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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Main Author:	Braun Lukas
Format:	Article
Language:	English
Published:	Sciendo 2019-06-01
Series:	Communications in Mathematics
Subjects:	hilbert series of the grassmannian narayana numbers euler’s hypergeometric transform 14m15 13d40 33c90
Online Access:	https://doi.org/10.2478/cm-2019-0003

id	doaj-333a9dc87c59432fb6391719410f65f1
record_format	Article
spelling	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

Hilbert series of the Grassmannian and k-Narayana numbers

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