Singularities of normalized R-matrices and E-invariants for Dynkin quivers

We study the singularities of normalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> -matrices between arbitrary simple modules over the quantum loop algebra of type ADE in Hernandez–Leclerc’s level-one subcategory using equivariant perverse sheaves, following the previous works by Nakajima [Kyoto J. Math. 51 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mspace width="1.111pt"/> </mml:math> (1), 2011] and Kimura–Qin [Adv. Math. 262, 2014]. We show that the pole orders of these <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> -matrices coincide with the dimensions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> -invariants between the corresponding decorated representations of Dynkin quivers. This result can be seen as a correspondence of numerical characteristics between additive and monoidal categorifications of cluster algebras of finite ADE type.