Generating functions for the Hochschild cohomology of symmetric groups

City, University of London, UK

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We start out by reviewing some classical material on the representation theory of symmetric groups and the Hochschild cohomology of finite-dimensional algebras. We describe generating functions for the dimensions of the Hochschild cohomology of symmetric groups in each degree.

While it is known, using the classification of finite simple groups, that the first Hochschild cohomology of a non-semisimple finite group algebra is non-zero, it remains an open question whether this is true for non-simple blocks of finite groups.

We use generating functions to show that the first Hochschild cohomology of any non-simple block of a symmetric group algebra is non-zero. This is joint work with Dave Benson and Radha Kessar.