### Unitriangularity of Decomposition Matrices of Unipotent Blocks

Algebra and Geometry Seminar

5th February 2020, 2:30 pm – 3:30 pm

Fry Building, 2.04

One of the distinguished features of the representation theory of finite groups is the ability to take a representation in characteristic zero and reduce it to obtain a representation over a fixed field of positive characteristic (a modular representation). If one starts with a representation that is irreducible in characteristic zero then its modular reduction can fail to be irreducible. The decomposition matrix encodes the multiplicities of the modular irreducible representations in this reduction.

In this talk I will present recent joint work with Olivier Brunat and Olivier Dudas establishing a fundamental property of the decomposition matrix for finite reductive groups, namely that it has a unitriangular shape. The solution to this problem involves the interplay between Lusztig’s geometric theory of character sheaves and a family of representations whose construction was originally proposed by Kawanaka.

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