH POLYCYCLIC GROUPS Polycyclic Presentations Examples of Polycyclic Groups Subgroups and Membership Testing Factor Groups and Homomorphisms Subgroup Series Orbit-Stabilizer Methods Complements and Extensions Intersections, Centralizers, and Normalizers Automorphism Groups The Structure of Finite Solvable Groups COMPUTING QUOTIENTS OF FINITELY PRESENTED GROUPS Finite Quotients and Automorphism Groups of Finite Groups Abelian Quotients Practical Computation of the HNF and SNF P-Quotients of Finitely-Presented Groups ADVANCED COMPUTATIONS IN FINITE GROUPS Some Useful Subgroups Computing Composition and Chief Series Applications of the Solvable Radical Method Computing the Subgroups of a Finite Group Appication - Enumerating Finite Unlabelled Structures LIBRARIES AND DATABASES Primitive Permutation Groups Transitive Permutation Groups Perfect Groups The Small Groups Library Crystallorgraphic Groups Other Databases REWRITING SYSTEMS Monoid Systems Rewriting Systems Rewriting Systems in Monoids and Groups Rewriting Systems for Polycyclic Groups Verifying Nilpotency Applications FINITE STATE AUTOMATA AND AUTOMATIC GROUPS Finite State Automata Automatic Groups The Algorithm to Compute Shortlex Automatic Structures Related Algorithms Applications

On the Representation Theory of the Symmetric Groups

Abstract: We present here a new approach to the description of finite-dimensional complex irreducible representations of the symmetric groups due to A. Okounkov and A. Vershik. It gives an alternative construction to the combinatorial one, which uses tabloids, polytabloids, and Specht modules. Its aim is to show how the combinatorial objects of the theory (Young diagrams and tableaux) arise from the internal structure of the symmetric group. Bibliography: 9 titles.

Group Actions on Stacks and Applications

Abstract: The motivation at the origin of this article is to investigate some ways of constructing moduli for curves, and covers above them, using tools from stack theory. This idea arose from reading Bertin–Mezard [BeM, esp. Sec. 5] and Abramovich– Corti–Vistoli [ACV]. Our approach is in the spirit of most recent works, where one uses the flexibility of the language of algebraic stacks. This language has two (twin) aspects: category-theoretic on one side and geometric on the other. Some of our arguments, especially in Section 8, are formal arguments involving general constructions concerning group actions on algebraic stacks (this is more on the categoric side). They are, intrinsically, natural enough to preserve the “modular” aspect. In trying to isolate these arguments, we were led to write results of independent interest. It seemed therefore more adequate to present them in a separate, self-contained part. Thus the article is split into two parts of comparable size. More specifically, groups are ubiquitous in algebraic geometry (when one focuses on curves and maps between them, examples include the automorphism group, fundamental group, monodromy group, permutation group of the ramification points, . . .). It is natural to ask whether we can handle group actions on stacks in the same fashion as we do on schemes. For example, we expect: that the quotient of the stack of curves with ordered marked points Mg,n by the symmetric group should classify curves with unordered marked points; that if G acts on a scheme X then the fixed points of the stack Pic(X) under G should be related to G-linearized line bundles on X; and that the quotient of the modular stack curve X1(N ) by (Z/NZ)× should be X0(N ) (the notation is, we hope, well known to the reader). Other important examples appear in the literature: action of tori on stacks of stable maps in Gromov–Witten theory [Ko; GrPa], and action of the symmetric group Sd on a stack of multisections in [L-MB, (6.6)]. Our aim is to provide the material necessary to handle the questions raised here and then answer them, as well as to give other applications. Let us now explain in more detail the structure and results of this paper. In Part A we discuss the notion of a group action on a stack. We are mainly interested in giving general conditions under which the fixed points and the quotient of an algebraic stack are algebraic. In Sections 1 and 2 we give definitions and basics on actions. For simplicity let us now consider a flat group scheme G and an algebraic stack M, both of finite presentation (abbreviated fp) over some

Cohomological aspects of magnus expansions.

Abstract: We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group Aut(Fn).The extended ones are not homomor- phisms, but satisfy an infinite sequence of coboundary relations, so that we call them the Johnson maps.In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group Aut(Fn) and the mapping class groups for surfaces.The first one means that the first Johnson map is a twisted 1-cocycle of the group Aut(Fn).Its cohomology class coincides with "the unique elementary particle" of all the Morita-Mumford classes on the mapping class group for a surface (Ka1) (KM1).The second one restricted to the mapping class group is equal to a fundamental relation among twisted Morita-Mumford classes pro- posed by Garoufalidis and Nakamura (GN) and established by Morita and the author (KM2).This means we give a coherent proof of the fundamental relation.The first Johnson map gives the abelianization of the induced automorphism group IAn of a free group in an explicit way.

Noncrossing Partitions for the Group D n

Abstract: The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W. It is a self-dual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2,...,n} defined by Kreweras in 1972 when W is the symmetric group Sn, and to its type B analogue defined by the second author in 1997 when W is the hyperoctahedral group. We give a combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type Dn, thus answering a question of Bessis. Using this description, we compute a number of fundamental enumerative invariants of this lattice, such as the rank sizes, number of maximal chains, and Mobius function. We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B, and C. This leads to a (case-by-case) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to W-orbits.

On the Hall algebra of an elliptic curve, I

Abstract: In this article we describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, we study a certain natural subalgebra U_X of DH_X for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants C[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}}, i.e. the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.

A New Approach to the Representation Theory of the Symmetric Groups. II

Abstract: The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles.

Monoids and groups of I-type

Abstract: A monoid S generated by {x 1,. . .,x n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaMn of rank n generated by {u 1,. . .,u n} to S so that for all a∈FaMn one has {v(u 1 a),. . .,v(u n a)}={x 1 v(a),. . .,x n v(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fan and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups.

On the noncommuting graph associated with a finite group.

Abstract: Let G be a finite group. We define the noncommuting graph ∇(G) as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. We study some properties of ∇(G) and prove that, for many groups G, if H is a group with ∇(G) isomorphic to ∇(H) then |G| = |H|.

Enriched $P$-partitions and peak algebras

Abstract: We develop a more general view of Stembridge's enriched $P$-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Initially we focus on commutative peak algebras, spanned by sums of permutations with the same number of peaks, where we consider several variations on the definition of "peak." Whereas Stembridge's enriched $P$-partitions are related to quasisymmetric functions (the dual coalgebra of Solomon's type A descent algebra), our generalized enriched $P$-partitions are related to type B quasisymmetric functions (the dual coalgebra of Solomon's type B descent algebra). Using these functions, we move on to explore (non-commutative) peak algebras spanned by sums of permutations with the same set of peaks. While some of these algebras have been studied before, our approach gives explicit structure constants with a combinatorial description.

Cohomological structure of the mapping class group and beyond

Abstract: In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

Group Laws and Free Subgroups in Topological Groups

Abstract: A proof is given that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property, almost all finite subsets of the group are shown to generate free subgroups. Consequences of these theorems are derived for: Thompson's group , weakly branch groups, automorphism groups of regular trees, and profinite groups with alternating composition factors of unbounded degree.

The symmetric group defies strong Fourier sampling

Abstract: We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem in general groups. Specifically, we show that the hidden subgroup problem in the symmetric group cannot be efficiently solved by strong Fourier sampling. Indeed we prove the stronger statement that no measurement of a single coset state can reveal more than an exponentially small amount of information about the identity of the hidden subgroup, in the special case relevant to the graph isomorphism problem.

The hidden subgroup problem and permutation group theory

Abstract: We employ concepts and tools from the theory of finite permutation groups in order to analyse the Hidden Subgroup Problem via Quantum Fourier Sampling (QFS) for the symmetric group. We show that under very general conditions both the weak and the random-strong form (strong form with random choices of basis) of QFS fail to provide any advantage over classical exhaustive search. In particular we give a complete characterisation of polynomial size subgroups, and of primitive subgroups, that can be distinguished from the identity subgroup with the above methods. Furthermore, assuming a plausible group theoretic conjecture for which we give supporting evidence, we show that weak and random-strong QFS for the symmetric group have no advantage whatsoever over classical search.

On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function

Abstract: We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding ``zeta function'' ζ G s = ∑ M max ⁡ G G : M - s , which appears in many probabilistic applications. More specifically, we are able to show that for simple groups G and for any fixed real number s > 1 , ζ G s → 0 as G → ∞ . This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343, that the symmetric group S n has n o ⁡ 1 conjugacy classes of primitive maximal subgroups.

Symmetric Groups and Expander Graphs

Abstract: We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling and other areas.

The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution

Abstract: In this paper, we compute all the moments of the real Wishart distribution. To do so, we use the Gelfand pair (S2k,H), where H is the hyperoctahedral group, the representation theory of H and some techniques based on graphs.

Uncountable cofinalities of permutation groups

Abstract: A sufficient criterion is found for certain permutation groups to have uncountable strong cofinality, that is, they cannot be expressed as the union of a countable, ascending chain of proper subsets such that and . This is a strong form of uncountable cofinality for , where each is a subgroup of . This basic tool comes from a recent paper by Bergman on generating systems of the infinite symmetric groups, which is discussed in the introduction. The main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. Thus the result also unifies various known results about cofinalities. A notable example is the group BSym () of all bounded permutations of the rationals which has uncountable cofinality but countable strong cofinality.

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

Abstract: In a recent paper, Backelin, West and Xin describe a map ?* that recursively replaces all occurrences of the pattern k... 21 in a permutation ? by occurrences of the pattern (k?1)... 21 k. The resulting permutation ?*(?) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map ?* commutes with taking the inverse of a permutation. In the BWX paper, the definition of ?* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map ?* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T? be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group $${\cal S}$$ n that avoid T equals the number of permutations of $${\cal S}$$ n that avoid T?. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of $${\cal S}$$ n that avoid T is equal to the number of involutions of $${\cal S}$$ n avoiding T?, as recently conjectured by Jaggard.

Signed words and permutations, ii;

Abstract: As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.

The N-eigenvalue problem and two applications

Abstract: Consider the following two questions. (1) When does a compact Lie group G ⊂ U(n) have an element g ∈ G possessing exactly two eigenvalues? (2) When does a compact Lie group G ⊂ U(n) have a cocharacter U(1) → G such that the compositionU(1) → U(n) is a representation ofU(1)with exactly two weights? A solution to the second problem gives a family of solutions to the first, by choosing g to be almost any element of the image of U(1). The converse is not true. For one thing, any noncentral element of order 2 in G has exactly two eigenvalues. To eliminate these essentially trivial solutions,we can insist that the ratio between the two eigenvalues is not −1. There remain interesting cases of finite groups G satisfying the first (but obviously not the second) condition, especially when the ratio of eigenvalues is a third or fourth root of unity (see [3, 18, 35] for classification results). On the other hand,when G is infinite modulo center, the solutions of the two problems are essentially the same, though the historical reasons for considering them were quite different. The first problem was recently solved in the infinite-mod-center case by Freedman, Larsen, and Wang [13] with an eye toward understanding representations of Hecke algebras. The second problem was solved by Serre [27] nearly thirty years ago in order to classify representations arising from Hodge-Tate modules of weight 1.

The Berry-Esseen bound for character ratios

Abstract: Let A be a partition of n chosen from the Plancherel measure of the symmetric group S n , let Χ λ (12) be the irreducible character of the symmetric group parameterized by A evaluated on the transposition (12), and let dim(A) be the dimension of the irreducible representation parameterized by A. Fulman recently obtained the convergence rate of O(n -s ) for any 0 < s < 1 2 in the central limit theorem for character ratios (n-1) 2 Χ λ (12) dim(λ) by developing a connection between martingale and character ratios, and he conjectures that the correct speed is O(n -1/2 ). In this paper we confirm the conjecture via a refinement of Stein's method for exchangeable pairs.

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. While the former admits a deep geometric interpretation, the latter leads to a purely combinatorial construction of the representations of $H$, including the Dipper--James theory of Specht modules. In this paper, 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.