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

Full-text PDF

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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*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**, DOI 10.5802/aif.2996
- 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, 38. 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**
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*The Classical Groups. Their Invariants and Representations*, Princeton University Press, Princeton, NJ, 1939. MR **255**

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.