Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type

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Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type. / Vavilov, N. A.

In: Journal of Soviet Mathematics, Vol. 20, No. 6, 12.1982, p. 2546-2555.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{6ec6351313ea4cbf82eec77377ffae71,

title = "Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type",

abstract = "Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S≥2, R is generated by its invertible elements, and the ideal of R generated by the differences ε-1 for all invertible e{open} coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net b{cyrillic} of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e(b{cyrillic})≤P≤G(b{cyrillic}), where G is the net subgroup of (b{cyrillic}) and E(b{cyrillic}) is the subgroup generated by the transvections in G(b{cyrillic}). It is shown that E(b{cyrillic}) is a normal subgroup of G(b{cyrillic}). The factor group G(b{cyrillic}/E(b{cyrillic})) is studied. The case of the special linear group is also considered.",

author = "Vavilov, {N. A.}",

year = "1982",

month = dec,

doi = "10.1007/BF01681471",

language = "English",

volume = "20",

pages = "2546--2555",

journal = "Journal of Mathematical Sciences",

issn = "1072-3374",

publisher = "Springer Nature",

number = "6",

}

RIS

TY - JOUR

T1 - Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type

AU - Vavilov, N. A.

PY - 1982/12

Y1 - 1982/12

N2 - Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S≥2, R is generated by its invertible elements, and the ideal of R generated by the differences ε-1 for all invertible e{open} coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net b{cyrillic} of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e(b{cyrillic})≤P≤G(b{cyrillic}), where G is the net subgroup of (b{cyrillic}) and E(b{cyrillic}) is the subgroup generated by the transvections in G(b{cyrillic}). It is shown that E(b{cyrillic}) is a normal subgroup of G(b{cyrillic}). The factor group G(b{cyrillic}/E(b{cyrillic})) is studied. The case of the special linear group is also considered.

AB - Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S≥2, R is generated by its invertible elements, and the ideal of R generated by the differences ε-1 for all invertible e{open} coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net b{cyrillic} of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e(b{cyrillic})≤P≤G(b{cyrillic}), where G is the net subgroup of (b{cyrillic}) and E(b{cyrillic}) is the subgroup generated by the transvections in G(b{cyrillic}). It is shown that E(b{cyrillic}) is a normal subgroup of G(b{cyrillic}). The factor group G(b{cyrillic}/E(b{cyrillic})) is studied. The case of the special linear group is also considered.

UR - http://www.scopus.com/inward/record.url?scp=34250224481&partnerID=8YFLogxK

U2 - 10.1007/BF01681471

DO - 10.1007/BF01681471

M3 - Article

AN - SCOPUS:34250224481

VL - 20

SP - 2546

EP - 2555

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -

ID: 76482878