Hilbert series of the Grassmannian and k-Narayana numbers
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...
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Online Access: | https://doi.org/10.2478/cm-2019-0003 |
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doaj-333a9dc87c59432fb6391719410f65f12021-09-06T19:22:06ZengSciendoCommunications in Mathematics2336-12982019-06-01271274110.2478/cm-2019-0003cm-2019-0003Hilbert series of the Grassmannian and k-Narayana numbersBraun Lukas0Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, GermanyWe 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.https://doi.org/10.2478/cm-2019-0003hilbert series of the grassmanniannarayana numberseuler’s hypergeometric transform14m1513d4033c90 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Braun Lukas |
spellingShingle |
Braun Lukas Hilbert series of the Grassmannian and k-Narayana numbers Communications in Mathematics hilbert series of the grassmannian narayana numbers euler’s hypergeometric transform 14m15 13d40 33c90 |
author_facet |
Braun Lukas |
author_sort |
Braun Lukas |
title |
Hilbert series of the Grassmannian and k-Narayana numbers |
title_short |
Hilbert series of the Grassmannian and k-Narayana numbers |
title_full |
Hilbert series of the Grassmannian and k-Narayana numbers |
title_fullStr |
Hilbert series of the Grassmannian and k-Narayana numbers |
title_full_unstemmed |
Hilbert series of the Grassmannian and k-Narayana numbers |
title_sort |
hilbert series of the grassmannian and k-narayana numbers |
publisher |
Sciendo |
series |
Communications in Mathematics |
issn |
2336-1298 |
publishDate |
2019-06-01 |
description |
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. |
topic |
hilbert series of the grassmannian narayana numbers euler’s hypergeometric transform 14m15 13d40 33c90 |
url |
https://doi.org/10.2478/cm-2019-0003 |
work_keys_str_mv |
AT braunlukas hilbertseriesofthegrassmannianandknarayananumbers |
_version_ |
1717772663146938368 |