Stable rationality of certain \(PGL_ n\)-quotients. (English) Zbl 0741.14032
Consider the action of \(PGL_ n\) on the space \(M_ n\times M_ n\) of pairs of complex \(n\times n\) matrices by simultaneous conjugation, and let \(\mathbb{C}(M_ n\times M_ n)^{PGL_ n}\) be the field of \(PGL_ n\)- invariant complex rational functions on this space. It is known that this field is rational over \(\mathbb{C}\) if \(n\leq 4\) [case \(n=2\) dates back to the last century, cases \(n=3,4\) were considered by E. Formanek, Linear Multilinear Algebra 7, 203-212 (1979; Zbl 0419.16010) and J. Algebra 62, 304-319 (1980; Zbl 0437.16013), respectively]. For \(n\geq 5\) the question whether this field is (stably) rational is open.
The main result of the paper under review is the affirmative answer to this question for \(n=5\) and \(n=7\) (the stable rationality is proved in these cases). — In combination with some known results this shows that \(\mathbb{C}(V)^{PGL_ n}\) is stably rational whenever \(V\) is an almost free representation of \(PGL_ n\) and \(n\) divides \(420=2^ 2\cdot 3\cdot 5\cdot 7\).
The main result of the paper under review is the affirmative answer to this question for \(n=5\) and \(n=7\) (the stable rationality is proved in these cases). — In combination with some known results this shows that \(\mathbb{C}(V)^{PGL_ n}\) is stably rational whenever \(V\) is an almost free representation of \(PGL_ n\) and \(n\) divides \(420=2^ 2\cdot 3\cdot 5\cdot 7\).
Reviewer: V.L.Popov (Moskva)
MSC:
| 14M20 | Rational and unirational varieties |
| 14L24 | Geometric invariant theory |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14M17 | Homogeneous spaces and generalizations |
References:
| [1] | Benson, D.: Modular Representation Theory: New Trends and Methods. (Lect Notes Math., vol. 1081) Berlin Heidelberg, New York: Springer 1984 · Zbl 0564.20004 |
| [2] | Benson, D.J., Parker, R.A.: The Green ring of a finite group. J. Algebra87, 290-331 (1984) · Zbl 0539.20009 · doi:10.1016/0021-8693(84)90139-X |
| [3] | Bogomolov, F.A.: The stable rationality of quotient spaces for simply connected groups. Math. USSR Sb.58, 1-14 (1987) · Zbl 0681.14030 · doi:10.1070/SM1987v058n01ABEH003089 |
| [4] | Colliot-Thélène, J-L., Sansuc, J-J.: LaR-équivalence sur les tores. Ann. Sci. Éc. Norm. Super.10, 175-229 (1977) · Zbl 0356.14007 |
| [5] | Colliot-Thélène, J-L., Sansuc, J-J.: Principal homogeneous spaces under flasque tori: applications. J. Algebra106, 148-205 (1987) · Zbl 0597.14014 · doi:10.1016/0021-8693(87)90026-3 |
| [6] | Curtis, C.W., Reiner, I.: Methods of Representation Theory I. New York: Wiley-Interscience 1981 · Zbl 0469.20001 |
| [7] | Curtis, C.W., Reiner, I.: Methods of Representation Theory II. New York: Wiley-Interscience 1981 · Zbl 0469.20001 |
| [8] | Dolgachev, I.V.: Rationality of fields of invariants. In: Bloch, S.J. )ed.) Algebraic Geometry Bowdoin 1985, Proc. Symp. Pure Math.46, Part 2, 3-16 (1987) |
| [9] | Dress, A.: The permutation class group of a finite group. J. Pure Appl. Algebra6, 1-12 (1975) · Zbl 0302.20010 · doi:10.1016/0022-4049(75)90008-0 |
| [10] | Endo, S., Miyata, T.: On a classification of the function fields of algebraic tori. Nagoya Math. J.56, 85-104 (1974) · Zbl 0301.14008 |
| [11] | Endo, S., Miyata, T.: On the projective class group of finite groups. Osaka J. Math.13, 109-122 (1976) · Zbl 0364.20009 |
| [12] | Formanek, E.: The center of the ring of 3{\(\times\)}3 generic matrices. Linear Multilinear Algebra7, 203-212 (1979) · Zbl 0419.16010 · doi:10.1080/03081087908817278 |
| [13] | Formanek, E.: The center of the ring of 4{\(\times\)}4 generic matrices. J. Algebra62, 304-319 (1980) · Zbl 0437.16013 · doi:10.1016/0021-8693(80)90184-2 |
| [14] | Hulek, K.: On the classification of stable rankr vector bundles over the projective plane. Prog. Math.7, 113-144 (1980) |
| [15] | Kempf, G. et al.: Toroidal Embeddings. (Lect Notes Math., vol. 339) Berlin, Heidelberg, New York: Spinger 1973 |
| [16] | Le Bruyn, L., Schofield, A.: Rational invariants of quivers and the ring of matrixinvariants. NATO-ASI Ser. C-233, 21-30 (1988) |
| [17] | Lenstra, H.W.: Rational functions invariant under a finite abelian group. Invent. Math.25, 299-325 (1974) · Zbl 0292.20010 · doi:10.1007/BF01389732 |
| [18] | Mumford, D., Fogarty, J.: Geometric Invariant Theory (2nd edition). Berlin, Heidelberg, New York: Springer 1981 · Zbl 0504.14008 |
| [19] | Oliver, R.:G-actions on disks and permutation representations. J. Algebra50, 44-62 (1978) · Zbl 0386.20002 · doi:10.1016/0021-8693(78)90173-4 |
| [20] | Procesi, C.: Non-commutative affine rings. Atti Accad. Naz. Lincei, VIII. Ser., v. VIII, fo.6, 239-255 (1967) · Zbl 0204.04802 |
| [21] | Procesi, C.: The invariant theory ofn{\(\times\)}n matrices. Adv. Math.19, 306-381 (1976) · Zbl 0331.15021 · doi:10.1016/0001-8708(76)90027-X |
| [22] | Reiner, I.: Maximal Orders. London: Academic Press 1975 · Zbl 0305.16001 |
| [23] | Saltman, D.J.: Retract rational fields and cyclic Galois extensions. Isr. J. Math.47, 165-215 (1984) · Zbl 0546.14013 · doi:10.1007/BF02760515 |
| [24] | Saltman, D.J.: The Brauer group and the center of generic matrices. J. Algebra97, 53-67 (1985) · Zbl 0586.13005 · doi:10.1016/0021-8693(85)90073-0 |
| [25] | Schofield, A.: Matrix invariants of composite size. 1989 (Preprint) · Zbl 0785.14030 |
| [26] | Serre, J.P.: Corps Locaux. Paris: Hermann 1962 · Zbl 0137.02501 |
| [27] | Swan, R.G.: Invariant rational functions and a problem of Steenrod. Invent. Math.7, 148-158 (1969) · Zbl 0186.07601 · doi:10.1007/BF01389798 |
| [28] | Swan, R.G.: Noether’s Problem in Galois theory. In: Srinivasan, B., Sally, J. (eds.) Emmy Noether in Bryn Mawr. Berlin, Heidelberg, New York: Springer 1983, pp. 21-40 · Zbl 0538.12012 |
| [29] | Sylvester, J.: On the involution of two matrices of the second order. Southport: British Association Report 1883, pp. 430-432 |
| [30] | Voskresenskiî, V.E.: Rationality of certain algebraic tori. Math. USSR, Izv.5, 1049-1056 (1971) · Zbl 0251.14008 · doi:10.1070/IM1971v005n05ABEH001200 |
| [31] | Voskresenskiî, V.E.: The birational invariants of algebraic tori. Usp. Mat. Nauk30/3n2, 207-208 (1975) · Zbl 0315.14002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
