A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP

Markus Linckelmann

doi:10.24330/ieja.440216

EN

A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP

Abstract

By results of Boltje and Kulshammer, if a source algebra A of a

principal p-block of a nite group with a defect group P with inertial quotient

E is a depth two extension of the group algebra of P, then A is isomorphic

to a twisted group algebra of the group P o E. We show in this note that

this is true for arbitrary blocks. We observe further that the results of Boltje

and Kulshammer imply that A is a depth two extension of its hyperfocal

subalgebra, with a criterion for when this is a depth one extension. By a

result of Watanabe, this criterion is satised if the defect groups are abelian.

Keywords

Source algebra

References

R. Boltje and B. Kulshammer, On the depth 2 condition for group algebra and Hopf algebra extensions, J. Algebra, 323(6) (2010), 1783-1796.
R. Boltje and B. Kulshammer, Group algebra extensions of depth one, Algebra Number Theory, 5(1) (2011), 63-73.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. II, with applications to nite groups and orders, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.
L. Kadison and K. Szlachanyi, Bialgebroid actions on depth two extensions, Adv. Math., 179(1) (2003), 75-121.
M. Linckelmann, On splendid derived and stable equivalences between blocks of nite groups, J. Algebra, 242(2) (2001), 819-843.
L. Puig, Nilpotent blocks and their source algebras, Invent. Math., 93(1) (1988), 77-116.
L. Puig, Pointed groups and construction of modules, J. Algebra, 116(1) (1988), 7-129.
[L. Puig, The hyperfocal subalgebra of a block, Invent. Math., 141(2) (2000), 365-397.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Markus Linckelmann ^* This is me

Publication Date

July 5, 2018

Submission Date

October 30, 2017

Acceptance Date

-

Published in Issue

Year 2018 Volume: 24 Number: 24

DOI

https://doi.org/10.24330/ieja.440216

IZ

https://izlik.org/JA28DH84NJ

Cite

RIS / Bibtex

APA

Linckelmann, M. (2018). A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. International Electronic Journal of Algebra, 24(24), 68-72. https://doi.org/10.24330/ieja.440216

AMA

1.Linckelmann M. A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. IEJA. 2018;24(24):68-72. doi:10.24330/ieja.440216

Chicago

Linckelmann, Markus. 2018. “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”. International Electronic Journal of Algebra 24 (24): 68-72. https://doi.org/10.24330/ieja.440216.

EndNote

Linckelmann M (July 1, 2018) A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. International Electronic Journal of Algebra 24 24 68–72.

IEEE

[1]M. Linckelmann, “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”, IEJA, vol. 24, no. 24, pp. 68–72, July 2018, doi: 10.24330/ieja.440216.

ISNAD

Linckelmann, Markus. “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”. International Electronic Journal of Algebra 24/24 (July 1, 2018): 68-72. https://doi.org/10.24330/ieja.440216.

JAMA

1.Linckelmann M. A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. IEJA. 2018;24:68–72.

MLA

Linckelmann, Markus. “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”. International Electronic Journal of Algebra, vol. 24, no. 24, July 2018, pp. 68-72, doi:10.24330/ieja.440216.

Vancouver

1.Markus Linckelmann. A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. IEJA. 2018 Jul. 1;24(24):68-72. doi:10.24330/ieja.440216

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https://doi.org/10.1007/s10998-022-00474-6