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Schur asymptotics of Veronese syzygies

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Abstract

We study the asymptotic behavior of Veronese syzygies as representations of the general linear group. For a fixed homological degree p of the syzygies, we describe the exact asymptotic growth for the number of distinct irreducible representations and for the number of irreducible representations also counting multiplicities. This shows that asymptotically Veronese syzygies have a very rich algebraic and representation-theoretic structure as the degree of the embedding grows.

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Notes

  1. Plethysm functors are polynomial. Rubei in [21, §2] checks that the general linear group decompositions of syzygies are functorial.

  2. Such tableaux can be described by Gelfand–Tsetlin patterns (see [23, p133] for a definition).

  3. In the language of [12], we are determining the dimension of the moment body and of the multiplicity body (which in our case is also the classical Gelfand–Tsetlin polytope) for the \(p\)-th product of a sufficiently large dimension projective space.

  4. To see this, it is enough to produce a point of \(\mathcal Y\) that satisfies strictly all the defining inequalities. Setting \(t_{ij}=p^i \epsilon \) for all \(i<j\), and for sufficiently small \(\epsilon \), defines such a point.

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Acknowledgments

We have benefited from useful discussions with Samuel Altschul, Igor Dolgachev, Daniel Erman, William Fulton, Roger Howe, Thomas Lam, and Mircea Mustaţă. We thank Claudiu Raicu, David Speyer, and the anonymous referee for providing many comments, suggestions, and improvements. We are especially grateful to our advisor Robert Lazarsfeld for posing the problem, and for numerous suggestions and encouragements.

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  1. Mihai Fulger

    Present address: Department of Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Mihai Fulger & Xin Zhou

  2. Institute of Mathematics of the Romanian Academy, P. O. Box 1-764, Bucharest, 014700, Romania

    Mihai Fulger

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  1. Mihai Fulger
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Correspondence to Mihai Fulger.

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Fulger, M., Zhou, X. Schur asymptotics of Veronese syzygies. Math. Ann. 362, 529–540 (2015). https://doi.org/10.1007/s00208-014-1125-4

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