On Lusztig’s Character Formula for Chevalley Groups of Type <i>A<sub>l</sub></i>

On Lusztig’s Character Formula for Chevalley Groups of Type <i>A<sub>l</sub></i>

For a Chevalley group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> over an algebraically closed field <inline...

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Bibliographic Details
Main Authors: Sherali S. Ibraev, Larissa Kainbaeva, Gulzat M. Yensebayeva, Anar A. Ibrayeva, Manat Z. Parmenova, Gulnur K. Yeshmurat
Format: Article
Language:English
Published: MDPI AG 2024-11-01
Series:Mathematics
Subjects:
Chevalley group
simple module
Kazhdan–Lusztig polynomial
Lusztig’s character formula
Jantzen filtration
Jantzen sum formula
Online Access:https://www.mdpi.com/2227-7390/12/23/3791
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Summary:For a Chevalley group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> over an algebraically closed field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi></mrow></semantics></math></inline-formula> of characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> with the irreducible root system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>,</mo></mrow></semantics></math></inline-formula> Lusztig’s character formula expresses the formal character of a simple <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula>-module by the formal characters of the Weyl modules and the values of the Kazhdan–Lusztig polynomials at 1. It is known that, for a sufficiently large characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula> of the field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>,</mo></mrow></semantics></math></inline-formula> Lusztig’s character formula holds. The known lower bound of the characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula> is much larger than the Coxeter number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi></mrow></semantics></math></inline-formula> of the root system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>.</mo></mrow></semantics></math></inline-formula> Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></semantics></math></inline-formula>, Lusztig’s character formula holds for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>≥</mo><mi>h</mi></mrow></semantics></math></inline-formula>. For large Chevalley groups, no other examples are known. In this paper, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> of type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo></mrow></semantics></math></inline-formula> we give some series of simple modules for which Lusztig’s character formula holds for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>≥</mo><mi>h</mi></mrow></semantics></math></inline-formula>. Using this result, we compute the cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> with coefficients in these simple modules. To prove the results, Jantzen’s filtration properties for Weyl modules and the properties of Kazhdan–Lusztig polynomials are used.
ISSN:2227-7390

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