On the Popov-Pommerening conjecture for groups of type 𝐴_{𝑛}

Tan, Lin

doi:10.1090/S0002-9939-1989-0969528-5

On the Popov-Pommerening conjecture for groups of type $A_ n$
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Proc. Amer. Math. Soc. 106 (1989), 611-616 Request permission

Abstract:

The present paper gives an affirmative answer to the Popov-Pommerening conjecture in the case where the reductive group $G$ in the conjecture is of type ${A_n}$ with $n \leq 4$ , and provides a subgroup $H$ of $G{L_5}(k)$ such that the algebra ${A^H}$ is finitely generated, but is not spanned by the invariant standard bitableaux.

References

Frank Grosshans, Observable groups and Hilbert’s fourteenth problem, Amer. J. Math. 95 (1973), 229–253. MR 325628, DOI 10.2307/2373655
F. D. Grosshans, The invariants of unipotent radicals of parabolic subgroups, Invent. Math. 73 (1983), no. 1, 1–9. MR 707345, DOI 10.1007/BF01393822
Frank D. Grosshans, Hilbert’s fourteenth problem for nonreductive groups, Math. Z. 193 (1986), no. 1, 95–103. MR 852912, DOI 10.1007/BF01163357
W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102 (1975), no. 1, 67–83. MR 382294, DOI 10.2307/1970974
G. Hochschild and G. D. Mostow, Unipotent groups in invariant theory, Proc. Nat. Acad. Sci. U.S.A. 70 (1973), 646–648. MR 320174, DOI 10.1073/pnas.70.3.646
David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
Masayoshi Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3 (1963/64), 369–377. MR 179268, DOI 10.1215/kjm/1250524787
M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828
Klaus Pommerening, Invarianten unipotenter Gruppen, Math. Z. 176 (1981), no. 3, 359–374 (German). MR 610216, DOI 10.1007/BF01214612
Klaus Pommerening, Ordered sets with the standardizing property and straightening laws for algebras of invariants, Adv. in Math. 63 (1987), no. 3, 271–290. MR 877787, DOI 10.1016/0001-8708(87)90057-0

Invariants of unipotent groups—a survey

1278

Hilbert’s theorem on invariants

20

C. S. Seshadri, On a theorem of Weitzenböck in invariant theory, J. Math. Kyoto Univ. 1 (1961/62), 403–409. MR 144914, DOI 10.1215/kjm/1250525012
Robert Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, Vol. 366, Springer-Verlag, Berlin-New York, 1974. Notes by Vinay V. Deodhar. MR 0352279, DOI 10.1007/BFb0067854

The invariant theory of unipotent subgroups of reductive algebraic groups

Some recent developments in the Popov-Pommerening Conjecture

R. Weitzenböck, Über die Invarianten von linearen Gruppen, Acta Math. 58 (1932), no. 1, 231–293 (German). MR 1555349, DOI 10.1007/BF02547779