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Torus fixed points in Schubert varieties and Genocchi numbers

Fang, X. and Fourier, G. (2016) Torus fixed points in Schubert varieties and Genocchi numbers. Séminaire Lotharingien de Combinatoire, 75, B75f.

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Abstract

We give a new proof for the fact that the number of torus fixed points for the degenerated flag variety is equal to the normalized median Genocchi number, using the identification with a certain Schubert variety. We further study the torus fixed points for the symplectic degenerated flag variety and develop a combinatorial model, symplectic Dellac configurations, so parametrize them. The number of these symplectic fixed points is conjectured to be the median Euler number.

Item Type:	Articles
Status:	Published
Refereed:	Yes
Glasgow Author(s) Enlighten ID:	Fourier, Dr Ghislain
Authors:	Fang, X., and Fourier, G.
College/School:	College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Research Group:	Algebra
Journal Name:	Séminaire Lotharingien de Combinatoire
Publisher:	Université Louis Pasteur
ISSN:	1286-4889
Related URLs:	arXiv

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References

[GTM05] A. Bj¨rner and F. Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, o 231. Springer, New York, 2005. xiv+363 pp. [CL14] G. Cerulli Irelli and M. Lanini, Degenerate flag varieties of type A and C are Schubert varieties. Int. Math. Res. Notices. first published online August 7, 2014, doi:10.1093/imrn/rnu128, 2014. [CLL15] G. Cerulli Irelli, M. Lanini, and P. Littelmann, Degenerate flag varieties and Schubert vari eties: a characteristic free approach. arXiv:1502.04590, 2015 [CFR12] G. Cerulli Irelli, E. Feigin, and M. Reineke, Quiver Grassmannians and degenerate flag vari eties. Algebra Number Theory 6 (2012), no. 1, 165-194. [DR94] D. Dumont and A. Randrianarivony, D´rangements et nombres de Genocchi. Discrete Math. e 132 (1994), no. 1-3, 37-49. [FFR15] X. Fang, G. Fourier, and M. Reineke, PBW-type filtration on quantum groups of type An preprint arXiv:1503.05428 [Fei11] E. Feigin, Degenerate flag varieties and the median Genocchi numbers. Math. Res. Lett. 18 (2011), no. 6, 1163-1178. [Fei12] E. Feigin, The median Genocchi numbers, q-analogues and continued fractions. European J. Combin. 33 (2012), no. 8, 1913-1918. [FFiL14] E. Feigin, M. Finkelberg, and P. Littelmann, Symplectic degenerate flag varieties. Canad. J. Math. 66 (2014), no. 6, 1250-1286. [FFoL11a] E. Feigin, G. Fourier, and P. Littelmann, PBW filtration and bases for irreducible modules in type An , Transformation Groups: Volume 16, Issue 1 (2011), 71-89. [FFoL11b] E. Feigin, G. Fourier, and P. Littelmann, PBW filtration and bases for symplectic Lie algebras, International Mathematics Research Notices 2011 (24), pp. 5760-5784. [FFoL13] E. Feigin, G. Fourier, and P. Littelmann, Favourable modules: Filtrations, polytopes, NewtonOkounkov bodies and flat degenerations, preprint arXiv:1306.1292. [Fou15] G. Fourier, New homogeneous ideals for current algebras: filtrations, fusion products, and the Pieri rules, Moscow Mathematical Journal, Volume 15 (1), 2015 pp. 49-72. [Fou14] G. Fourier, Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equiv alence, preprint, arXiv:1410.8744. [Hag14] C. Hague, Degenerate coordinate rings of flag varieties and Frobenius splitting, Selecta Math. (N.S.) 20 (2014), no. 3, 823-838. [K97] G. Kreweras, Sur les permutations comptés par les nombres de Genocchi de 1-i`re et 2-i`me e e e esp`ce. European J. Combin. 18 (1997), no. 1, 49-58. e [RZ96] A. Randrianarivony and J. Zeng, Une famille de polynˆmes qui interpole plusieurs suites clas o siques de nombres. Adv. in Appl. Math. 17 (1996), no. 1, 1-26. [Sjo07] J. Sj¨strand, Bruhat intervals as rooks on skew Ferrers boards. Journal of Combinatorial Theory, o Series A 114 (2007), 1182-1198.

Deposit and Record Details

ID Code:	108502
Depositing User:	Dr Ghislain Fourier
Datestamp:	05 Aug 2015 13:29
Last Modified:	14 Jan 2019 16:28
Date of acceptance:	25 May 2016
Date of first online publication:	2016