Showing papers on "Symmetric group published in 2001"

The Symmetric Group

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Three-point functions for M^N/S^N orbifolds with N=4 supersymmetry

Abstract: The D1-D5 system is believed to have an `orbifold point' in its moduli space where its low energy theory is a N=4 supersymmetric sigma model with target space M^N/S^N, where M is T^4 or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are `universal' in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space.

Rank-1 phenomena for mapping class groups

Abstract: We prove that every element of the mapping class group Γg has linear growth (confirming a conjecture of N. Ivanov) and that Γg is not boundedly generated. We also provide restrictions on linear representations of Γg and its finite index subgroups.

Sur l'anneau de cohomologie du schéma de Hilbert de ?

Abstract: We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

Abstract: In (Deodhar, i>Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials i>Px,w in the case where i>W is any Coxeter group. We explicitly describe the combinatorics in the case where W=\hbox{\ca}_n (the symmetric group on i>n letters) and the permutation i>w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for i>w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincare polynomial of the intersection cohomology of the Schubert variety corresponding to i>w is (1+q)^{l(w)} if and only if i>w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety i>Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of i>Xw when i>w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (i>Bn, i>F4, i>G2).

Approximate factorization and concentration for characters of symmetric groups

Abstract: We prove a factorization-concentration result for characters of symmetric groups. This is then applied to the asymptotic behaviour of the decomposition of the tensor representations. There are connections with the Pastur-Marcenko distribution of random matrix theory, and freely infinitely divisible distributions.

The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations

Abstract: We prove the convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool, we introduce a new semigroup of partial permutations. We describe its structure, representations, and characters. We also discuss filtrations on the subalgebra of invariants in the semigroup algebra. Bibliography: 10 titles.

The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes

Abstract: The Kronecker product of two Schur functions i>sμ and i>sν, denoted by i>sμ a i>sν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions μ and ν The coefficient of i>sλ in this product is denoted by γλμν, and corresponds to the multiplicity of the irreducible character χλ in χμχν We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for i>sλ[i>XY] to find closed formulas for the Kronecker coefficients γλμν when λ is an arbitrary shape and μ and ν are hook shapes or two-row shapes Remmel (JB Remmel, i>J Algebra 120 (1989), 100–118; i>Discrete Math 99 (1992), 265–287) and Remmel and Whitehead (JB Remmel and T Whitehead, i>Bull Belg Math Soc Simon Stiven 1 (1994), 649–683) derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach We believe that the approach of this paper is more natural The formulas obtained are simpler and reflect the symmetry of the Kronecker product

Order structure on the algebra of permutations and of planar binary trees

Abstract: In "Hopf algebra of the planar binary trees", Adv. Math. 139 (1998), no. 2, 293--309, we constructed by induction a graded associative product on the vector space generated by the planar binary trees (resp. the permutations). In the present paper we prove an explicit formula for this product in terms of a partial order on the set of planar binary trees (resp. the weak Bruhat order on the symmetric groups).

Symmetric and Exterior Powers, Linear Source Modules and Representations of Schur Superalgebras

Abstract: We study modules for the general linear group (over an infinite field of arbitrary characteristic) which are direct summands of tensor products of exterior powers and symmetric powers of the natural module. These modules, which we call listing modules, include the tilting modules and the injective modules for Schur algebras. The modules are studied via their relationship to linear source modules for symmetric groups on the one hand, and simple modules for Schur superalgebras on the other. Listing modules are parametrized by certain pairs of partitions. They are used to describe, by generators and relations, the Grothendieck ring of polynomial functors generated by the symmetric and exterior powers. We also (continuing work of J. Grabmeier) describe the vertices and sources of linear source modules for symmetric groups.

Hecke-Clifford superalgebras, crystals of type _{2ℓ}⁽²⁾ and modular branching rules for ̂_{}

Abstract: This paper is concerned with the modular representation theory of the ane Hecke-Cliord superalgebra, the cyclotomic Hecke-Cliord superalgebras, and projective representations of the symmetric group Our approach exploits crystal graphs of ane Kac-Moody algebras

Hyperimaginaries and automorphism groups

Abstract: A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M).In Section 2 we show that if T is simple and canonical bases of Lascar strong types exist in Meq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In Section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In Section 4 we study a certain group introduced in [5], which we call the Galois group of T, develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

Strong multiplicity one theorems for affine Hecke algebras of type A

Abstract: Given an irreducible module for the affine Hecke algebraH n of type A, we consider its restriction toH n−1. We prove that the socle of restriction is multiplicity free and moreover that the summands lie in distinct blocks. This is true regardless of the characteristic of the field or of the order of the parameterq in the definition ofH n . The result generalizes and implies the classical “branching rules” that describe the restriction of an irreducible representation of the symmetric groupS n toS n−1.

Representations of the Symmetric Group

Abstract: In this chapter we construct all the irreducible representations of the symmetric group. We know that the number of such representations is equal to the number of conjugacy classes (Proposition 1.10.1), which in the case of S n is the number of partitions of n. It may not be obvious how to associate an irreducible with each partition λ = (λ1, λ2,...., λl), but it is easy to find a corresponding subgroup S λ that is an isomorphic copy of S λl x Sλ2 x · · · x S λl, inside S n . We can now produce the right number of representations by inducing the trivial representation on each Sλ up to S n .

Young modules for symmetric groups

Abstract: Let K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

Irreducible Symmetric Group Characters of Rectangular Shape

Abstract: We give a new formula for the values of an irreducible character of the symmetric group S_n indexed by a partition of rectangular shape. Some observations and a conjecture are given concerning a generalization to arbitrary shapes.

Spectral Theorem for Convex Monotone Homogeneous Maps, and Ergodic Control

Abstract: We consider convex maps f:R^n -> R^n that are monotone (i.e., that preserve the product ordering of R^n), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is non-empty, is isomorphic to a convex inf-subsemilattice of R^n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation

Abstract: The notion of a geometric crystal was introduced by A.Berenstein and D.Kazhdan, motivated by the needs of representation theory of p-adic groups. It was shown by A.Braverman, A.Berenstein, and D.Kazhdan that some particular geometric crystals give rise to an interesting birational automorphism R of the Cartesian square of an n-dimensional torus, which satisfies the quantum Yang-Baxter equation and the unitarity condition. On the other hand, unitary set-theoretical solutions of the quantum Yang-Baxter equation were studied by Schedler, Soloviev, and the author. It was shown that the theory is especially nice if the solution satisfies an additional nondegeneracy condition. In particular, in this situation one can define the so called reduced structure group, whose complexity characterizes the complexity of the solution. In this note we show that the map R is nondegenerate, and give a new proof that it satisfies the quantum Yang-Baxter equation and the unitarity condition. Then we calculate the reduced structure group of R, and show that it is a subgroup of the "loop group" PGL(n,C(t))$. We also give a new, direct proof of a Theorem of Braverman and Kazhdan on the commutativity of two symmetric group actions on the space of matrices.

Ordering the Affine Symmetric Group

Abstract: We review several descriptions of the affine symmetric group We explicit the basis of its Bruhat order

Parking Functions of Types A and B

Abstract: The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use an analogous embedding for type B non-crossing partitions in order to answer a question raised by R. Stanley on the edge labeling of the type B non-crossing partitions lattice.

Conjugation in tree automorphism groups

Abstract: It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

Z-Measures on Partitions, Robinson-Schensted-Knuth Correspondence, and beta=2 Random Matrix Ensembles

Abstract: We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.

Representations of the symmetric group which are irreducible over subgroups

Abstract: Let F be an algebraically closed field of characteristic p, and Σn be the symmetric group on n letters. In this paper we classify all pairs (G,D), where D is an irreducible FΣn-module of dimension greater than 1 and G is a proper subgroup of Σn, such that the restriction D↓G is irreducible, provided p > 3.

Primitive Rank 3 Groups on Symmetric Designs

Abstract: We determine the symmetric designs {\cal D} which admit a group G\leq \Aut( {\cal D}) such that G has a nonabelian socle and is a primitive rank 3 group on points (and blocks).

Configurations of points

Abstract: Berry & Robbins, in their discussion of the spin–statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of n distinct particles in 3–space to the flag manifold of the unitary group U ( n )? I shall discuss this problem and various generalizations of it. In particular, there is a version in which U ( n ) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm9s equations, which are an integrable system of ordinary differential equations arising from the self–dual Yang-Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.

Estimates for homological dimension of configuration spaces of graphs

Abstract: We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b. 0. Introduction. Let Γ be a finite graph and n a natural number. The marked n-point configuration space of Γ is a subspace CnΓ in the nth cartesian power of Γ defined by CnΓ := {(x1, . . . , xn) ∈ Γ n : xi 6= xj for i 6= j}. Consider the natural free action of the symmetric group Sn on the space CnΓ defined by σ(x1, . . . , xn) = (xσ(1), . . . , xσ(n)) and put CnΓ := CnΓ/Sn. The space CnΓ is called the (unmarked) n-point configuration space of Γ . This paper reports on partial progress towards understanding the homology of configuration spaces of graphs, or even more generally of compact polyhedra. For another recent result in that direction, see [G]. We call a vertex v of Γ branched if it is adjacent to at least three edges. We denote by b = b(Γ ) the number of branched vertices in Γ . The main result of this paper is the following. 0.1. Theorem. Let Γ be a finite graph and n a natural number. (1) There exists a cube complex KnΓ of dimension min(b(Γ ), n) which embeds as a deformation retract into the configuration space CnΓ . (2) The fundamental group π1(CnΓ ) contains a subgroup isomorphic to the free abelian group Z with k = min(b(Γ ), [n/2]), where [x] denotes the integer part of x. 2000 Mathematics Subject Classification: Primary 55M10; Secondary 20J05, 51F99. The author was supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 023 14.

A Specht Module Analog for the Rook Monoid

Abstract: The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

Prime Power Degree Representations of the Symmetric and Alternating Groups

Abstract: In 1998, the second author of this paper raised the problem of classifying the irreducible characters of S n of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of S n of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskii's problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p -power degree > 1 and all of the irreducible characters of G have degrees that are relatively prime to p or are powers of p ? The case of the double covers of the symmetric and alternating groups will be dealt with in a forthcoming paper; in particular, this completes the answer to Zalesskii's problem. The paper is organized as follows. In Section 2, some results on hook lengths in partitions are proved. These results lead to an algorithm which allows us to show that every irreducible representation of S n with prime power degree is labelled by a partition having a large hook. In Section 3, we obtain a new result concerning the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove Theorem 2.4, the main result. To do so, we combine the algorithm above with Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for large n . To complete the proof, we check that the algorithm terminates appropriately for small n (that is, those n [les ] 9.25 · 10 8 ) with the aid of a computer. In the last section we derive the classification of irreducible characters of A n of prime power degree, and we solve Huppert's question for alternating groups.