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Noether numbers for subrepresentations of cyclic groups of prime order
 BULL. LONDON MATH. SOC
, 2002
"... Let W be a finitedimensional �/pmodule over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W] �/p, is called the Noether number of the representation, and is denoted by β(W). A lower bound for β(W) is derived, and it is shown that i ..."
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Let W be a finitedimensional �/pmodule over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W] �/p, is called the Noether number of the representation, and is denoted by β(W). A lower bound for β(W) is derived, and it is shown that if U is a �/p submodule of W, then β(U) � β(W). A set of generators, in fact a SAGBI basis, is constructed for k[V2 ⊕ V3] �/p, where Vn is the indecomposable �/pmodule of dimension n.
Polynomial Invariants of Finite Groups: A Survey of Recent Developments
 Bull. Amer. Math. Soc
, 1997
"... Abstract. The polynomial invariants of finite groups have been studied for more than a century now and continue to find new applications and generate interesting problems. In this article we will survey some of the recent developments coming primarily from algebraic topology and the rediscovery of o ..."
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Cited by 6 (0 self)
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Abstract. The polynomial invariants of finite groups have been studied for more than a century now and continue to find new applications and generate interesting problems. In this article we will survey some of the recent developments coming primarily from algebraic topology and the rediscovery of old open problems. It has been almost two decades since the Bulletin of the AMS published the marvelous survey article [111] of R. P. Stanley. Since then the invariant theory of finite groups has taken on a central role in many problems of algebraic topology, such as e.g. [22], [2], [101], [65], [105], [84], [106] chapter 11, and the references there. It has received new impetus as a subject of study in its own right, [72]–[81], [3], [43], and several textbooks with varying viewpoints [9], [114], and [106], as well as a reprint of venerable old lecture notes [48], have recently appeared. In this survey article I will try to discuss some of these developments as seen through the eyes of one who came to the subject from algebraic topology. That means that finite groups and finite fields will play a central role, and the modular case, i.e. where the
On CohenMacaulay Rings of Invariants
 J. Algebra
, 2001
"... Abstract. We investigate the transfer of the CohenMacaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we briefly discuss the s ..."
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Cited by 6 (1 self)
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Abstract. We investigate the transfer of the CohenMacaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we briefly discuss the special case of multiplicative actions, that is, actions on group algebras k[Z n] via an action on Z n.
The Noether numbers for cyclic groups of prime order
, 2005
"... Abstract. The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p − ..."
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Cited by 4 (1 self)
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Abstract. The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p − 3 conjecture”. 1.
Depth Of Modular Invariant Rings
 Advances in Math
, 2000
"... . It is wellknown that the ring of invariants associated to a nonmodular representation of a finite group is CohenMacaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be CohenMacaulay and computing the de ..."
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. It is wellknown that the ring of invariants associated to a nonmodular representation of a finite group is CohenMacaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be CohenMacaulay and computing the depth is often very difficult. In this paper 1 we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6]. Introduction We consider a faithful representation of a finite group G on a vector space V of dimension n over a field K. The representation is said to be modular if the characteristic of K divides the order of the group G. Otherwise it is called nonmodular. The action Received July 28, 1998. Accepted May 10, 1999. 1 Research partially supported by grants from ARP an...