A combinatorial approach to Harada's conjecture II of the groups related to the symmetric groups

Fischer Matrices for Generalised Symmetric Groups—A Combinatorial Approach

Factorizations of large cycles in the symmetric group

To restructure stereochemistry into a systematic format, enantiomeric and diastereomeric relationships have been investigated by using ethylene derivatives as examples in the light of a new group-theoretical and combinatorial approach. On one hand, enantiomeric relationship for ethylene derivatives has been characterized by means of a point group of order 8 (D 2h ), where chirality fittingness based on the sphericity concept has been applied to the enumeration of stereoisomers. On the other hand, diastereomeric relationship for ethylene derivatives has been examined by a permutation group of order 8 (S 9 [4]), which is a subgroup of the symmetric group of order 4 (S [4]) and isomorphic to a point group D 2d . The subgroups of S 9 [4] have been classified into stereogenic and astereogenic ones. A stereogenic subgroup corresponds to a pair of diastereomers, while an astereogenic subgroup is assigned to a self-diastereomer. The relationship between diastereomers and constitutional isomers have been also discussed.

A versatile combinatorial approach of studying products of long cycles in symmetric groups

A (left) quandle is connected if its left translations generate a group that acts transitively on the underlying set. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to constant quandle cocycles of Andruskiewitsch and Graña. A connected quandle is simply connected if it has no nontrivial coverings, or, equivalently, if all its second constant cohomology sets with coefficients in symmetric groups are trivial. In this paper, we develop a combinatorial approach to constant cohomology. We prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from [Formula: see text] are simply connected. We also consider constant cohomology with coefficients in arbitrary groups.

The Mobius (8 4 ) configuration is generalized in a purely combinatorial approach. We consider (2 n n ) configurations M ( n , φ ) depending on a permutation φ in the symmetric group S n . Classes of non-isomorphic configurations of this type are determined. The parametric characterization of M ( n , φ ) is given. The uniqueness of the decomposition of M ( n , φ ) into two mutually inscribed n -simplices is discussed. The automorphisms of M ( n , φ ) are characterized for n ≥ 3 .

Stability properties of the plethysm: A combinatorial approach

Kerov's polynomials give irreducible character values of the symmetric group in term of the free cumulants of the associated Young diagram. Using a combinatorial approach with maps, we prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Les polynômes de Kerov expriment les valeurs des caractères irréductibles du groupe symétrique en fonction des cumulants libres du diagramme de Young associé. Grâce à une approche combinatoire à base de cartes, nous prouvons dans cet article un résultat de positivité sur leurs coefficients, qui généralise une conjecture de S. Kerov.

A Combinatorial Approach to the Fusion Process for the Symmetric Group

A Combinatorial Approach to the Double Cosets of the Symmetric Group with respect to Young Subgroups

Character decomposition of Potts model partition functions, II: Toroidal geometry

The theory of Pi-algebras has been developed in the last decade mainly through combinatorial methods pertaining to the representation theory of the symmetric group in characteristic zero. These methods together with the study of the asymptotic behavior of a numerical sequence related to a PI-algebra have lead to the notion of the PIexponent of an algebra. Through the scale determined by the exponent many results have been obtained in the last few years shedding new light on the understanding of the polynomial identities of an algebra.

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.