Let $ G = C_p\times H $ be a finite group, where $ C_p $ is a cyclic group of prime order $ p $ and $ H $ is a $ p^{\prime} $-group. Let $ \mathbb{F} $ be an algebraically closed field in characteristic $ p $. Let $ V $ be a direct sum of $ m $ non-trivial indecomposable $ G $-modules such that the norm polynomials of the simple $ H $-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring $ \mathbb{F}[V] $ with actions of $ G $. In this paper, by the periodicity property, we showed that $ \mathbb{F}[V]^G $ is generated by $ m $ norm polynomials together with homogeneous invariants of degree at most $ m|G|-{\rm dim}(V) $ and transfer invariants, which yields the well-known degree bound $ {\rm dim} $$ (V)\cdot(|G|-1) $. More precisely, we found that this bound gets less sharp as the dimensions of simple $ H $-modules increase.
Citation: Yang Zhang, Jizhu Nan. A note on the degree bounds of the invariant ring[J]. AIMS Mathematics, 2024, 9(5): 10869-10881. doi: 10.3934/math.2024530
Let $ G = C_p\times H $ be a finite group, where $ C_p $ is a cyclic group of prime order $ p $ and $ H $ is a $ p^{\prime} $-group. Let $ \mathbb{F} $ be an algebraically closed field in characteristic $ p $. Let $ V $ be a direct sum of $ m $ non-trivial indecomposable $ G $-modules such that the norm polynomials of the simple $ H $-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring $ \mathbb{F}[V] $ with actions of $ G $. In this paper, by the periodicity property, we showed that $ \mathbb{F}[V]^G $ is generated by $ m $ norm polynomials together with homogeneous invariants of degree at most $ m|G|-{\rm dim}(V) $ and transfer invariants, which yields the well-known degree bound $ {\rm dim} $$ (V)\cdot(|G|-1) $. More precisely, we found that this bound gets less sharp as the dimensions of simple $ H $-modules increase.
| [1] |
E. Noether, Der endlichkeitssatz der invarianten endlicher gruppen, Math. Ann., 77 (1915), 89–92. https://doi.org/10.1007/BF01456821 doi: 10.1007/BF01456821
|
| [2] |
P. Fleischmann, The Noether bound in invariant theory of finite groups, Adv. Math., 156 (2000), 23–32. https://doi.org/10.1006/aima.2000.1952 doi: 10.1006/aima.2000.1952
|
| [3] |
J. Fogarty, On Noether's bound for polynomial invariants of a finite group, Electron. Res. Announc. Amer. Math. Soc., 7 (2001), 5–7. https://doi.org/10.1090/S1079-6762-01-00088-9 doi: 10.1090/S1079-6762-01-00088-9
|
| [4] |
D. R. Richman, On vector invariants over finite fields, Adv. Math., 81 (1990), 30–65. https://doi.org/10.1016/0001-8708(90)90003-6 doi: 10.1016/0001-8708(90)90003-6
|
| [5] |
D. R. Richman, Invariants of finite groups over fields of characteristic $p$, Adv. Math., 124 (1996), 25–48. https://doi.org/10.1006/aima.1996.0076 doi: 10.1006/aima.1996.0076
|
| [6] | G. Kemper, Hilbert series and degree bounds in invariant theory, In: B. H. Matzat, G. M. Greuel, G. Hiss, Algorithmic algebra and number theory, Springer-Verlag, 1999,249–263. https://doi.org/10.1007/978-3-642-59932-3_12 |
| [7] |
H. E. A. Campbell, A. V. Geramita, I. P. Hughes, R. J. Shank, D. L. Wehlau, Non Cohen-Macaulay vector invariants and a Noether bound for a Gorenstein ring of invariants, Canad. Math. Bull., 42 (1999), 155–161. https://doi.org/10.4153/CMB-1999-018-4 doi: 10.4153/CMB-1999-018-4
|
| [8] | A. Broer, Remarks on invariant theory of finite groups, Université de Montréal, 1997. |
| [9] |
P. Symonds, On the Castelnuovo-Mumford regularity of rings of polynomial invariants, Ann. Math., 174 (2011), 499–517. https://doi.org/10.4007/annals.2011.174.1.14 doi: 10.4007/annals.2011.174.1.14
|
| [10] |
P. Fleischmann, M. Sezer, R. J. Shank, C. F. Woodcock, The Noether numbers for cyclic groups of prime order, Adv. Math., 207 (2006), 149–155. https://doi.org/10.1016/j.aim.2005.11.009 doi: 10.1016/j.aim.2005.11.009
|
| [11] |
M. D. Neusel, M. Sezer, The invariants of modular indecomposable representations of $Z_{p^2}$, Math. Ann., 341 (2008), 575–587. https://doi.org/10.1007/s00208-007-0203-2 doi: 10.1007/s00208-007-0203-2
|
| [12] |
M. Sezer, Coinvariants and the regular representation of a cyclic $P$-group, Math. Z., 273 (2013), 539–546. https://doi.org/10.1007/s00209-012-1018-8 doi: 10.1007/s00209-012-1018-8
|
| [13] |
Y. Zhang, J. Z. Nan, H. X. Chen, A periodicity property of symmetric algebras with actions of metacyclic groups in the modular case, J. Math. Res. Appl., 43 (2023), 665–672. https://doi.org/10.3770/j.issn:2095-2651.2023.06.003 doi: 10.3770/j.issn:2095-2651.2023.06.003
|
| [14] |
I. Hughes, G. Kemper, Symmetric powers of modular representations, hilbert series and degree bounds, Commun. Algebra, 28 (2000), 2059–2088. https://doi.org/10.1080/00927870008826944 doi: 10.1080/00927870008826944
|
| [15] | B. Huppert, N. Blackburn, Finite groups II, Springer-Verlag, 1982. https://doi.org/10.1007/978-3-642-67994-0 |
| [16] |
R. P. Stanley, Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc., 1 (1979), 475–511. https://doi.org/10.1090/S0273-0979-1979-14597-X doi: 10.1090/S0273-0979-1979-14597-X
|
| [17] | L. Smith, Polynomial invariants of finite groups, AK Peters/CRC Press, 1995. https://doi.org/10.1201/9781439864470 |
| [18] | M. D. Neusel, L. Smith, Invariant theory of finite groups, American Mathematical Society, 2002. |
| [19] | J. L. Alperin, Local representation theory, Cambridge University Press, 1986. https://doi.org/10.1017/CBO9780511623592 |
Yang Zhang, Jizhu Nan. A note on the degree bounds of the invariant ring[J]. AIMS Mathematics, 2024, 9(5): 10869-10881. doi: 10.3934/math.2024530
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