scholarly journals REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

2012 ◽  
Vol 21 (10) ◽  
pp. 1250071 ◽  
Author(s):  
CLAIRE LEVAILLANT
Keyword(s):  
Root System ◽  
Linear Representation ◽  
Computer Assisted ◽  
Artin Group ◽  
Type D ◽  
Faithful Linear Representation ◽  
Positive Roots

We construct a linear representation of the CGW algebra of type Dn. This representation has degree n2 - n, the number of positive roots of a root system of type Dn. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type Dn, this representation is equivalent to the faithful linear representation of Cohen–Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type Dn. Our proof is computer-assisted using Mathematica.

scholarly journals Accelerating root system phenotyping of seedlings through a computer-assisted processing pipeline

Plant Methods ◽  
2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Lionel X. Dupuy ◽  
Gladys Wright ◽  
Jacqueline A. Thompson ◽  
Anna Taylor ◽  
Sebastien Dekeyser ◽  
...  
Keyword(s):  
Root System ◽  
Computer Assisted ◽  
Processing Pipeline

scholarly journals Linear representations of random groups

2019 ◽  
Vol 09 (03) ◽  
pp. 1950016
Author(s):  
Gady Kozma ◽  
Alexander Lubotzky
Keyword(s):  
Linear Representation ◽  
Linear Representations ◽  
Faithful Linear Representation ◽  
Random Groups

We show that for a fixed [Formula: see text], Gromov random groups with any density [Formula: see text] have no nontrivial degree [Formula: see text] representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when [Formula: see text] such groups have a faithful linear representation over [Formula: see text], a.a.s.

scholarly journals On the characters of the Sylow -subgroups of untwisted Chevalley groups

2016 ◽  
Vol 19 (2) ◽  
pp. 303-359 ◽  
Author(s):  
Frank Himstedt ◽  
Tung Le ◽  
Kay Magaard
Keyword(s):  
Root System ◽  
Chevalley Group ◽  
Positive Root ◽  
Chevalley Groups ◽  
Irreducible Characters ◽  
Single Root ◽  
The Family ◽  
Positive Roots ◽  
Root Character

Let$UY_{n}(q)$be a Sylow$p$-subgroup of an untwisted Chevalley group$Y_{n}(q)$of rank$n$defined over $\mathbb{F}_{q}$where$q$is a power of a prime$p$. We partition the set$\text{Irr}(UY_{n}(q))$of irreducible characters of$UY_{n}(q)$into families indexed by antichains of positive roots of the root system of type$Y_{n}$. We focus our attention on the families of characters of$UY_{n}(q)$which are indexed by antichains of length$1$. Then for each positive root$\unicode[STIX]{x1D6FC}$we establish a one-to-one correspondence between the minimal degree members of the family indexed by$\unicode[STIX]{x1D6FC}$and the linear characters of a certain subquotient$\overline{T}_{\unicode[STIX]{x1D6FC}}$of$UY_{n}(q)$. For$Y_{n}=A_{n}$our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of$\text{Irr}(UE_{i}(q))$,$6\leqslant i\leqslant 8$, and$\text{Irr}(UF_{4}(q))$.

scholarly journals Arrangements of ideal type are inductively free

2019 ◽  
Vol 29 (05) ◽  
pp. 761-773 ◽  
Author(s):  
Michael Cuntz ◽  
Gerhard Röhrle ◽  
Anne Schauenburg
Keyword(s):  
Large Class ◽  
Root System ◽  
Ideal Type ◽  
Ideal Formula ◽  
Type Formula ◽  
Positive Roots

Extending earlier work by Sommers and Tymoczko, in 2016, Abe, Barakat, Cuntz, Hoge, and Terao established that each arrangement of ideal type [Formula: see text] stemming from an ideal [Formula: see text] in the set of positive roots of a reduced root system is free. Recently, Röhrle showed that a large class of the [Formula: see text] satisfy the stronger property of inductive freeness and conjectured that this property holds for all [Formula: see text]. In this paper, we confirm this conjecture.

scholarly journals Faithful linear representations of certain free nilpotent groups

1995 ◽  
Vol 37 (1) ◽  
pp. 33-36 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Free Groups ◽  
Linear Representation ◽  
Nilpotent Groups ◽  
Finite Degree ◽  
Final Comment ◽  
Linear Representations ◽  
Variety Of Groups ◽  
Countable Rank ◽  
Faithful Linear Representation ◽  
Finitary Linear

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

scholarly journals Gradedness of the set of rook placements in A n−1

2021 ◽  
Vol 29 (2) ◽  
pp. 171-182
Author(s):  
Mikhail V. Ignatev
Keyword(s):  
Algebraic Group ◽  
Root System ◽  
Borel Subgroup ◽  
Coadjoint Orbit ◽  
Combinatorial Structure ◽  
Natural Partial Order ◽  
Reductive Algebraic Group ◽  
Rook Placements ◽  
Positive Roots ◽  
Different Order

Abstract A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in An− 1 with respect to a slightly different order and prove that this poset is graded.

scholarly journals The freeness of ideal subarrangements of Weyl arrangements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Mohamed Barakat ◽  
Michael Cuntz ◽  
Torsten Hoge ◽  
Hiroaki Terao
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Height Distribution ◽  
Free Arrangement ◽  
Positive Roots ◽  
International Audience

International audience A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula. Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule.

scholarly journals Noncomplete affine structures on Lie algebras of maximal class

2002 ◽  
Vol 29 (2) ◽  
pp. 71-77 ◽  
Author(s):  
E. Remm ◽  
Michel Goze
Keyword(s):  
Lie Algebra ◽  
Linear Representation ◽  
Heisenberg Algebra ◽  
Maximal Class ◽  
Affine Structure ◽  
Nilpotent Lie Algebra ◽  
3 Dimensional ◽  
Faithful Linear Representation ◽  
Affine Structures ◽  
Filiform Lie Algebra

Every affine structure on Lie algebra𝔤defines a representation of𝔤inaff(ℝn). If𝔤is a nilpotent Lie algebra provided with a complete affine structure then the corresponding representation is nilpotent. We describe noncomplete affine structures on the filiform Lie algebraLn. As a consequence we give a nonnilpotent faithful linear representation of the 3-dimensional Heisenberg algebra.

Sign in / Sign up

Export Citation Format

Share Document