Generalized amoebas for subvarieties of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi>
Abstract
<jats:p> The paper addresses the problem of generalizing theory of amoebas from tropical geometry in algebraic torus to other reductive algebraic groups. The logarithm of singular values of a matrix has been suggested as an analogue of the logarithm map on the algebraic torus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> for the general linear group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We call the image of a subvariety under this logarithm map a matrix amoeba. In this paper, we establish the main properties of these matrix amoebas, in particular, extending the notion of Ronkin function to this setting. For hypersurfaces, we show how to describe the asymptotic directions of the matrix amoebas using a notion of Newton polytope and study the relationship between matrix amoebas and spherical tropical varieties. Our matrix amoebas should be considered as the Archimedean version of the spherical tropicalization of Tevelev–Vogiannou for the variety <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> regarded as a spherical homogeneous space for the left-right action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . </jats:p>