Standard monomials and invariant theory of arc spaces II: Symplectic group
Authors:
Andrew R. Linshaw and Bailin Song
Journal:
J. Algebraic Geom. 33 (2024), 601-628
DOI:
https://doi.org/10.1090/jag/834
Published electronically:
June 13, 2024
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Abstract |
References |
Additional Information
Abstract: This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the Pfaffian variety over $K$. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group.
References
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- Andrew R. Linshaw and Bailin Song, The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds II, Comm. Math. Phys. 399 (2023), no. 1, 189–202. MR 4567372, DOI 10.1007/s00220-022-04554-z
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- Fyodor Malikov, Vadim Schechtman, and Arkady Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999), no. 2, 439–473. MR 1704283, DOI 10.1007/s002200050653
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References
- Peter Bardsley and R. W. Richardson, Étale slices for algebraic transformation groups in characteristic $p$, Proc. London Math. Soc. (3) 51 (1985), no. 2, 295–317. MR 794118, DOI 10.1112/plms/s3-51.2.295
- Bhargav Bhatt, Algebraization and Tannaka duality, Camb. J. Math. 4 (2016), no. 4, 403–461. MR 3572635, DOI 10.4310/CJM.2016.v4.n4.a1
- T. Creutzig, A. Linshaw, and B. Song, Classical freeness of orthosymplectic affine vertex superalgebras, arXiv:2211.15032v2, To appear in Proc. Am. Math. Soc.
- C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), no. 3, 330–354. MR 422314, DOI 10.1016/S0001-8708(76)80003-5
- Lawrence Ein and Mircea Mustaţă, Jet schemes and singularities, Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 505–546. MR 2483946, DOI 10.1090/pspum/080.2/2483946
- W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Cambridge Philos. Soc. 39 (1943), 22–30. MR 7739, DOI 10.1017/s0305004100017631
- E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 568864
- V. Lakshmibai and C. S. Seshadri, Geometry of $G/P$. II. The work of de Concini and Procesi and the basic conjectures, Proc. Indian Acad. Sci. Sect. A 87 (1978), no. 2, 1–54. MR 490244
- V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of $G/P$. III. Standard monomial theory for a quasi-minuscule $P$, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 3, 93–177. MR 561813
- V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of $G/P$. IV. Standard monomial theory for classical types, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 4, 279–362. MR 553746
- Venkatramani Lakshmibai and Komaranapuram N. Raghavan, Standard monomial theory, Encyclopaedia of Mathematical Sciences, vol. 137, Springer-Verlag, Berlin, 2008. Invariant theoretic approach; Invariant Theory and Algebraic Transformation Groups, 8. MR 2388163
- Andrew R. Linshaw, Gerald W. Schwarz, and Bailin Song, Jet schemes and invariant theory, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2571–2599 (English, with English and French summaries). MR 3449590
- Andrew R. Linshaw and Bailin Song, Standard monomials and invariant theory for arc spaces I: General linear group, Commun. Contemp. Math. 26 (2024), no. 4, Paper No. 2350013. MR 4730616, DOI 10.1142/S021919972350013X
- Andrew R. Linshaw and Bailin Song, The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds II, Comm. Math. Phys. 399 (2023), no. 1, 189–202. MR 4567372, DOI 10.1007/s00220-022-04554-z
- Andrew R. Linshaw and Bailin Song, Cosets of free field algebras via arc spaces, Int. Math. Res. Not. IMRN 1 (2024), 47–114. MR 4686646, DOI 10.1093/imrn/rnac367
- Fyodor Malikov, Vadim Schechtman, and Arkady Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999), no. 2, 439–473. MR 1704283, DOI 10.1007/s002200050653
- C. S. Seshadri, Geometry of $G/P$. I. Theory of standard monomials for minuscule representations, C. P. Ramanujam—a tribute, Tata Inst. Fundam. Res. Stud. Math., vol. 8, Springer, Berlin-New York, 1978, pp. 207–239. MR 541023
- Hermann Weyl, The classical groups. Their invariants and representations, Princeton University Press, Princeton, NJ, 1939. MR 255
Additional Information
Andrew R. Linshaw
Affiliation:
Department of Mathematics, University of Denver, Denver, CO 80208
MR Author ID:
791304
ORCID:
0000-0002-8497-8721
Email:
andrew.linshaw@du.edu
Bailin Song
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
MR Author ID:
826622
Email:
bailinso@ustc.edu.cn
Received by editor(s):
October 7, 2021
Received by editor(s) in revised form:
October 18, 2023, and February 2, 2024
Published electronically:
June 13, 2024
Additional Notes:
The first author was supported by Simons Foundation Grant #635650 and NSF Grant DMS-2001484. The second author was supported by National Natural Science Foundation of China Grant #12171447
Communicated by:
David Nadler
Article copyright:
© Copyright 2024
University Press, Inc.
