Group Bn Research Articles

Generating Functions for Alternating Descents and Alternating Major Index

In 2008, Chebikin introduced the alternating descent set, AltDes(σ), of a permutation σ = σ1 ··· σn in the symmetric group Sn as the set of all i such that either i is odd and σi > σi+1 or i is even and σi < σi+1. We can then define altdes(σ) = |AltDes(σ)| and \({{\rm altmaj}(\sigma) = \sum_{i \in AltDes(\sigma)}i}\). In this paper, we compute a generating function for the joint distribution of altdes(σ) and altmaj(σ) over Sn. Our formula is similar to the formula for the joint distribution of des and maj over the symmetric group that was first proved by Gessel. We also compute similar generating functions for the groups Bn and Dn and for r-tuples of permutations in Sn. Finally we prove a general extension of these formulas in cases where we keep track of descents only at positions r, 2r, . . ..

Formula omitted] dimensions of spaces of braid-invariant harmonic forms

Let X be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L2 spaces of harmonic vector-valued forms on the product manifold XN that are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions (the braided L2-Betti numbers).

FINITE THURSTON-TYPE ORDERINGS ON DUAL BRAID MONOIDS

For a finite Thurston-type ordering &lt; of the braid group Bn, we introduce a new normal form of a dual positive braid which we call the [Formula: see text]-normal form, which is useful to compute the ordering. This normal form extends Fromentin's rotating normal form and the author's [Formula: see text]-normal form of positive braids. Using the [Formula: see text]-normal form, we give a combinatorial description of the restriction of the ordering &lt; to the dual braid monoids [Formula: see text]. We prove that the restriction to the dual positive monoid [Formula: see text] is a well-ordered set of the order type ωωn-2.

A loopless implementation of a gray code for signed permutations

Conway, Sloane and Wilks [1989] proved the existence of a Gray code for the reflection group Bn. The elements of this group are the signed permutations of the set 1, 2, . . . ,n. Here we give a loopless algorithm which generates a specific Gray code for Bn.

Counting derangements, involutions and unimodal elements in the wreath product C r ≀ S n

We count derangements, involutions and unimodal elements in the wreath product Cr ≀ Sn by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group Sn and on the hyperoctahedral group Bn when r = 1, 2, respectively.

Efficacy of bupivacaine‐neostigmine and bupivacaine‐tramadol in caudal block in pediatric inguinal herniorrhaphy

Limited duration of analgesia is among the limitations of single caudal injection with local anesthetics. Therefore, the purpose of this study was to evaluate the effectiveness and safety of bupivacaine in combination with either neostigmine or tramadol for caudal block in children undergoing inguinal herniorrhaphy. In a double-blinded randomized trial, sixty children undergoing inguinal herniorrhaphy were enrolled to receive a caudal block with either 0.25% bupivacaine (1 ml x kg(-1)) with neostigmine (2 microg x kg(-1)) (group BN) or tramadol (1 mg x kg(-1)) (group BT). Hemodynamic variables, pain and sedation scores, additional analgesic requirements, and side effects were compared between two groups. Duration of analgesia was longer in group BT (17.30 +/- 8.24 h) compared with group BN (13.98 +/- 10.03 h) (P = 0.03). Total consumption of rescue analgesic was significantly lower in group BT compared with group BN (P = 0.04). There were no significant differences in heart rate, mean arterial pressure, and oxygen saturation between groups. Adverse effects excluding the vomiting were not observed in any patients. In conclusion, tramadol (1 mg x kg(-1)) compared with neostigmine (2 microg x kg(-1)) might provide both prolonged duration of analgesia and extended time to first analgesic in caudal block.

Quasi-Coxeter Algebras, Dynkin Diagram Cohomology, and Quantum Weyl Groups

The author, and independently, De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra is described by the quantum Weyl group operators of the quantum group ⁠. The aim of this article, and of its sequel [47], is to prove this conjecture. The proof relies upon the use of quasi-Coxeter algebras, which are to generalized braid groups what Drinfeld's quasitriangular quasibialgebras are to the Artin braid groups Bn. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra which interpolates between the quasi-Coxeter algebra structure underlying the Casimir connection, and the quasitriangular quasibialgebra structure underlying the Knizhnik–Zamolodchikov equations. The existence of this structure will be proved in [47].

DUAL PRESENTATION AND LINEAR BASIS OF THE TEMPERLEY-LIEB ALGEBRAS

The braid group Bn maps homomorphically into the Temper- ley-Lieb algebra TLn. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group Bn form a basis for the vector space underlying the Temperley-Lieb algebra TLn. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno's theorem, using the interpretation of simple elements as non-crossing partitions.

SOME IRREDUCIBLE REPRESENTATIONS OF THE BRAID GROUP 𝔹n OF DIMENSION GREATER THAN n

For any n ≥ 3, we construct a family of finite-dimensional irreducible representations of the braid group 𝔹n. Moreover, we give necessary conditions for a member of this family to be irreducible. In particular we give a explicitly irreducible subfamily (ϕm, Vm), 1 ≤ m &lt; n, where [Formula: see text]. The representation obtained in the case m = 1 is equivalent to the standard representation.

PALINDROMES AND ORDERINGS IN ARTIN GROUPS

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn→Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element).More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

ISOTROPY SUBGROUP OF HURWITZ ACTION OF THE 3-BRAID GROUP ON THE BRAID SYSTEMS

The Hurwitz action of the n-braid group Bn on the n-fold direct product Bmn of the m-braid group Bm is studied. We show that the isotropy subgroup of the Hurwitz group action of B3 at the triple of the standard generators of B4 has index 16, by explicitly describing a complete system of coset representatives. An application to the braided surfaces in 4-space is also given.

Hydrogen bonding interaction of CpCo(Dithiolene) complex with monocyclic 2-pyridonyl substituent and unexpected formation of dithiolene-fused tricyclic pyridone derivative

One-pot reaction of [CpCo(CO) 2], elemental sulfur with some heterocycle-substituted alkynes (R–C C–HET) produced [CpCo(dithiolene)] complexes with 2PyOBn ( 2), with both 2PyOBn and 2-hydroxy-2-propyl groups (C(OH)Me 2) ( 5), both 2Py and C(OH)Me 2 ( 8), both 4Py and C(OH)Me 2 ( 11), and with 4Py substituent ( 13). A deprotection of benzyl group (Bn) from 2 with trimethylsilyl iodide formed [CpCo(dithiolene)] with 2-pyridonyl substituent ( 3). Heating reaction of 8 without any base resulted in the C(OH)Me 2 group elimination to form the 2-pyridylethylenedithiolate complex ( 9), but 11 underwent only dehydration at the C(OH)Me 2 under heating. While the preparation of 5, the benzyl free complex ( 6) was obtained as a main product. 6 has a dithiolene-fused tricyclic pyridone skeleton. The structures of 3, 5, 6, 8, and 11 were determined by X-ray diffraction studies. Intramolecular OH⋯N( 2Py) hydrogen bondings are found in 5 and 8, and an intermolecular OH⋯N( 4Py) one is found in 11 at solid state. In the 2-pyridonyl complex 3, intermolecular NH⋯O and CH(dithiolene)⋯O hydrogen bondings are observed. 8 showed an intermolecular Cp⋯Cp face-to-face interaction. The tricyclic complex 6 exhibited lower energy electronic absorption ( λ max = 668 nm) compared with the others ( λ max = 562–614 nm), due to an extended π-conjugation of aromatic cobaltadithiolene ring.

Intrathecal neostigmine with bupivacaine for infants undergoing lower abdominal and urogenital procedures: dose response

Intrathecal (IT) neostigmine produces dose-dependent analgesia in adults. However, the dose of spinal neostigmine has not been investigated in infants. The purpose of this study was to assess spinal anesthesia (SA) duration provided by four doses of spinal neostigmine added to bupivacaine for lower abdominal and urogenital procedures in infants. Seventy-five infants were randomized into five groups. The control group B received IT plain 0.5% hyperbaric bupivacaine. Groups BN.25, BN.50, BN.75, and BN1.0 received bupivacaine with 0.25, 0.5, 0.75, and 1 microg/kg of neostigmine, respectively. The primary variable was the duration of anesthesia assessed by recovery of hip flexion. Postoperative pain with facial expression, leg activity, arm activity, crying and consolability scale score, and rescue analgesic requirements were the secondary variables measured, and the side effects were noted. Seventy-three infants completed the study. There was a significant linear increase in SA duration with IT neostigmine to 65.2 (4.3) min with 0.5 microg/kg (P<0.01), 88.2 (5.1) with 0.75 microg/kg (P<0.001) and 92 (4.3) with 1 microg/kg (P<0.001) from 52.4 (4.3) min with bupivacaine alone. SA duration showed no significant difference between plain bupivacaine and BN.25 (P=0.100) or between groups BN.75 and BN1.0 (P=0.451). Groups BN.75 and BN1.0 had significantly reduced pain scores, and the median duration before the first dose rescue analgesic was requested prolonged significantly (P<0.001) compared with the other three groups. IT neostigmine at a dose of 0.75 microg/kg added to bupivacaine significantly prolonged SA duration with reduced postoperative pain scores and rescue analgesic requirements in infants undergoing lower abdominal and urogenital procedures. No additional benefits were provided on increasing it to 1 microg/kg.

A CHARACTERIZATION OF Bn(q) BY THE SET OF ORDERS OF MAXIMAL ABELIAN SUBGROUPS

Let n = 2m ≥ 4 and let q be a power of a prime p. We prove that the simple group Bn(q) is uniquely determined by the set of orders of its maximal abelian subgroups.

Subgroups of free groups generated by conjugates of powers of the generators

For a free group Fn = 〈x1, . . . , xn〉 we investigate when a finite index subgroup of Fn is generated by a set 𝒳 of powers of conjugates of the generators x1, . . . , xn. For the braid group Bn = 〈σ1, . . . , σn–1〉 we similarly consider when a finite index subgroup is generated by a set of powers of Dehn twists, i.e., conjugates of the elements . We indicate a connection of these problems with the theory of c-groups developed by Burnside, Schur and Wielandt.

GENERALIZED LONG-MOODY REPRESENTATIONS OF BRAID GROUPS

Long and Moody give a method of constructing representations of the braid groups Bn. We discuss some ways to generalize their construction. One of these gives representations of subgroups of Bn, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra.

Desempenho e características de carcaça de bovinos de diferentes grupos genéticos

Avaliou-se o desempenho de bovinos castrados de três grupos genéticos: Nelore (NN); ½ Guzerá × ½ Nelore (GN); e ½ Brahman × ½ Nelore (BN). Foram utilizados 41 bovinos com 24 meses de idade, mantidos em pastagem de capim-braquiária. Todos os animais eram de mesmo rebanho e foram criados sob as mesmas condições e abatidos aos 3 anos de idade. No início do experimento e no abate, os animais cruzados com Brahman foram mais pesados que os animais dos outros dois grupamentos genéticos. As médias para peso corporal dos grupos Nelore, Guzerá × Nelore e Brahman × Nelore no início do experimento foram: 324, 320 e 343 kg e no abate de: 474, 470 e 499 kg, respectivamente. Os ganhos médios diários, no entanto, foram similares entre os três grupos (0,388; 0,386 e 0,409 kg/dia, respectivamente). Animais do grupo Brahman × Nelore produziram carcaças quentes mais pesadas que as obtidas no grupo Guzerá × Nelore (253 × 238 kg), porém, estes dois grupos não diferiram estatisticamente do grupo Nelore (242 kg). As porcentagens de músculo, gordura e ossos na carcaça foram similares entre os grupos. Outras características da carcaça (rendimento, área de olho-de-lombo, gordura de cobertura e marmoreio) e a maciez da carne, avaliada por um painel de avaliadores e por um texturômetro, foram também similares entre os três grupamentos genéticos. O cruzamento de outras raças zebuínas (Brahman ou Guzerá) com o Nelore não melhorou as características qualitativas da carcaça e da carne, embora o cruzamento com Brahman tenha resultado em animais mais pesados e com carcaças mais pesadas.

SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS

We construct an action of the braid group BN on the twisted quantized enveloping algebra [Formula: see text] where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra [Formula: see text]. We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

Signed Involutions Avoiding 2-Letter Signed Patterns

Let In be the class of all signed involutions in the hyperoctahedral group Bn and let In(T) be the set of involutions in In which avoid a set T of signed patterns. In this paper, we complete a further case of the program initiated by Simion and Schmidt (6) by enumerating In(T) for all signed permutations T B2.