Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic

DOI10.1016/J.JPAA.2021.106694arXiv2010.16173MaRDI QIDQ6352635

Publication date: 30 October 2020

Abstract: Let

G

be a classical group with natural module

V

and Lie algebra

m a t h f r a k g

over an algebraically closed field

K

of good characteristic. For rational irreducible representations

f : G i g h t a r r o w o p e r a t o r n a m e G L (W)

occurring as composition factors of

V o t i m e s V^{*}

,

w e d g e^{2} (V)

, and

S^{2} (V)

, we describe the Jordan normal form of

m a t h r m d f (e)

for all nilpotent elements

e i n m a t h f r a k g

. The description is given in terms of the Jordan block sizes of the action of

e

on

V o t i m e s V^{*}

,

w e d g e^{2} (V)

, and

S^{2} (V)

, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of

f (u)

for every unipotent element

u i n G

.

Mathematics Subject Classification ID

Representation theory for linear algebraic groups (20G05)

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