Hilbert series of the Grassmannian and k -Narayana numbers

Lukas Braun

Communications in Mathematics (2019)

  • Volume: 27, Issue: 1, page 27-41
  • ISSN: 1804-1388

Abstract

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We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the q -Hilbert series is a Vandermonde-like determinant. We show that the h -polynomial of the Grassmannian coincides with the k -Narayana polynomial. A simplified formula for the h -polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k -Narayana numbers, i.e. the h -polynomial of the Grassmannian.

How to cite

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Braun, Lukas. "Hilbert series of the Grassmannian and $k$-Narayana numbers." Communications in Mathematics 27.1 (2019): 27-41. <http://eudml.org/doc/294371>.

@article{Braun2019,
abstract = {We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$-Hilbert series is a Vandermonde-like determinant. We show that the $h$-polynomial of the Grassmannian coincides with the $k$-Narayana polynomial. A simplified formula for the $h$-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$-Narayana numbers, i.e. the $h$-polynomial of the Grassmannian.},
author = {Braun, Lukas},
journal = {Communications in Mathematics},
keywords = {Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform},
language = {eng},
number = {1},
pages = {27-41},
publisher = {University of Ostrava},
title = {Hilbert series of the Grassmannian and $k$-Narayana numbers},
url = {http://eudml.org/doc/294371},
volume = {27},
year = {2019},
}

TY - JOUR
AU - Braun, Lukas
TI - Hilbert series of the Grassmannian and $k$-Narayana numbers
JO - Communications in Mathematics
PY - 2019
PB - University of Ostrava
VL - 27
IS - 1
SP - 27
EP - 41
AB - We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$-Hilbert series is a Vandermonde-like determinant. We show that the $h$-polynomial of the Grassmannian coincides with the $k$-Narayana polynomial. A simplified formula for the $h$-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$-Narayana numbers, i.e. the $h$-polynomial of the Grassmannian.
LA - eng
KW - Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform
UR - http://eudml.org/doc/294371
ER -

References

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