A simple character formula Article Swipe

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Coxeter element Mathematics Simple (philosophy) Character (mathematics) Conjecture Coxeter group Algebraic number Pure mathematics Field (mathematics) Verma module Algebraic group Group (periodic table) Base (topology) Algebra over a field Combinatorics Geometry Physics Mathematical analysis Lie algebra Philosophy Epistemology Quantum mechanics

Simon Riche , Geordie Williamson ·

YOU? · · 2021 · Open Access · · DOI: https://doi.org/10.5802/ahl.79 · OA: W3088472702

In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mi>h</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi></mml:math> is the Coxeter number of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>. This provides a replacement for Lusztig’s conjecture, valid under a reasonable assumption on the characteristic.