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12
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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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.
Computing Modular Invariants of pgroups
, 2002
"... Let V be a finite dimensional representation of a pgroup, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V] G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V] ..."
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Cited by 8 (7 self)
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Let V be a finite dimensional representation of a pgroup, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V] G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V] G. We use these methods to analyse k[2V3] U3 where U3 is the pSylow subgroup of GL3(Fp) and 2V3 is the sum of two copies of the canonical representation. We give a generating set for k[2V3] U3 for p = 3 and prove that the invariants fail to be Cohen–Macaulay for p> 2. We also give a minimal generating set for k[mV2] Z/p were V2 is the twodimensional indecomposable representation of the cyclic group Z/p.
THE NOETHER NUMBER IN INVARIANT THEORY
, 2006
"... Let F be any field. Let G be any reductive linear algebraic group and consider a finite dimensional rational representation V of G. Then the Falgebra F[V] G of polynomial invariants for G acting on V is finitely generated. The Noether Number β(G, V) is the highest degree of an element of a minimal ..."
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Cited by 6 (0 self)
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Let F be any field. Let G be any reductive linear algebraic group and consider a finite dimensional rational representation V of G. Then the Falgebra F[V] G of polynomial invariants for G acting on V is finitely generated. The Noether Number β(G, V) is the highest degree of an element of a minimal homogeneous generating set for F[V] G. We survey what is known about Noether Numbers, in particular describing various upper and lower bounds for them. Both finite and infinite groups and both characteristic 0 and positive characteristic are considered.
VECTOR INVARIANTS FOR THE TWO DIMENSIONAL MODULAR REPRESENTATION OF A CYCLIC GROUP OF PRIME ORDER
, 2009
"... In this paper, we study the vector invariants, F[m V2] Cp, of the 2dimensional indecomposable representation V2 of the cylic group, Cp, of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman [18] who showed that this ring required a generator of d ..."
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Cited by 6 (4 self)
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In this paper, we study the vector invariants, F[m V2] Cp, of the 2dimensional indecomposable representation V2 of the cylic group, Cp, of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman [18] who showed that this ring required a generator of degree m(p−1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p = 2. This conjecture was proved by Campbell and Hughes in [2]. Later, Shank and Wehlau in [21] determined which elements in Richman’s generating set were redundant thereby producing a minimal generating set. We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the
COINVARIANTS FOR MODULAR REPRESENTATIONS OF CYCLIC GROUPS OF PRIME ORDER
, 2004
"... Abstract. We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariant ..."
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Cited by 4 (3 self)
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Abstract. We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three dimensional indecomposable representation. 1.
Classical covariants and modular invariants
 Invariant Theory in All Characteristics, CRM Proc. Lecture Notes 35, Amer. Math. Soc., 241–249 (2004) Zbl 1094.13008 MR 2066471
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Invariants for the modular cyclic group of prime order via classical invariant theory
 J. EUR. MATH. SOC. 15, 775–803
, 2013
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Invariants of the Diagonal Cpaction on V3
, 2005
"... Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by V3 the indecomposable three dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V3 ⊕ V3] Cp. Our main result confirms th ..."
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Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by V3 the indecomposable three dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V3 ⊕ V3] Cp. Our main result confirms the conjecture of Shank[15], for this example, that all modular rings of invariants of Cp are generated by rational invariants, norms and transfers.
DECOMPOSING SYMMETRIC POWERS OF CERTAIN MODULAR REPRESENTATIONS OF CYCLIC GROUPS
, 2008
"... For a prime number p, we construct a generating set for the ring of invariants for the p + 1 dimensional indecomposable modular representation of a cyclic group of order p². We then use the constructed invariants to describe the decomposition of the symmetric algebra as a module over the group ring ..."
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For a prime number p, we construct a generating set for the ring of invariants for the p + 1 dimensional indecomposable modular representation of a cyclic group of order p². We then use the constructed invariants to describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case.