TY - JOUR

T1 - Overgroups of E(m,R) ⊗ E(n,R) i. Levels and normalizers

AU - Anańevskiǐ, A. S.

AU - Vavilov, N. A.

AU - Sinchuk, S. S.

PY - 2012/12/28

Y1 - 2012/12/28

N2 - The study of subgroups H such that E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) is started, provided that the ring R is commutative and m, n ≥ 3. The principal results of this part can be summarized as follows. The group GLm ⊗GLn is described by equations, and it is proved that the elementary subgroup E(m,R) ⊗ E(n,R) is normal in (GLm ⊗GLn)(R). Moreover, when m ≠ n, the normalizers of all three subgroups E(m,R) ⊗ e, e ⊗ E(n,R), and E(m,R) ⊗ E(n,R) in GL(mn,R) coincide with (GLm ⊗GLn)(R). With each such intermediate subgroup H, a uniquely defined level (A,B,C) is associated where A,B,C are ideals in R such that mA,A2 ≤ B ≤ A and nA,A2 ≤ C ≤ A. Conversely, a level determines a perfect intermediate subgroup EE(m, n, R, A,B,C). It is shown that each intermediate subgroup contains a unique largest subgroup of this type. Next, the normalizer NG(EE(m, n, R,A)) of these perfect intermediate subgroups is calculated completely in the crucial case, where A = B = C. The standard answer to the above problem can now be stated as follows. Every such intermediate subgroup H is contained in the normalizer NG(EE(m, n, R, A,B,C)). In the special case where n ≥ m + 2, such a standard description will be established in the second part of the present work.

AB - The study of subgroups H such that E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) is started, provided that the ring R is commutative and m, n ≥ 3. The principal results of this part can be summarized as follows. The group GLm ⊗GLn is described by equations, and it is proved that the elementary subgroup E(m,R) ⊗ E(n,R) is normal in (GLm ⊗GLn)(R). Moreover, when m ≠ n, the normalizers of all three subgroups E(m,R) ⊗ e, e ⊗ E(n,R), and E(m,R) ⊗ E(n,R) in GL(mn,R) coincide with (GLm ⊗GLn)(R). With each such intermediate subgroup H, a uniquely defined level (A,B,C) is associated where A,B,C are ideals in R such that mA,A2 ≤ B ≤ A and nA,A2 ≤ C ≤ A. Conversely, a level determines a perfect intermediate subgroup EE(m, n, R, A,B,C). It is shown that each intermediate subgroup contains a unique largest subgroup of this type. Next, the normalizer NG(EE(m, n, R,A)) of these perfect intermediate subgroups is calculated completely in the crucial case, where A = B = C. The standard answer to the above problem can now be stated as follows. Every such intermediate subgroup H is contained in the normalizer NG(EE(m, n, R, A,B,C)). In the special case where n ≥ m + 2, such a standard description will be established in the second part of the present work.

KW - Affine group schemes

KW - Automorphisms

KW - Elementary subgroup

KW - Elementary transvections

KW - Form parameters

KW - General linear group

KW - Intermediate subgroups

KW - Lower level

KW - Normalizer

KW - Standard description

KW - Tensor product

UR - http://www.scopus.com/inward/record.url?scp=84871477980&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-2012-01219-7

DO - 10.1090/S1061-0022-2012-01219-7

M3 - Article

AN - SCOPUS:84871477980

VL - 23

SP - 819

EP - 849

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 5

ER -