Parabolic subgroups of Vershik-Kerov’s group

Abstract

In this note we show that all parabolic subgroups of Vershik-Kerov’s group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L upper B left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GLB(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (i.e. subgroups containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis normal infinity comma upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(\infty ,R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> —the group of infinite dimensional upper triangular matrices) are net subgroups for a wide class of semilocal rings <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .