On the S^k-Fixed Points of Springer Varieties in Type A
Keywords:
Flag Variety, Springer Varieties, Bruhat Order, Coxeter Group, General Linear Group
Abstract
Springer varieties are sub-varieties of the full (complete) flag variety Fℓn(C)Fℓn(C), which can be thought of as the fiber over XX of the Springer resolution of singularities of cone of nilpotent endomorphism X:V⟶VX:V⟶V, where XX is a nilpotent endomorphism in its Jordan canonical form of type λλ and VV is a nn-dimensional vector space over CC (V=CnV=Cn for convenience). The geometry of Springer varieties is reviewed in this article, along with their SkSk-fixed points. We accomplish this by briefly reviewing nilpotent orbits in type AA within the framework of integer partition.
References
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[15] Julianna S Tymoczko. Linear conditions imposed on flag varieties. American Journal of Mathematics, 128(6):1587-1604, 2006.
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s
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s
1
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(
n
k
,
k
)
(nk,k) springer varieties. Osaka Journal of Mathematics, 52(4):1051-1063, 2015.
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[20] Tonny Albert Springer. A construction of representations of weyl groups. Inventiones mathematicae, 44(3):279-293, 1978.
[21] Naohisa Shimomura. The fixed point subvarieties of unipotent transformations on the flag varieties. Journal of the Mathematical Society of Japan, 37(3):537-556, 1985.
[22] Francis YC Fung. On the topology of components of some springer fibers and their relation to kazhdan-lusztig theory. Advances in Mathematics, 178(2):244-276, 2003.
[23] Mikhail Khovanov. Crossingless matchings and the cohomology of (n, n) springer varieties. Communications in Contemporary Mathematics, 6(04):561-577, 2004.
[24] Heather Russell. A topological construction for all two-row springer varieties. Pacific journal of mathematics, 253(1):221-255, 2011.
[25] Lucas Fresse, Ronit Mansour, and Anna Melnikov. Unimodality of the distribution of betti numbers for some springer fibers. Journal of Algebra, 391:284-304, 2013.
[26] Lucas Fresse, Anna Melnikov, and Sammar Sakas-Obeid. On the structure of smooth components of springer fibers. Proceedings of the American Mathematical Society, 143(6):2301-2315, 2015.
[27] Martha Precup and Julianna Tymoczko. Springer fibers and schubert points. arXiv preprint arXiv:1701.03502, 2017.
[28] Felemu Olasupo and Praise Adeyemo. On the tymoczko codes for standard young tableaux. Earthline Journal of Mathematical Sciences, 14(6):1173-1193, 2024.
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[32] Andrew Baker. Matrix groups: An introduction to Lie group theory. Springer Science & Business Media, 2012.
[33] Vinoth Nandakumar. Nilpotent cones. 2010.
[34] Venkatramani Lakshmibai and Justin Brown. Flag varieties: an interplay of geometry, combinatorics, and representation theory, volume 53. Springer, 2018.
[35] Michel Brion. Lectures on the geometry of flag varieties. In Topics in cohomological studies of algebraic varieties, pages 33-85. Springer, 2005.
[36] David H Collingwood and William M McGovern. Nilpotent orbits in semisimple Lie algebra: an introduction. CRC Press, 1993.
[2] OJ Omidire. Common fixed point theorems of some certain generalized contractive conditions in convex metric space settings. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(3):1-9, 2024.
[3] Filippo De Mari, Claudio Procesi, and Mark A Shayman. Hessenberg varieties. Transactions of the American Mathematical Society, 332(2):529-534, 1992.
[4] Julianna S Tymoczko. Hessenberg varieties are not pure dimensional. Pure and Applied Mathematics Quarterly, 2(3), 2006.
[5] Jens Jordan and Uwe Helmke. Controllability of the qr-algorithm on hessenberg flags. In Proceeding of the Fifteenth International Symposium on Mathematical Theory of Network and Systems (MTNS 2002), 2002.
[6] Praise Adeyemo. Some combinatorial aspects of s1 fixed points of peterson varieties. In International Mathematical Forum, volume 12, pages 339-350, 2017.
[7] Yukiko Fukukawa, Megumi Harada, and Mikiya Masuda. The equivariant cohomology rings of peterson varieties. Journal of the Mathematical Society of Japan, 67(3):1147-1159, 2015.
[8] Erik Insko. Schubert calculus and the homology of the peterson variety. The Electronic Journal of Combinatorics, 22(2):2-26, 2015.
[9] Erik Insko and Alexander Yong. Patch ideals and peterson varieties. Transformation Groups, 17(4):1011-1036, 2012.
[10] Megumi Harada and Julianna Tymoczko. A positive monk formula in the s1-equivariant cohomology of type a peterson varieties. Proceedings of the London Mathematical Society, 103(1):40-72, 2011.
[11] Darius Bayegan and Megumi Harada. Poset pinball, the dimension pair algorithm, and type a regular nilpotent hessenberg varieties. ISRN Geometry, 2012, 2012.
[12] Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda. The cohomology rings of regular nilpotent hessenberg varieties in lie type a. International Mathematics Research Notices, 2016.
[13] Martha Precup and Julianna Tymoczko. Springer fibers and schubert points. European Journal of Combinatorics, 76:10-26, 2019.
[14] Lucas Fresse. Betti numbers of springer fibers in type a. Journal of Algebra, 322(7):2566-2579, 2009.
[15] Julianna S Tymoczko. Linear conditions imposed on flag varieties. American Journal of Mathematics, 128(6):1587-1604, 2006.
[16] Felemu Olasupo and Adetunji Patience. On the inversion and dimension pairs of row-strict tableaux. Journal of Nepal Mathematical Society, 7(1):32-39, 2024.
[17] Lucas Fresse. Singular components of springer fibers in the two-column case (composantes singulières des fibres de springer dans le cas deux-colonnes). In Annales de l'institut Fourier, volume 59, pages 2429-2444, 2009.
[18] Tatsuya Horiguchi et al. The
s
1
s
1
equivariant cohomology ring of
(
n
k
,
k
)
(nk,k) springer varieties. Osaka Journal of Mathematics, 52(4):1051-1063, 2015.
[19] Julianna Tymoczko. The geometry and combinatorics of springer fibers. arXiv preprint arXiv:1606.02760, 2016.
[20] Tonny Albert Springer. A construction of representations of weyl groups. Inventiones mathematicae, 44(3):279-293, 1978.
[21] Naohisa Shimomura. The fixed point subvarieties of unipotent transformations on the flag varieties. Journal of the Mathematical Society of Japan, 37(3):537-556, 1985.
[22] Francis YC Fung. On the topology of components of some springer fibers and their relation to kazhdan-lusztig theory. Advances in Mathematics, 178(2):244-276, 2003.
[23] Mikhail Khovanov. Crossingless matchings and the cohomology of (n, n) springer varieties. Communications in Contemporary Mathematics, 6(04):561-577, 2004.
[24] Heather Russell. A topological construction for all two-row springer varieties. Pacific journal of mathematics, 253(1):221-255, 2011.
[25] Lucas Fresse, Ronit Mansour, and Anna Melnikov. Unimodality of the distribution of betti numbers for some springer fibers. Journal of Algebra, 391:284-304, 2013.
[26] Lucas Fresse, Anna Melnikov, and Sammar Sakas-Obeid. On the structure of smooth components of springer fibers. Proceedings of the American Mathematical Society, 143(6):2301-2315, 2015.
[27] Martha Precup and Julianna Tymoczko. Springer fibers and schubert points. arXiv preprint arXiv:1701.03502, 2017.
[28] Felemu Olasupo and Praise Adeyemo. On the tymoczko codes for standard young tableaux. Earthline Journal of Mathematical Sciences, 14(6):1173-1193, 2024.
[29] Ian Grant Macdonald. Symmetric functions and Hall polynomials. Oxford university press, 1998.
[30] George E Andrews and Kimmo Eriksson. Integer partitions. Cambridge University Press, 2004.
[31] Anders Bjorner and Francesco Brenti. Combinatorics of Coxeter groups, volume 231. Springer Science & Business Media, 2006.
[32] Andrew Baker. Matrix groups: An introduction to Lie group theory. Springer Science & Business Media, 2012.
[33] Vinoth Nandakumar. Nilpotent cones. 2010.
[34] Venkatramani Lakshmibai and Justin Brown. Flag varieties: an interplay of geometry, combinatorics, and representation theory, volume 53. Springer, 2018.
[35] Michel Brion. Lectures on the geometry of flag varieties. In Topics in cohomological studies of algebraic varieties, pages 33-85. Springer, 2005.
[36] David H Collingwood and William M McGovern. Nilpotent orbits in semisimple Lie algebra: an introduction. CRC Press, 1993.
Published
2025-05-30
How to Cite
Felemu, O. J. (2025). On the S^k-Fixed Points of Springer Varieties in Type A. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 11(2), 74 - 92. Retrieved from https://ijmso.unilag.edu.ng/article/view/2895
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