# Manchester Algebra Seminar - John MacQuarrie

Starts: | 13:00 8 Mar 2022 |
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Ends: | 14:00 8 Mar 2022 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, Current University students |

Speaker: John MacQuarrie - Universidade Federal de Minas Gerais

Title: A variant of a theorem of Weiss characterising permutation modules

Abstract: In 1988, Weiss gave a powerful sufficient condition guaranteeing that a lattice U for a finite p-group G over the p-adic integers be a permutation module, in terms of modules for groups smaller than G. Weiss' Theorem is not a characterization, however. Working with Pavel Zalesski, we give a characterization of permutation Z_pG-modules for a finite p-group G. The theorem has the form: "U is a permutation module if, and only if, the G/N-modules A, B and C are permutation modules". Easy (rank 3) examples show that one cannot remove the demands on A or B from the theorem, but we didn't know if the final condition is redundant. Working with Marlon Stefano, we show that we cannot remove this condition, by constructing a non-permutation lattice such that A and B are permutation modules. The methods used to construct the example are interesting in their own right, utilizing an ingenious description of ZpG-lattices (G abelian) due to Butler. I'll explain all this.

Place: Frank Adams (and to be streamed online*)

- subject to equipment and connection

Tea and biscuits 12:45 in the foyer