Brauer's height zero conjecture for quasi-simple groups

We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Sp\"ath's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups

Lusztig induction and ℓ-blocks of finite reductive groups

We present a unified parametrisation of ℓ-blocks of quasi-simple finite groups of Lie type in non-defining characteristic via Lusztig's induction functor in terms of e-Jordan-cuspidal pairs and e-Jordan quasi-central cuspidal pairs

Quasi-isolated blocks and Brauer's height zero conjecture

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer’s long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups

The Discrete Fundamental Group of the Associahedron, and the Exchange Module

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.Comment: 16 pages, 4 figure

Schur elements for the Ariki-Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki-Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type $G(l,p,n)$ in characteristic 0.Comment: The paper contains the results of arXiv:1101.146

On zeros of irreducible characters lying in a normal subgroup

[EN] Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that x(g)¿0 for all irreducible characters x of G. Such an element is said to be non-vanishing in G. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we also study certain arithmetical properties of the G-conjugacy class sizes of the elements of N which are zeros of some irreducible character of G. In particular, if N=G, then new contributions are obtained.The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain). The research of the second author is partially funded by the Istituto Nazionale di Alta Matematica - INdAM. The third author acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain). The first and third authors are also supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovacion y Universidades (Spain).Felipe Román, MJ.; Grittini, N.; Ortiz-Sotomayor, VM. (2020). On zeros of irreducible characters lying in a normal subgroup. Annali di Matematica Pura ed Applicata (1923 -). 199:1777-1789. https://doi.org/10.1007/s10231-020-00942-1S17771789199Beltrán, A., Felipe, M.J.: Prime powers as conjugacy class lengths of $$\pi$$-elements. Bull. Aust. Math. Soc. 69, 317–325 (2004)Beltrán, A., Felipe, M.J., Malle, G., Moretó, A., Navarro, G., Sanus, L., Solomon, R., Tiep, P.H.: Nilpotent and abelian Hall subgroups in finite groups. Trans. Am. Math. Soc. 368, 2497–2513 (2016)Berkovich, Y., Kazarin, L.S.: Indices of elements and normal structure of finite groups. J. Algebra 283, 564–583 (2005)Bianchi, M., Chillag, D., Lewis, M.L., Pacifici, E.: Character degree graphs that are complete graphs. Proc. Am. Math. Soc. 135, 671–676 (2007)Brough, J., Kong, Q.: On vanishing criteria that control finite group structure II. Bull. Aust. Math. Soc. 98, 251–257 (2018)Brough, J.: Non-vanishing elements in finite groups. J. Algebra 460, 387–391 (2016)Dolfi, S., Pacifici, E., Sanus, L., Spiga, P.: On the orders of zeros of irreducible characters. J. Algebra 321, 345–352 (2009)Grüninger, M.: Two remarks about non-vanishing elements in finite groups. J. Algebra 460, 366–369 (2016)Isaacs, I.M.: Character Theory of Finite Groups. Academic Press Inc., London (1976)Isaacs, I.M., Navarro, G., Wolf, T.R.: Finite group elements where no irreducible character vanishes. J. Algebra 222, 413–423 (1999)Malle, G., Navarro, G.: Characterizing normal Sylow $$p$$-subgroups by character degrees. J. Algebra 370, 402–406 (2012)Malle, G., Navarro, G., Olsson, J.B.: Zeros of characters of finite groups. J. Group Theory 3, 353–368 (2000)The GAP Group: GAP—Groups, Algorithms, and Programming. Version 4.10.0 (2018). http://www.gap-system.or

PGL\u3csub\u3e2\u3c/sub\u3e(F\u3csub\u3el\u3c/sub\u3e) Number Fields with Rational Companion Forms

We give a list of PGL2(Fl) number fields for ℓ ≥ 11 which have rational companion forms. Our list has fifty-three fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group

Fifty-kDa Hyaluronic Acid Upregulates Some Epidermal Genes without Changing TNF-α Expression in Reconstituted Epidermis

Background: Due to its strong water binding potential, hyaluronic acid (HA) is a well-known active ingredient for cosmetic applications. However, based on its varying molecular size, skin penetration of HA may be limited. Recent studies have demonstrated that low-molecular-weight HA (LMW HA) may show a certain proinflammatory activity. We thus aimed to characterize an LMW-sized HA molecule that combines strong anti-aging abilities with efficient skin penetration but lacks potential proinflammatory effects. Methods: Total RNA and total protein were isolated from reconstituted human epidermis following incubation with HAs of various molecular weights (20, 50, 130, 300, 800 and 1,500 kDa). Tumor necrosis factor-alpha expression was determined using quantitative PCR. Genonnic and proteomic expression of various junctional proteins was determined using Affymetrix and common Western blotting techniques. Results: LMW HA of approximately 50 kDa did not significantly alter tumor necrosis factor-alpha expression compared to 20-kDa HA, but revealed significantly higher skin penetration rates than larger sized HA associated with increased expression of genes and proteins known to be involved in tight junction formation and keratinocyte cohesion. Conclusion: LMW HA of approximately 50 kDa shows better penetration abilities than larger-sized HA. In addition, LMW HA influences the expression of various genes including those contributing to keratinocyte differentiation and formation of intercellular tight junction complexes without showing proinflammatory activity. These observations contribute to current knowledge on the effects of LMW HA on keratinocyte biology and cutaneous physiology. Copyright (C) 2011 S. Karger AG, Base

Extraction And Identification Of Sulfated Polysaccharide From Gracilaria sp.

Polysaccharide from Gracilaria sp (locally known as “Sayur Karangâ€) was extracted using hot alkaline solutions (NaOH and KOH). Fractionation using DEAE-Sepharose Fast Flow column showed that the polysaccharide discharged after being eluted with 0.5 – 1.0 M NaCl. Spectroscopic analysis using FTIR showed that the polysaccharide contained hydroxyl  (3420 cm-1 (NaOH frcation) dan 3446 cm-1 (KOH frcation)), carbonyl (1642 cm-1 (NaOH frcation) and 1644 cm-1 (KOH frcation)) and sulphate (1271 cm-1 (NaOH frcation) and 1262 cm-1 (KOH frcation)) groups proving that the polysaccharide is a sulphated polysaccharide. A further study is needed to elucidate the structure of the sulphated polysaccharide as well as physiochemical properties of the polysaccharide

Extraction And Identification Of Sulfated Polysaccharide From Gracilaria sp.

Polysaccharide from Gracilaria sp (locally known as “Sayur Karang”) was extracted using hot alkaline solutions (NaOH and KOH). Fractionation using DEAE-Sepharose Fast Flow column showed that the polysaccharide discharged after being eluted with 0.5 – 1.0 M NaCl. Spectroscopic analysis using FTIR showed that the polysaccharide contained hydroxyl  (3420 cm-1 (NaOH frcation) dan 3446 cm-1 (KOH frcation)), carbonyl (1642 cm-1 (NaOH frcation) and 1644 cm-1 (KOH frcation)) and sulphate (1271 cm-1 (NaOH frcation) and 1262 cm-1 (KOH frcation)) groups proving that the polysaccharide is a sulphated polysaccharide. A further study is needed to elucidate the structure of the sulphated polysaccharide as well as physiochemical properties of the polysaccharide