A note on a cluster structure of the coordinate ring of a simple algebraic group

We show that the coordinate ring of a simply-connected simple algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over the complex number field coincides with the Berenstein–Fomin–Zelevinsky cluster algebra and its upper cluster algebra, at least when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 4"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .