Commutator subgroup of Vershik–Kerov group

Gupta, C. K.; Hołubowski, Waldemar

doi:10.1016/j.laa.2012.01.032

Cited by 15 publications

(12 citation statements)

References 12 publications

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“…Hence we may put The results discussed above can be applied also to improve the statements of Theorem 1 in [6]. For the proof of the next theorem we assume that R = K is a field.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Compiling Theorem 1 with results of [6], we characterize the derived subgroup of the Vershik-Kerov group and improve the upper bound of the commutator width of this group. Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [3,4,5,6,7,8,9] (for infinite matrices) and [11,12,13,14,15,16] (for finite matrices) and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…The Vershik-Kerov group GL V K (∞, R) is a group of all invertible infinite matrices with finite number of nonzero entries below the main diagonal (see [8,6]). Let n be a natural number.…”

Section: Introductionmentioning

confidence: 99%

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A note on commutators in the group of infinite triangular matrices over a ring

Bier

Hołubowski

2015

Linear and Multilinear Algebra

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We investigate the commutators of elements of the group UT(∞, R) of infinite unitriangular matrices over an associative ring R with 1 and a commutative group R * of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence we give a complete characterization of the lower central series of the group UT(∞, R) including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring R, we show that the derived subgroup of T(∞, R) coincides with the group UT(∞, R). The obtained results generalize the results obtained for triangular groups over a field.

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“…Hence we may put The results discussed above can be applied also to improve the statements of Theorem 1 in [6]. For the proof of the next theorem we assume that R = K is a field.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Compiling Theorem 1 with results of [6], we characterize the derived subgroup of the Vershik-Kerov group and improve the upper bound of the commutator width of this group. Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [3,4,5,6,7,8,9] (for infinite matrices) and [11,12,13,14,15,16] (for finite matrices) and references therein.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A note on commutators in the group of infinite triangular matrices over a ring

Bier

Hołubowski

2015

Linear and Multilinear Algebra

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“…In recent years, some important subgroups of Vershik-Kerov group have been studied. Gupta and Hołubowski determined the commutator subgroup of Vershik-Kerov group over an infinite field 4 and a wide class of associative rings 5 . Parabolic subgroups of VershikKerov group are described in Refs.…”

Section: Introductionmentioning

confidence: 99%

Commutator subgroups of Vershik-Kerov groups for infinite symplectic groups

Hou¹,

Li²,

Yang³

2017

ScienceAsia

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Finite traces and representations of the group of infinite matrices over a finite field

Gorin

Kerov²,

Vershik

2014

Advances in Mathematics

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Finite traces and representations of the group of infinite matrices over a finite field. Vadim Gorin Massachusetts Institute of Technology, Cambridge, MA, USA and Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, RussiaSergei Kerov Anatoly Vershik St.Petersburg Department of V. A. Steklov Institute of Mathematics of Russian Academy of Sciences, Saint Petersburg, Russia AbstractThe article is devoted to the representation theory of locally compact infinitedimensional group GLB of almost upper-triangular infinite matrices over the finite field with q elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate n = ∞ analogue of general linear groups GL(n, q). It serves as an alternative to GL(∞, q), whose representation theory is poor. Our most important results are the description of semi-finite unipotent traces (characters) of the group GLB via certain probability measures on the Borel subgroup B and the construction of the corresponding von Neumann factor representations of type II ∞ .As a main tool we use the subalgebra A(GLB) of smooth functions in the group algebra L 1 (GLB). This subalgebra is an inductive limit of the finitedimensional group algebras C(GL(n, q)) under parabolic embeddings.As in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups, like multiplicativity of indecomposable characters or connections to probabilistic concepts.The infinite dimensional Iwahori-Hecke algebra H q (∞) plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group S(∞) which Email addresses: vadicgor@gmail.com (Vadim Gorin), avershik@gmail.com (Anatoly Vershik) MSC 2010: 22D10, 22D25, 20G40. Keywords: Infinite-dimensional group; finite field; factor representation; Hecke algebra Preprint submitted to ElsevierDecember 19, 2013 had been intensively studied in numerous previous papers.To the memory of Andrei Zelevinsky. Contents Historical prefaceMy joint work with S. Kerov on the asymptotic representation theory of the matrix groups GL(n, q) over finite field as the rank n grows to infinity, 2 was started at the beginning of 80s as a continuation of our papers devoted to analogous problems for symmetric groups of growing ranks at the end of 70-th. It is a part of what I called "the asymptotic representation theory".The "trivial" embedding GL(n, q) ֒→ GL(n + 1, q) does not lead to an interesting or useful theory. However, another "true" (i.e. parabolic) embedding of the group algebras of GL(n, q) was well-known starting from the very first papers on the representation theory of GL(n, q) (see [Gr], [Zel], [F], etc). It was used by A. Zelevinsky and us (see [V82]) to define a natural limit object (i.e. inductive limit) which is the group (named GLB) of infinite matrices over finite field with finitely many non-zero elements below the main diagon...

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Commutator subgroup of Vershik–Kerov group

Cited by 15 publications

References 12 publications

A note on commutators in the group of infinite triangular matrices over a ring

A note on commutators in the group of infinite triangular matrices over a ring

Commutator subgroups of Vershik-Kerov groups for infinite symplectic groups

Finite traces and representations of the group of infinite matrices over a finite field

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