Journal of Lie Theory

no. 1
Linear embeddings of Grassmannians and ind-Grassmannians
Penkov, Ivan1; Tsanov, Valdemar2, 1
1Constructor University, 28759 Bremen, Germany
2Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia 1113, Bulgaria
Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 213-268

By a Grassmannian we understand a usual complex Grassmannian or possibly an orthogonal or symplectic Grassmannian. We classify, with few exceptions, linear embeddings of Grassmannians into larger Grassmannians, where the linearity requirement is the condition that the embedding induces an isomorphism on Picard groups. This classification implies that most linear embeddings of Grassmannians are equivariant.

A linear ind-Grassmannian is the direct limit of a chain of linear embeddings of Grassmannians. We conclude the paper by classifying linear embeddings of linear ind-Grassmannians.

Received:
Accepted:
Published online:
DOI: 10.5802/jlt.1425
Classification: 14E25, 14L30, 14M15
Keywords: Grassmannian, isotropic Grassmannian, ind-Grassmannian, linear embedding, equivariant embedding

Penkov, Ivan  1 ; Tsanov, Valdemar  2 , 1

1 Constructor University, 28759 Bremen, Germany
2 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia 1113, Bulgaria 
Penkov, Ivan; Tsanov, Valdemar. Linear embeddings of Grassmannians and ind-Grassmannians. Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 213-268. doi: 10.5802/jlt.1425
@article{10_5802_jlt_1425,
     author = {Penkov, Ivan and Tsanov, Valdemar},
     title = {Linear embeddings of {Grassmannians} and {ind-Grassmannians}},
     journal = {Journal of Lie Theory},
     pages = {213--268},
     year = {2026},
     publisher = {XXXX},
     volume = {36},
     number = {1},
     doi = {10.5802/jlt.1425},
     language = {en},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jlt.1425/}
}
TY  - JOUR
AU  - Penkov, Ivan
AU  - Tsanov, Valdemar
TI  - Linear embeddings of Grassmannians and ind-Grassmannians
JO  - Journal of Lie Theory
PY  - 2026
SP  - 213
EP  - 268
VL  - 36
IS  - 1
PB  - XXXX
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jlt.1425/
DO  - 10.5802/jlt.1425
LA  - en
ID  - 10_5802_jlt_1425
ER  - 
%0 Journal Article
%A Penkov, Ivan
%A Tsanov, Valdemar
%T Linear embeddings of Grassmannians and ind-Grassmannians
%J Journal of Lie Theory
%D 2026
%P 213-268
%V 36
%N 1
%I XXXX
%U https://jolt.centre-mersenne.org/articles/10.5802/jlt.1425/
%R 10.5802/jlt.1425
%G en
%F 10_5802_jlt_1425

[1] Dimitrov, Ivan; Penkov, Ivan Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups, Int. Math. Res. Not., Volume 2004 (2004) no. 55, pp. 2935-2953 | Zbl | DOI | MR

[2] Dynkin, Evgeniĭ B. Maximal subgroups of classical group, Tr. Mosk. Mat. O.-va, Volume 1 (1952), pp. 39-166 | Zbl | MR

[3] Dynkin, Evgeniĭ B. Semisimple subalgebras of semisimple Lie algebras, Mat. Sb., N. Ser., Volume 30 (1952), pp. 349-462 English translation in Am. Math. Soc. Transl., Series 2 6 (1957), pp. 111–244 | Zbl | MR

[4] Humphreys, James E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer, 1975, xv+247 pages | Zbl | MR

[5] Knop, Friedrich; Kraft, Hanspeter; Vust, Thierry The Picard group of a G-variety, Algebraische Transformationsgruppen und Invariantentheorie (DMV Seminar), Volume 13, Birkhäuser, 1989, pp. 77-87 | Zbl | DOI

[6] Landsberg, Joseph M. Tensors: Geometry and Applications, Graduate Studies in Mathematics, 128, American Mathematical Society, 2012, xx+439 pages | Zbl | MR

[7] Landsberg, Joseph M.; Manivel, Laurent On the projective geometry of rational homogeneous varieties, Comment. Math. Helv., Volume 78 (2003) no. 1, pp. 65-100 | Zbl | DOI | MR

[8] Onishchik, Arkadiĭ L. Inclusion relations between transitive compact transformation groups, Tr. Mosk. Mat. O.-va, Volume 11 (1962), pp. 199-242 English translation in Am. Math. Soc. Transl., Series 2 50 (1966), pp. 5–58 | Zbl | MR

[9] Penkov, Ivan; Tikhomirov, Alexander S. Linear ind-Grassmannians, Pure Appl. Math. Q., Volume 10 (2014) no. 2, pp. 289-323 | Zbl | MR | DOI

[10] Sato, Mikio The KP hierarchy and infinite-dimensional Grassmann manifolds, Theta functions, Bowdoin 1987 (Proceedings of Symposia in Pure Mathematics), Volume 49.1, American Mathematical Society, 1987, pp. 51-66 | Zbl | DOI | MR

[11] Tits, Jacques Sur les analogues algébriques des groupes semi-simples complexes, Colloque d’algèbre supérieure, Bruxelles, 19-22 décembre 1956, Centre Belge de Recherches Mathématiques, 1957, pp. 261-289 | Zbl | MR

Cited by Sources: