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A Survey of Recent Developments on Hessenberg Varieties

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Schubert Calculus and Its Applications in Combinatorics and Representation Theory (ICTSC 2017)

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Abstract

This article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups, and Stanley’s chromatic symmetric functions are related to the cohomology rings of Hessenberg varieties. We also include several other topics on Hessenberg varieties to cover recent developments.

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Notes

  1. 1.

    For the origin of the name Hessenberg varieties, see [22].

  2. 2.

    A paving by affines means a “(complex) cellular decomposition” in algebraic geometry. See [72, Definition 2.1] for the details.

  3. 3.

    For \(w\in S_n\), the associated permutation flag \(V_{\bullet }\) is given by \(V_i:={ \mathrm span }_{\mathbb {C}}\{e_{w(1)},\ldots ,e_{w(i)}\}\) where \(\{e_{1},\ldots ,e_{n}\}\) is the standard basis of \(\mathbb {C}^n\).

  4. 4.

    For regular semisimple matrices S and \(S'\), the associated Hessenberg varieties \(\mathrm{{Hess}}(S,h)\) and \(\mathrm{{Hess}}(S',h)\) with a same Hessenberg function h are diffeomorphic.

  5. 5.

    A complete graph is a graph in which every pair of distinct vertices is connected by an edge.

  6. 6.

    A map \(\kappa :[n]\rightarrow \mathbb {N}\) is called a proper coloring of G if \(\kappa (i)\ne \kappa (j)\) for all pair of vertices i and j which are connected by an edge.

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Acknowledgements

We offer our immense gratitude to those who helped in preparation of the talks and this article including Takuro Abe, Peter Crooks, Mikiya Masuda, Haozhi Zeng, and all of the organizers of International Festival in Schubert Calculus Jianxun Hu, Changzheng Li, and Leonardo C. Mihalcea. We also want to thank the students in Sun Yat-sen University who helped organize the conference, the audience who attended the talks at the conference, and the readers of this article. The first author is supported in part by JSPS Grant-in-Aid for Early-Career Scientists: 18K13413. The second author is supported in part by JSPS Grant-in-Aid for JSPS Fellows: 17J04330.

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Authors and Affiliations

  1. Faculty of Liberal Arts and Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Osaka, Sakai, 599-8531, Japan

    Hiraku Abe

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5, Yamadaoka, Suita, Osaka, 565-0871, Japan

    Tatsuya Horiguchi

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  1. Hiraku Abe
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Correspondence to Hiraku Abe .

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Editors and Affiliations

  1. School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong, China

    Jianxun Hu

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong, China

    Changzheng Li

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Leonardo C. Mihalcea

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Abe, H., Horiguchi, T. (2020). A Survey of Recent Developments on Hessenberg Varieties. In: Hu, J., Li, C., Mihalcea, L.C. (eds) Schubert Calculus and Its Applications in Combinatorics and Representation Theory. ICTSC 2017. Springer Proceedings in Mathematics & Statistics, vol 332. Springer, Singapore. https://doi.org/10.1007/978-981-15-7451-1_10

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