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Journal of Siberian Federal University. Mathematics & Physics 2022, 15(5), 610—614
DOI: 10.17516/1997-1397-2022-15-5-610-614 УДК 512.54
Irreducible Carpets of Additive Subgroups of Type G2 Over a Field of Characteristic 0
Yakov N. Nuzhin* Elizaveta N. Troyanskaya^
Siberian Federal University Krasnoyarsk, Russian Federation
Received 08.02.2022, received in revised form 23.04.2022, accepted 27.06.2022 Abstract. It is proved that any irreducible carpet of type G2 over a field F of characteristic 0, at least one additive subgroup of which is an R-module, where F is an algebraic extension of the field R, up to conjugation by a diagonal element defines a Chevalley group of type G2 over an intermediate subfield between R and F.
Keywords: Chevalley group, carpet of additive subgroups, carpet subgroup.
Citation: Ya.N. Nuzhin, E.N. Troyanskaya, Irreducible Carpets of Additive Subgroups of Type G2 Over a Field of Characteristic 0, J. Sib. Fed. Univ. Math. Phys., 2022, 15(5), 610-614. DOI: 10.17516/1997-1397-2022-15-5-610-614.
1. Introduction
Let $ be a reduced indecomposable root system, $(F) be a Chevalley group of type $ over the field F generated by the root subgroups
xr(F) = {xr(t) | t G F}, r G $.
We call a carpet of type $ of rank l over F a collection of additive subgroups A = {Ar | r G $} of the field F with the condition
Cij,rsAirAS ç Air+js, r, s, ir + js G $, i,j > 0, (1)
where Ar = {aO1 | a G Ar}, and constants Cij,rs are equal to ±1, ±2 or ±3. Inclusions (1) come from the Chevalley commutator formula
[xs(u),xr (t)] = JJ Xir+js(Cij,rs(-t)iuj), r,s,ir + js G $. (2)
i,j>0
Every carpet A defines a carpet subgroup $(A) generated by the subgroups xr(Ar), r G $. A carpet A is called closed if its carpet subgroup $(A) has no new root elements, i.e., if
$(A) n xr (F) = xr (Ar).
* nuzhin2008@rambler.ru ttroyanskaya.elizaveta@yandex.ru © Siberian Federal University. All rights reserved
The definition of a carpet used here was given by V. M. Levchuk [1] (see also [2, question 7.28]), and in [3] he described irreducible carpets of rank greater than 1 over field F, at least one additive subgroup of which is an R-module, where F is an algebraic extension of the field R, under the assumption that the characteristic of the field F is different from 0 and 2 for types Bi, Cl, F4, and for the type G2 is different from 0, 2 and 3. It turned out that, up to conjugation by a diagonal element, all additive subgroups of the carpet coincide with one intermediate subfield between R and F. We call such carpets constant. A similar problem for carpets of type G2 over a field of characteristic 2 and 3 was considered by S.K.Franchuk and she established that non-constant carpets appear in characteristic 3 [4]. We have proved that in the remaining case of characteristic 0 for the type G2 only constant carpets are possible.
Theorem 1. Let A = {Ar | r € be an irreducible carpet of type G2 over a field F of characteristic 0; with at least one additive subgroup Ar which is an R-module, where F is an algebraic extension of the field R. Then, up to conjugation by a diagonal element, all additive subgroups Ar coincide with some intermediate subfield P between the fields R and F.
2. Preliminary results
The group $(F) increasing to the extended Chevalley group $(F) by all diagonal elements h(x), where x is a F-character integral root lattice Z$, that is, a homomorphism of the additive group Z$ into the multiplicative group F* of the field F [5, Sec. 7.1]. Any F-character x is uniquely defined by the values at the fundamental roots, so for any r € $ and t € F
h(x)xr (t)h(x)-1 = xr (x(r)t). (3)
The next lemma states that the equality (3) fits naturally with the definition of carpet.
Lemma 1 ([6], Lemma 1). Conjugating the carpet subgroup $(A) with the diagonal element h(x), we obtain the carpet subgroup
h(x)m)Hx)-1 = w,
defined by the carpet
A = {A'r | r G where A'r = x(r)Ar
It is natural to call the carpet A' from Lemma 1 conjugate to the original carpet A, and we can talk about conjugate carpets without relating them to carpet subgroups. Therefore, such statements are permissible. "Up to conjugation by a diagonal element, the carpet A coincides with the carpet A'."
For a root system of type A2 (see Fig. 1), there is one kind of commutator formula
[xa(t), xb(u)] = xa+b(±tu).
Therefore, the carpet conditions have only one form AaAb Ç Aa+b.
a + b
b
—a
a
—a — b —b
Fig. 1
For a root system of type G2 (see Fig. 2), there are four kinds of commutator formulas
[Xa(t),Xb(u)] = Xa+b(±tu)x2a+b(±t2u)x3a+b(±t3u)x3a+2b(±t3u2 ), (4)
[xa(t),xa+b(u)] = x2a+b(±2tu)x3a+b(±3t2u)x3a+2b(±3tu2), [xa (t),x2a+b(u)] = x3a+b(±3tu), [xb(t),x3a+b(u)} = x3a+2b(±tu).
(5)
(6) (7)
So that, in this case, the carpet conditions look more impressive than for other types of root systems, and the formulas (4), (5), (6), (7) provide, respectively, the following forms
AaAb C Aa+b, A2aAb C A2a+b, A^Ab C Asa+b, A^ C Asa+2b, 2AaAa+b C A2a+b, ^Aa+b C Asa+b, 3AaA2a+b C Asa+2b, 3AaA2a+b C Asa+b, AbAsa+b C Asa+2b-
3a + 2b
2a + b , 3a + b
3a b
b
—3a — 2b Fig. 2
The proof of the following lemma is elementary, so we omit it.
b
Lemma 2. Let F be an algebraic extension of the field R and A is a subring of the field F which is an R-module. Then A is the field between R and F.
Lemma 3. Let A = {Ar | r € $} be an irreducible carpet of type A2 over a field F, {a, b} is the fundamental system for $ and let 1 € A-a n A-b and the additive subgroup Aa+b is an R-module, where F is an algebraic extension of the field R. Then all Ar coincide with some fixed subfield of the field F.
Proof. By [3, Lemma 3] all Ar coincide with some fixed subring of the field F, and by Lemma 2 this subring is a field. The lemma is proved. □
3. Proof of Theorem 1
Up to conjugation, diagonal elements can be assumed to be 1 € A-a n A-b. Then, by virtue of the carpet conditions, from the commutator formula (4) we obtain 1 € Ar for all r € $-. Without loss of generality, we can assume that A2a+b or A3a+2b is an R-module. Since the field R has characteristic 0, then for any non-zero integer n we use the equality nAr = Ar without mentioning in case when the additive subgroup Ar is an R-module.
Let A2a+b be an R-module. Due to the carpet conditions 2A-a-bA2a+b C Aa and 2A-aA2a+b C Aa+b we get the inclusions A2a+b C Aa and A2a+b C Aa+b respectively. Hence, due to the carpet condition 2AaAa+b C A2a+b it follows that A2a+b is a ring, and by virtue of Lemma 2 it is a field. In particular, 1 € A2a+b. Therefore, due to the carpet conditions from the commutator formula (4), replacing the pair of roots (a, b) with the pairs (2a + b, -3a — b) and (2a + b, —3a — 2b) we obtain 1 € Ar for all r € $. Let A2a+b = P. From the six carpet conditions of type 2AaAa+b C A2a+b we obtain the equalities Ar = P for all short roots of r. By Lemma 3, all additive subgroups Ar indexed by long roots r coincide with some fixed field Q. Now, from the carpet conditions AaAb C Aa+b and AaA2a+b C A3a+b we obtain the inclusions Q C P and P C Q respectively. Thus, in this case we have established that all additive subgroups of the carpet coincide with the field P.
Let A3a+2b be an R-module. By Lemma 3, all additive subgroups Ar indexed by long roots r coincide with some fixed field P. In particular, 1 € A3a+2b. Therefore, due to the carpet conditions from the commutator formula (4), when the pair of roots (a, b) is replaced by the pairs (—2a — b, 3a + 2b) and (—a — b, 3a + 2b) we get 1 € Ar for all r € $. Further, just as in the previous case, we obtain that all additive subgroups of the carpet coincide with the field P. The theorem is proved.
This work is supported by Russian Science Foundation, project 22-21-00733.
References
[1] V.M.Levchuk, Parabolic subgroups of certain ABA-groups, Mathematical Notes, 31(1982), no. 4, 509-525. DOI: 10.1007/BF01138934
[2] The Kourovka notebook: Unsolved Problems in Group Theory, Eds. V.D.Mazurov, E.I.Khukhro, Sobolev Institute of Mathematics, Novosibirsk, 2018, no. 19.
[3] V.M.Levchuk, On generating sets of root elements of Chevalley groups over a field, Algebra i Logika, 22(1983), no. 5, 504-517 (in Russian). DOI: 10.1007/BF01982113
[4] S.K.Franchuk, On irreducible carpets of additive subgroups of type G2 over fields of characteristic p > 0, Vladikavkaz. Matem. J., 22(2020), no. 1, 78-84 (in Russian).
DOI: 10.23671/VNC.2020.1.57590
[5] R.Carter, Simple groups of lie type, Wiley and Sons, London-New York-Sydney-Toronto, 1972.
[6] V.A.Koibaev, S.K.Kuklina, A.O.Likhacheva, Ya.N Nuzhin, Subgroups, of Chevalley Groups over a Locally Finite Field, Defined by a Family of Additive Subgroups, Math. Notes, 102(2017), no. 6, 792-798. DOI: 10.1134/S0001434617110190
Неприводимые ковры аддитивных подгрупп типа G2 над полем характеристики 0
Яков Н. Нужин Елизавета Н. Троянская
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Доказано, что любой неприводимый ковер типа 02 над полем Е характеристики 0, хотя бы одна аддитивная подгруппа которого является Д-модулем, где Е — алгебраическое расширение поля Д, с точностью до сопряжения диагональным элементом определяет группу Шевалле типа 02 над промежуточным подполем между Д и Е.
Ключевые слова: группа Шевалле, ковер аддитивных подгрупп, ковровая подгруппа.
