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Noetherianity up to Symmetry

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Combinatorial Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2108))

Abstract

These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations—whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few “personal favourites”: exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.

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Notes

  1. 1.

    Snowden chose the notion of \(\Delta \)-varieties contravariant in the linear maps f i so as to make defining ideals and more general \(\Delta \) -modules [Sno13] depend covariantly on them.

  2. 2.

    The reason for labelling with {0, …, p − 1} rather than [p] will become apparent soon.

  3. 3.

    The convenient fact that the sum of two finitary words is again finitary explains our choice of labelling x 0, …, x p−1.

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Acknowledgement

The author was supported by a Vidi grant from the Netherlands Organisation for Scientific Research (NWO).

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

    Jan Draisma

  2. Centrum Wiskunde en Informatica, Amsterdam, The Netherlands

    Jan Draisma

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Draisma, J. (2014). Noetherianity up to Symmetry. In: Combinatorial Algebraic Geometry. Lecture Notes in Mathematics(), vol 2108. Springer, Cham. https://doi.org/10.1007/978-3-319-04870-3_2

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