Showing papers on "Symmetric group published in 2012"

FI-modules and stability for representations of symmetric groups

Abstract: In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.

Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements

Abstract: We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).

FI-modules over Noetherian rings

Abstract: FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.

Totally Nonfree Actions and the Infinite Symmetric Group

Abstract: We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group SN. The problem is closely related to the theory of characters and factor representations of groups.

Construction of S8 Liu J S-boxes and their applications

Abstract: In this paper, we present a method for the construction of 8x8 substitution boxes used in the area of cryptography. A symmetric group permutation S"8 is applied on Galois field elements that originally belong to GF(2^8), and as a consequence, 40320 new substitution boxes are synthesized. The Liu J substitution box is used as a seed in the creation process of the new algebraically complex nonlinear components. The core design of this new algorithm relies on the symmetric group permutation operation which is embedded in the algebraic structure of the new substitution box. We study the characteristics of the newly created substitution boxes and highlight the improved performance parameters and their usefulness in practical applications. In particular, we focus on the nonlinear properties and the behavior of input/output bits and determine the suitability of a particular substitution box for a specific type of encryption application. A comparison with some of the prevailing and popular substitution boxes is presented.

Hom-Lie Color Algebra Structures

Abstract: The aim of this article is to introduce the notion of Hom-Lie color algebras. This class of algebras is a natural generalization of the Hom-Lie algebras as well as a special case of the quasi-hom-Lie algebras. In the article, homomorphism relations between Hom-Lie color algebras are defined and studied. We present a way to obtain Hom-Lie color algebras from the classical Lie color algebras along with algebra endomorphisms and offer some applications. Also, we introduce a multiplier σ on the abelian group Γ and provide constructions of new Hom-Lie color algebras from old ones by the σ-twists. Finally, we explore some general classes of Hom-Lie color admissible algebras and describe all these classes via G–Hom-associative color algebras, where G is a subgroup of the symmetric group S 3.

Matrix product solution to an inhomogeneous multi-species TASEP

Abstract: We study a multi-species exclusion process with inhomogeneous hopping rates. This model is equivalent to a Markov chain on the symmetric group that corresponds to a random walk in the affine braid arrangement. We find a matrix product representation for the stationary state of this model. We also show that it is equivalent to a graphical construction proposed by Ayyer and Linusson, which generalizes Ferrari and Martin's construction.

On the classification of easy quantum groups

Abstract: In 2009, Banica and Speicher began to study the compact quantum subgroups of the free orthogonal quantum group containing the symmetric group S_n. They focused on those whose intertwiner spaces are induced by some partitions. These so-called easy quantum groups have a deep connection to combinatorics. We continue their work on classifying these objects introducing some new examples of easy quantum groups. In particular, we show that the six easy groups O_n, S_n, H_n, B_n, S_n' and B_n' split into seven cases on the side of free easy quantum groups. Also, we give a complete classification in the half-liberated case.

Computing Multiplicities of Lie Group Representations

Abstract: For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

Maximal subgroups of free idempotent-generated semigroups over the full transformation monoid

Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n− 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr. 2000 Mathematics Subject Classification: 20M05, 05E15, 20F05.

The partition algebra and the Kronecker coefficients

Abstract: We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behavior and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.

Polynomiality of monotone Hurwitz numbers in higher genera

Abstract: Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition.

On the base size for the symmetric group acting on subsets

Abstract: Let k, n be natural numbers with k ≦ n/2 and let Xn,k denote the set of k-element subsets of {1, 2, … n} The symmetric group Sn acts in a natural way on the set Xn,k Motivated by a question of Robert Guralnick, we investigate the size of a minimal base for this action We give constructions providing a minimal base if n = 2k or if n ≧ k2 We also describe a general process providing a base of size at most c times bigger than the size of a minimal base for some universal constant c

A simple model of trees for unicellular maps

Abstract: We consider unicellular maps, or polygon gluings, of fixed genus. In FPSAC '09 the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the ``recursive part'' of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, or the Lehman-Walsh/Goupil-Schaeffer formulas. Thanks to previous work of the second author this also leads us to a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group.

Combinatorial Markov chains on linear extensions

Abstract: We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.

On the distribution of galois groups

Abstract: Let G be a subgroup of the symmetric group Sn, and let δG=∣Sn/G∣−1 where ∣Sn/G∣ is the index of G in Sn. Then there are at most On,ϵ(Hn−1+δG+ϵ) monic integer polynomials of degree n that have Galois group G and height not exceeding H, so there are only a “few” polynomials having a “small” Galois group.

Groups with the same order and degree pattern

Abstract: The degree pattern of a finite group has been introduced in [18]. A group M is called k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. It is shown that the alternating groups A m and A m+1, for m = 27, 35, 51, 57, 65, 77, 87, 93 and 95, are OD-characterizable, while their automorphism groups are 3-fold OD-characterizable. It is also shown that the symmetric groups S m+2, for m = 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89 and 97, are 3-fold OD-characterizable. From this, the following theorem is derived. Let m be a natural number such that m ⩽ 100. Then one of the following holds: (a) if m ≠ 10, then the alternating groups A m are OD-characterizable, while the symmetric groups S m are OD-characterizable or 3-fold OD-characterizable; (b) the alternating group A 10 is 2-fold OD-characterizable; (c) the symmetric group S 10 is 8-fold OD-characterizable. This theorem completes the study of OD-characterizability of the alternating and symmetric groups A m and S m of degree m ⩽ 100.

Computing Multiplicities of Lie Group Representations

Abstract: For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

A Markov Chain on the Symmetric Group That Is Schubert Positive

Abstract: We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.

Singularities of symmetric hypersurfaces and Reed-Solomon codes

Abstract: We determine conditions on $q$ for the nonexistence of deep holes of the standard Reed-Solomon code of dimension $k$ over $\mathbb F_q$ generated by polynomials of degree $k+d$. Our conditions rely on the existence of $q$-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over $\mathbb F_q$. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of $q$-rational points is established.

Solving polynomial systems globally invariant under an action of the symmetric group and application to the equilibria of N vortices in the plane

Abstract: We propose an efficient algorithm to solve polynomial systems of which equations are globally invariant under an action of the symmetric group GN acting on the variable xi with σ(xi) = xσ(i) and the number of variables is a multiple of N. For instance, we can assume that swapping two variables (or two pairs of variables) in one equation gives rise to another equation of the system (perhaps changing the sign). The idea is to apply many times divided difference operators to the original system in order to obtain a new system of equations involving only the symmetric functions of a subset of the variables.The next step is to solve the system using Grobner techniques; this is usually several order faster than computing the Grobner basis of the original system since the number of solutions of the corresponding ideal, which is always finite has been divided by at least N!.To illustrate the algorithm and to demonstrate its efficiency, we apply the method to a well known physical problem called equilibria positions of vortices. This problem has been studied for almost 150 years and goes back to works by von Helmholtz and Lord Kelvin. Assuming that all vortices have same vorticity, the problem can be reformulated as a system of polynomial equations invariant under an action of GN. Using numerical methods, physicists have been able to compute solutions up to N ≤ 7 but it was an open challenge to check whether the set of solution is complete. Direct naive approach of Grobner bases techniques give rise to hard-to-solve polynomial system: for instance, when N = 5, it takes several days to compute the Grobner basis and the number of solutions is 2060. By contrast, applying the new algorithm to the same problem gives rise to a system of 17 solutions that can be solved in less than 0.1 sec. Moreover, we are able to compute all equilibria when N ≤ 7.