Showing papers on "Hyperoctahedral group published in 2011"

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Abstract: We study symmetric polynomials whose variables are odd-numbered Jucys–Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials.

Representation stability for the cohomology of the pure string motion groups

Abstract: The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups given by Jensen, McCammond, and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group vanish in positive degree. We also prove that the subgroup of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.

Enumerating wreath products via Garsia-Gessel bijections

Abstract: We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics (des"G,maj,@?"G,col) and (des"G,ides"G,maj,imaj,col,icol) over the wreath product of a symmetric group by a cyclic group. Here des"G, @?"G, maj, col, ides"G, imaj"G, and icol denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type A and B.

Some other algebraic properties of folded hypercubes

Abstract: We construct explicity the automorphism group of the folded hypercube $FQ_n$ of dimension $n>3$, as a semidirect product of $N$ by $M$, where $N$ is isomorphic to the Abelian group $Z_2^n$, and $M$ is isomorphic to $Sym(n+1)$, the symmetric group of degree $n+1$, then we will show that the folded hypercube $FQ_n$ is a symmetric graph.

Giambelli, Pieri, and tableau formulas via raising operators

Abstract: We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew Young diagram.

The absolute order on the hyperoctahedral group

Abstract: The absolute order on the hyperoctahedral group B n is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen---Macaulay. This method results in a new proof of Cohen---Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on B n is shellable and an example of a non-Cohen---Macaulay interval in the absolute order on D 4 is given. Finally, the closed intervals in the absolute order on B n and D n which are lattices are characterized and some of their important enumerative invariants are computed.

Bijective evaluation of the connection coefficients of the double coset algebra

Abstract: This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests.

Topics in computational group theory : primitive permutation groups and matrix group normalisers

Abstract: Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O’Nan–Scott Theorem and Aschbacher’s theorem. Tables of the groups G are given for each O’Nan– Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in Sd of G and the rank of N . Part II presents a new algorithm NormaliserGL for computing the normaliser in GLn(q) of a group G ≤ GLn(q). The algorithm is implemented in the computational algebra system Magma and employs Aschbacher’s theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over Magma’s existing algorithm.

Abelianization of subgroups of reflection groups and their braid groups: an application to Cohomology

Abstract: The final result of this article gives the order of the extension $$ 1 \longrightarrow P/[P,P]{\mathop{\longrightarrow} \limits^j} B/[P,P] {\mathop{\longrightarrow} \limits^p} W \longrightarrow 1$$ as an element of the cohomology group H 2(W, P/[P, P]) (where B and P stands for the braid group and the pure braid group associated to the complex reflection group W). To obtain this result, we first refine Stanley-Springer’s theorem on the abelianization of a reflection group to describe the abelianization of the stabilizer N H of a hyperplane H. The second step is to describe the abelianization of big subgroups of the braid group B of W. More precisely, we just need a group homomorphism from the inverse image of N H by p (where p: B → W is the canonical morphism) but a slight enhancement gives a complete description of the abelianization of p −1(W′) where W′ is a reflection subgroup of W or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection in W.

The Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16

Abstract: We show that the Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16 is true.