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## Noether numbers for subrepresentations of cyclic groups of prime order (2002)

Venue: | BULL. LONDON MATH. SOC |

Citations: | 12 - 10 self |

### Citations

2036 |
Commutative algebra with a view towards algebraic geometry, GTM 150
- Eisenbud
(Show Context)
Citation Context ...t regular sequence on M. The depth of a ring is bounded above by its Krull dimension. A ring is Cohen–Macaulay if the depth equals the dimension. For a detailed discussion of depth and dimension, see =-=[9]-=-. 2. Preliminaries In this paper we consider the invariant theory of �/p, the cyclic group of order p, over a field k of characteristic p. We denote by σ a fixed generator of �/p. In the group ring, k... |

1345 | The Magma algebra system. I. The user language
- Bosma, Cannon, et al.
- 1997
(Show Context)
Citation Context ...to � � p−1 2 +1 by Fleischmann [13, Proposition 12.3], and to (p + 3)(p − 1)/4+1 by Hughes and Kemper [16, Corollary 2.15]. Conjecture 6.1. β(Vp) =2p − 3. Direct calculation of Fp[Vp] �/p using MAGMA =-=[4]-=- confirms the conjecture for p � 7. A less direct method, outlined below, confirms the conjecture for the primes 11 and 13. Let z be a distinguished variable for Vp, and let N be the norm of z. Let I ... |

410 | Gröbner Bases: a computational approach to commutative algebra, volume 141 of Graduate Texts in Mathematics - Becker, Weispfenning - 1991 |

151 |
Polynomial invariants of finite groups
- Benson
- 1993
(Show Context)
Citation Context ...dular if the characteristic of k divides the order of G. We say the representation is non-modular if |G| is invertible in k. For an introduction to the invariant theory of finite groups, we recommend =-=[3]-=- or[22]. Suppose that R = ⊕∞ i=0Ri is a finitely generated graded algebra. Let R+ denote the augmentation ideal of R, that is, the ideal generated by the homogeneous elements of positive degree. We ca... |

90 |
The Cohomology of Groups
- Evens
- 1991
(Show Context)
Citation Context ...oetherian ring, and H 1 (�/p, k[V ]) is a quotient of a submodule of the finitely generated module k[V ]. Therefore H 1 (�/p, k[V ]) is a finitely generated k[V ] �/p -module. We direct the reader to =-=[11]-=- for a detailed discussion of group cohomology.s440 r. james shank and david l. wehlau Every �/p-module may be written as a direct sum of copies of indecomposable modules. Note that k[W ⊕ V1] �/p ∼ = ... |

90 |
Polynomial invariants of finite groups
- Smith
- 1995
(Show Context)
Citation Context ...if the characteristic of k divides the order of G. We say the representation is non-modular if |G| is invertible in k. For an introduction to the invariant theory of finite groups, we recommend [3] or=-=[22]-=-. Suppose that R = ⊕∞ i=0Ri is a finitely generated graded algebra. Let R+ denote the augmentation ideal of R, that is, the ideal generated by the homogeneous elements of positive degree. We call an e... |

73 |
Endlichkeitssatz der Invarianten endlicher Gruppen
- Noether, Der
- 1916
(Show Context)
Citation Context ...ther words, β(R) is the largest degree of a homogeneous indecomposable element of R. IfR = k[V ] G , we shall often write β(V ) in place of β(R) if the group G is clear from the context. Emmy Noether =-=[18]-=- proved that if the characteristic of k is zero, or if it exceeds |G|, then β(V ) � |G|. Fleischmann [14] and Fogarty [12] recently proved that the same bound holds for the general non-modular case. M... |

37 | A completion procedure for computing a canonical basis for a k-subalgebra. In: Computers and mathematics
- Kapur, Madlener
- 1989
(Show Context)
Citation Context ...n n. The term ‘SAGBI’ is an acronym for Subalgebra Analog to Gröbner Bases for Ideals, and was introduced by Robbiano and Sweedler [20]. The concept was introduced independently by Kapur and Madlener =-=[17]-=-. In the final section of the paper, we discuss a conjectured value for the Noether number of the regular modular representation of �/p. Let R be a finitely generated graded algebra. We shall denote b... |

35 |
Subalgebra bases
- Robbiano, Sweedler
- 1990
(Show Context)
Citation Context ... characteristic p and Vn is the indecomposable �/p-module of dimension n. The term ‘SAGBI’ is an acronym for Subalgebra Analog to Gröbner Bases for Ideals, and was introduced by Robbiano and Sweedler =-=[20]-=-. The concept was introduced independently by Kapur and Madlener [17]. In the final section of the paper, we discuss a conjectured value for the Noether number of the regular modular representation of... |

31 |
On Noether’s bound for polynomial invariants of a finite group
- Fogarty
(Show Context)
Citation Context ...β(V ) in place of β(R) if the group G is clear from the context. Emmy Noether [18] proved that if the characteristic of k is zero, or if it exceeds |G|, then β(V ) � |G|. Fleischmann [14] and Fogarty =-=[12]-=- recently proved that the same bound holds for the general non-modular case. Much less is known about β(V ) in the modular case. Göbel [15] proved that, for any characteristic, if G acts by permuting ... |

31 |
The Noether bound in invariant theory of finite groups
- Fleischmann
(Show Context)
Citation Context ...hall often write β(V ) in place of β(R) if the group G is clear from the context. Emmy Noether [18] proved that if the characteristic of k is zero, or if it exceeds |G|, then β(V ) � |G|. Fleischmann =-=[14]-=- and Fogarty [12] recently proved that the same bound holds for the general non-modular case. Much less is known about β(V ) in the modular case. Göbel [15] proved that, for any characteristic, if G a... |

22 |
Decompositions of exterior and symmetric powers of indecomposable Z/pZ-modules in characteristic p and relations to invariants
- Almkvist, Fossum
- 1977
(Show Context)
Citation Context ...] ♯ and k[W ] ♭ inherit this grading. It is easy to see that k[W ] ♭ (d1,...,dt) ∼ = k[W1] ♭ ♭ ♭ ⊗ k[W2] ⊗ ...⊗ k[Wt] d1 d2 dt . Furthermore, k[Vn] ♭ d is a free �/p-module for all d � p − n + 1 (see =-=[1]-=- or[16, Lemma 2.10]). Thus k[W ] ♭ (d1,...,dt) is free if any di � p − dim(Wi)+1. 3. Lower bounds It is well known (see, for example, [5] or[19]) that k[2 V2] �/p = k[x1,N1,x2,N2,u], where N1 = y p p−... |

18 |
On vector invariants over finite fields,
- Richman
- 1990
(Show Context)
Citation Context ...[Vn] ♭ d is a free �/p-module for all d � p − n + 1 (see [1] or[16, Lemma 2.10]). Thus k[W ] ♭ (d1,...,dt) is free if any di � p − dim(Wi)+1. 3. Lower bounds It is well known (see, for example, [5] or=-=[19]-=-) that k[2 V2] �/p = k[x1,N1,x2,N2,u], where N1 = y p p−1 1 − y1x1 , N2 = y p p−1 2 − y2x2 , u = x2y1 − x1y2, and {x1,y1,x2,y2} is a basis for (2 V2) ∗ with ∆y1 = x1, ∆x1 =0,∆y2 = x2, and ∆x2 = 0. In ... |

17 | Symmetric powers of modular representations, Hilbert series and degree bounds
- Hughes, Kemper
(Show Context)
Citation Context ... case. Much less is known about β(V ) in the modular case. Göbel [15] proved that, for any characteristic, if G acts by permuting a basis of V , then β(V ) � � � dim V 2 . Recently, Hughes and Kemper =-=[16]-=- found bounds for β(V ) for any modular representation of �/p. Here, we shall show that if W is a representation of �/p in characteristic p and U is a subrepresentation, then β(U) � β(W ). We shall al... |

13 |
Vector invariants of U2(Fp): A proof of a conjecture
- Campbell, Hughes
- 1997
(Show Context)
Citation Context ...ore, k[Vn] ♭ d is a free �/p-module for all d � p − n + 1 (see [1] or[16, Lemma 2.10]). Thus k[W ] ♭ (d1,...,dt) is free if any di � p − dim(Wi)+1. 3. Lower bounds It is well known (see, for example, =-=[5]-=- or[19]) that k[2 V2] �/p = k[x1,N1,x2,N2,u], where N1 = y p p−1 1 − y1x1 , N2 = y p p−1 2 − y2x2 , u = x2y1 − x1y2, and {x1,y1,x2,y2} is a basis for (2 V2) ∗ with ∆y1 = x1, ∆x1 =0,∆y2 = x2, and ∆x2 =... |

10 |
Profondeur d’anneaux d’invariants en caractéristique p
- Ellingsrud, Skjelbred
- 1980
(Show Context)
Citation Context ...re contained in A. Thus Tr�/p (y p−1 p−1 1 y2z2 )is indecomposable. Similarly, since LM(Tr �/p (y p−1 1 z p−1 2 )) = y p−1 1 yp−1 2 , we see that Tr �/p (y p−1 1 z p−1 2 )is also indecomposable. ✷ By =-=[10]-=- or[6], we know that k[V2 ⊕ V3] �/p is not Cohen–Macaulay. Here we shall show explicitly that the partial homogeneous system of parameters x1,x2,d in k[V2 ⊕ V3] �/p is not a regular sequence.s448 r. j... |

10 | Relative Trace Ideals and Cohen-Macaulay Quotients of Modular Invariant Rings, - Fleischmann - 1999 |

9 |
bases for rings of formal modular seminvariants [semi-invariants
- I
- 1998
(Show Context)
Citation Context ...1 y2z2 ) and Tr �/p (y p−1 1 z p−1 2 ) are indecomposable elements of k[V2 ⊕ V3] �/p . In particular, β(V2 ⊕ V3) � 2p − 1. To prove Theorem 5.1, we shall use the method which the first author used in =-=[21]-=- to compute k[V4] �/p and k[V5] �/p . The method involves constructing a subalgebra Q of k[V2 ⊕ V3] �/p , and then showing that Q = k[V2 ⊕ V3] �/p , by showing that the two algebras have the same Hilb... |

5 | Bases for rings of coinvariants, Transformation Groups 4 - Campbell, Hughes, et al. - 1996 |

4 |
Computing bases for rings of permutation invariant polynomials
- Göbel
- 1995
(Show Context)
Citation Context ... exceeds |G|, then β(V ) � |G|. Fleischmann [14] and Fogarty [12] recently proved that the same bound holds for the general non-modular case. Much less is known about β(V ) in the modular case. Göbel =-=[15]-=- proved that, for any characteristic, if G acts by permuting a basis of V , then β(V ) � � � dim V 2 . Recently, Hughes and Kemper [16] found bounds for β(V ) for any modular representation of �/p. He... |

2 |
Depth of modular invariant rings, Transformation Groups 5 (2000) no
- Campbell, Hughes, et al.
(Show Context)
Citation Context ...ained in A. Thus Tr�/p (y p−1 p−1 1 y2z2 )is indecomposable. Similarly, since LM(Tr �/p (y p−1 1 z p−1 2 )) = y p−1 1 yp−1 2 , we see that Tr �/p (y p−1 1 z p−1 2 )is also indecomposable. ✷ By [10] or=-=[6]-=-, we know that k[V2 ⊕ V3] �/p is not Cohen–Macaulay. Here we shall show explicitly that the partial homogeneous system of parameters x1,x2,d in k[V2 ⊕ V3] �/p is not a regular sequence.s448 r. james s... |

2 | Gröbner bases and convex polytopes. Univ. Lecture Ser. 8 - Sturmfels - 1996 |