Permutation Representations Research Articles

Degree-One Rational Cherednik Algebras for the Symmetric Group

Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of $\mathfrak{gl}_n$-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the $\mathfrak{sl}_n$-type rational Cherednik algebras $H_{0,c}$.

Finite totally k-closed groups

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\dots\times \Omega=\Omega^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $k\geq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.

Representations and the reduction theorem for ultragraph Leavitt path algebras

In this paper, we study representations of ultragraph Leavitt path algebras via branching systems and, using partial skew ring theory, prove the reduction theorem for these algebras. We apply the reduction theorem to show that ultragraph Leavitt path algebras are semiprime and to completely describe faithfulness of the representations arising from branching systems, in terms of the dynamics of the branching systems. Furthermore, we study permutative representations and provide a sufficient criteria for a permutative representation of an ultragraph Leavitt path algebra to be equivalent to a representation arising from a branching system. We apply this criteria to describe a class of ultragraphs for which every representation (satisfying a mild condition) is permutative and has a restriction that is equivalent to a representation arising from a branching system.

The degrees of toroidal regular proper hypermaps

Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type (3, 3, 3).

Skew characters and cyclic sieving

AbstractIn 2010, Rhoades proved that promotion on rectangular standard Young tableaux, together with the associated fake-degree polynomial, provides an instance of the cyclic sieving phenomenon. We extend this result tom-tuples of skew standard Young tableaux of the same shape, for fixedm, subject to the condition that themth power of the associated fake-degree polynomial evaluates to nonnegative integers at roots of unity. However, we are unable to specify an explicit group action. Put differently, we determine in which cases themth tensor power of a skew character of the symmetric group carries a permutation representation of the cyclic group.To do so, we use a method proposed by Amini and the first author, which amounts to establishing a bound on the number of border-strip tableaux of skew shape. Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group. In particular, we prove the existence of a bijection between permutations and Stembridge’s alternating tableaux, which intertwines rotation and promotion.

On the quotient quantum graph with respect to the regular representation

Given a quantum graph $ \Gamma $, a finite symmetry group $ G $ acting on it and a representation $ R $ of $ G $, the quotient quantum graph $ \Gamma /R $ is described and constructed in the literature [1, 2, 18]. In particular, it was shown that the quotient graph $ \Gamma/\mathbb{C}G $ is isospectral to $ \Gamma $ by using representation theory where $ \mathbb{C}G $ denotes the regular representation of $ G $ [18]. Further, it was conjectured that $ \Gamma $ can be obtained as a quotient $ \Gamma/\mathbb{C}G $ [18]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph $ \Gamma $ and a finite symmetry group $ G $ acting on it, the quotient quantum graph $ \Gamma / \mathbb{C}G $ is not only isospectral but rather identical to $ \Gamma $ for a particular choice of a basis for $ \mathbb{C}G $. Furthermore, we prove that, this result holds for an arbitrary permutation representation of $ G $ with degree $ |G| $, whereas it doesn't hold for a permutation representation of $ G $ with degree greater than $|G|. $

Lift of fractional D-brane charge to equivariant Cohomotopy theory

The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this Hypothesis H, here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman’s comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational D-brane charge has been perceived as a paradox in string theory, we conclude that Hypothesis H serves to resolve this paradox.Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of SU(2); and we find explicitly that for the three exceptional subgroups and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters.

Correction to “Faithful permutation representations of toroidal regular maps”

Correction to “Faithful permutation representations of toroidal regular maps”

An Algorithm for Constructing Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

We describe an algorithm for decomposing permutation representations of wreath products of finite groups into irreducible components. The algorithm is based on the construction of a complete set of mutually orthogonal projections to irreducible invariant subspaces of the Hilbert space of the representation under consideration. In constructive models of quantum mechanics, the invariant subspaces of representations of wreath products describe the states of multicomponent quantum systems. The suggested algorithm uses methods of computer algebra and computational group theory. The C implementation of the algorithm is capable of constructing irreducible decompositions of representations of wreath products of high dimensions and ranks. Examples of calculations are given. Bibliography: 15 titles.

An Experimental Comparison of Algebraic Crossover Operators for Permutation Problems

Crossover operators are very important components in Evolutionary Computation. Here we are interested in crossovers for the permutation representation that find applications in combinatorial optimization problems such as the permutation flowshop scheduling and the traveling salesman problem. We introduce three families of permutation crossovers based on algebraic properties of the permutation space. In particular, we exploit the group and lattice structures of the space. A total of 34 new crossovers is provided. Algebraic and semantic properties of the operators are discussed, while their performances are investigated by experimentally comparing them with known permutation crossovers on standard benchmarks from four popular permutation problems. Three different experimental scenarios are considered and the results clearly validate our proposals.

Representability of Permutation Representations on Coalgebras and the Isomorphism Problem

Let H be an arbitrary group and let $$\rho :H\rightarrow {\text {Sym}}(V)$$ be any permutation representation of H on a set V. We prove that there is a faithful H-coalgebra C such that H arises as the image of the restriction of $${\text {Aut}}(C)$$ to G(C), the set of grouplike elements of C. Furthermore, we show that V can be regarded as a subset of G(C) invariant under the H-action and that the composition of the inclusion $$H\hookrightarrow {\text {Aut}}(C)$$ with the restriction $${\text {Aut}}(C)\rightarrow {\text {Sym}}(V)$$ is precisely $$\rho $$ . We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully.

Representations of $C^*$-algebras of row-countable graphs and unitary equivalence

In this article we generalize the main results of [7] and [12]. More specifically, we show that there are branching systems (which induce representations of the graph C∗(E)) associated to each row-countable graph E. For row-countable graphs, we characterize the Condition (L) via branching systems. Moreover, we show that each permutative representation by operators in Hilbert spaces is unitarily equivalent to one induced by a branching system, even the spaces being not separable. Furthermore, under some hypothesis on the graph, we show that each representation of the graph C∗-algebra is permutative.

Transitive distance-regular graphs from linear groups $L(3,q)$, $q = 2,3,4,5$

In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(3,2)$‎, ‎$L(3,3)$‎, ‎$L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.

Slow Continued Fractions and Permutative Representations of

AbstractRepresentations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

Computation of Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

An algorithm for the computation of the complete set of primitive orthogonal idempotents of the centralizer ring of the permutation representation of the wreath product of finite groups is described. This set determines the decomposition of the representation into irreducible components. In the quantum mechanics formalism, the primitive idempotents are projection operators onto irreducible invariant subspaces of the Hilbert space describing the states of many-particle quantum systems. The proposed algorithm uses methods of computer algebra and computational group theory. The C implementation of this algorithm is able to construct decompositions of representations of high dimensions and ranks into irreducible components.

ON A DESIGN FROM PRIMITIVE REPRESENTATIONS OF THE FINITE SIMPLE GROUPS

In this paper we present a design construction from primitive permutation representations of a finite simple group G. The group G acts primitively onthe points and transitively on the blocks of the design. The construction has this property that with some conditions we can obtain t-design for t >=2. We examine our design for fourteen sporadic simple groups. As a result we found a 2-(176,5,4) design with full automorphism group M22.

Wreath products and multipartite quantum systems

A natural symmetry group of a multicomponent quantum system is a special combination of a symmetry group acting within a single component (“local group”) and a group that permutes the components (“spatial symmetry group”). This combination is called the wreath product. Unitary representations of wreath products describe quantum evolutions of multipartite systems. It is known that any unitary representation of a finite group is contained in some permutation representation. We describe an algorithm for decomposing permutation representations of wreath products into irreducible components. This decomposition makes it possible to study the quantum behavior (entanglement, non-local correlations, etc.) of multipartite systems in invariant subspaces of the permutation Hilbert space.

Irreducible and permutative representations of ultragraph Leavitt path algebras

Abstract We completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this, we extend to ultragraph Leavitt path algebras Chen’s construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching system and characterize irreducible representations associated to perfect branching systems. Along the way, we improve the characterization of faithfulness of Chen’s irreducible representations.

On some Ternary Linear Codes and Designs obtained from the Projective Special Linear Goup PSL3(4)

Let (G, ∗) be a group and X any set, an action of a group G on X, denoted as G×X → X, (g, x) 7→ g.x, assigns to each element in G a transformation of X that is compatible with the group structure of G. If G has a subgroup H then there is a transitive group action of G on the set (G/H) of the right co-sets of H by right multiplication. A representation of a group G on a vector space V carries the dimension of the vector space. Now, given a field F and a finite group G, there is a bijective correspondence between the representations of G on the finitedimensional F-vector spaces and finitely generated FG-modules. We use the FG -modules to construct linear ternary codes and combinatorial designs from the permutation representations of the group L3(4). We investigate the properties and parameters of these codes and designs.
 We further obtain the lattice structures of the sub-modules and compare these ternary codes with the binary codes constructed from the same group.

On the spectral gap of some Cayley graphs on the Weyl group W(Bn)

The Laplacian of a (weighted) Cayley graph on the Weyl group W(Bn) is a N×N matrix with N=2nn! equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a 2n×2n matrix associated to a 2n-dimensional permutation representation of Wn. This result can be viewed as an extension to W(Bn) of an analogous result valid for the symmetric group, known as “Aldous' spectral gap conjecture”, proven in 2010 by Caputo, Liggett and Richthammer.