Invariants of hyperplane groups and vanishing ideals of finite sets of points

2012 ◽  
Vol 55 (2) ◽  
pp. 355-367 ◽  
Author(s):  
H. E. A. Campbell ◽  
Jianjun Chuai
Keyword(s):  
Finite Group ◽  
Polynomial Ring ◽  
Constructive Proof ◽  
Additive Subgroup ◽  
Finite Sets ◽  
Vanishing Ideal ◽  
Invariant Ring ◽  
Invariant Differential ◽  
A Finite Group ◽  
Vanishing Ideals

AbstractWe define a hyperplane group to be a finite group generated by reflections fixing a single hyperplane pointwise. Landweber and Stong proved that the invariant ring of a hyperplane group is again a polynomial ring in any characteristic. Recently, Hartmann and Shepler gave a constructive proof of this result. By their algorithm, one can always construct generators that are additive. In this paper, we study hyperplane groups of order a power of a prime p in characteristic p and give a slightly different construction of the generators than Hartmann and Shepler. We then show that such generators have a particular form. Furthermore, we show that if the group is defined by a finite additive subgroup W ⊆ $W\subseteq\mathbb{F}^n$, the vanishing ideal of W is generated by polynomials obtained from a set of generators of the invariant ring that are additive. Finally, we give a shorter proof of the fact that the module of the invariant differential 1-forms is free in our situation.

scholarly journals Generating a Noetherian Normalization of the Invariant Ring of a Finite Group

1998 ◽  
Vol 25 (6) ◽  
pp. 727-731 ◽  
Author(s):  
Wolfram Decker ◽  
Agnes Eileen Heydtmann ◽  
Frank-Olaf Schreyer
Keyword(s):  
Finite Group ◽  
Invariant Ring ◽  
A Finite Group

scholarly journals Pseudo-reflection group actions on local rings

1982 ◽  
Vol 88 ◽  
pp. 161-180 ◽  
Author(s):  
Luchezar L. Avramov
Keyword(s):  
Finite Group ◽  
Polynomial Ring ◽  
Representation Theorem ◽  
Regular Representation ◽  
Reflection Group ◽  
The Other ◽  
Local Rings ◽  
Natural Representation ◽  
Classical Paper ◽  
A Finite Group

In a classical paper [C] Chevalley considered the invariants of a finite group H ⊂ GLk(S1) generated by pseudo-reflections, acting on the graded polynomial ring S = k[X1,…,Xn] over a field k of characteristic zero. He proved that S is free as a graded SH-module, hence SH is a graded polynomial ring (Theorem A), and that the natural representation of H in is equivalent to the regular representation (Theorem B). On the other hand, a theorem of Shephard and Todd shows that when SH is a polynomial ring, the (finite) group H is generated by pseudo-reflections. These results have been extended by Bourbaki [Bo2] to fields whose characteristic may be positive, but does not divide the order |H| of the group.

On arithmetically realizable classes

1995 ◽  
Vol 118 (3) ◽  
pp. 383-392 ◽  
Author(s):  
D. Burns
Keyword(s):  
Finite Group ◽  
Grothendieck Group ◽  
Number Fields ◽  
Additive Subgroup ◽  
Semi Group ◽  
Grothendieck Groups ◽  
Symplectic Characters ◽  
A Finite Group ◽  
Theoretic Description ◽  
Complex Characters

We fix a number field L and a finite group G, and write Cl (ℤL[G]) for the reduced Grothendieck group of the category of finitely generated projective ℤL[G]-modules. We let RG denote the ring of complex characters of G, with SG the additive subgroup which is generated by the irreducible symplectic characters. We shall say that an element c ∈ Cl (ℤL[G]) is ‘(arithmetically) realizable’ if there exists a tamely ramified Galois extension N/K of number fields with L ⊆ K and an identification Gal (N/K) →˜ G via which c is the class of some Gal (N/K)-stble ℤN-ideal. We let RL(G) denote the subgroup of Cl (ℤL[G]) which is generated by the realizable elements for varying N/K. Our interest in RL(G) arises from the fact that it is the largest subset of Cl (ℤL[G]) upon which the results of Chinburg and the author in [Bu, Ch] can be used to give an explicit module theoretic description of the action of the integral semi-group ring AL, G of the Adams-Cassou-Noguès-Taylor operators (ΨL, k): k ∈ ℤ, 2 × k if SG ≠ {0}}. Whilst the results of [Bu, Ch] can (at least partially) be understood ‘geometrically’ via the action of Bott cannibalistic elements on suitable Grothendieck groups (cf. [Ch, E, P, T], [Bu]), the underlying problem of finding an explicit module theoretic interpretation of the action of AL, G on all elements of Cl(ℤL[G]) is of course essentially algebraic in nature. It is in this context that we were originally motivated to investigate RL(G).

A NEW PROOF OF THE RHODES TYPE II CONJECTURE

2004 ◽  
Vol 14 (05n06) ◽  
pp. 551-568 ◽  
Author(s):  
K. AUINGER
Keyword(s):  
Finite Group ◽  
Finite Graph ◽  
Constructive Proof ◽  
Type Ii ◽  
Finite Monoid ◽  
A Finite Group ◽  
Relational Morphism

For a given finite monoid M we explicitly construct a finite group G and a relational morphism τ:M→G such that only elements of the type II construct Mc relate to 1 under τ. This provides an elementary and constructive proof of the type II conjecture of John Rhodes. The underlying idea is also used to modify the proof of Ash's celebrated theorem on inevitable graphs. For any finite monoid M and any finite graph Γ a finite group G is constructed which "spoils" all labelings of Γ over M which are not inevitable.

scholarly journals Invariant Theory of Abelian Transvection Groups

2010 ◽  
Vol 53 (3) ◽  
pp. 404-411
Author(s):  
Abraham Broer
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Direct Summand ◽  
Invariant Theory ◽  
Type Theorem ◽  
Linear Map ◽  
Arbitrary Characteristic ◽  
Invariant Ring ◽  
A Finite Group

AbstractLet G be a finite group acting linearly on the vector space V over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants, and the direct summand property holds if there is a surjective k[V]G-linear map π : k[V] → k[V]G.The following Chevalley–Shephard–Todd type theorem is proved. Suppose G is abelian. Then the action is coregular if and only if G is generated by pseudo-reflections and the direct summand property holds.

scholarly journals Algebraic construction of quasi-split algebraic tori

2019 ◽  
Vol 19 (11) ◽  
pp. 2050206
Author(s):  
Armin Jamshidpey ◽  
Nicole Lemire ◽  
Éric Schost
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Explicit Construction ◽  
Constructive Proof ◽  
Algebraic Construction ◽  
Quotient Field ◽  
Algebraic Tori ◽  
Transcendental Extension ◽  
A Finite Group

The main purpose of this work is to give a constructive proof for a particular case of the no-name lemma. Let [Formula: see text] be a finite group, [Formula: see text] a field that is equipped with a faithful [Formula: see text]-action, and [Formula: see text] a sign permutation [Formula: see text]-lattice (see the Introduction for the definition). Then [Formula: see text] acts naturally on the group algebra [Formula: see text] of [Formula: see text] over [Formula: see text], and hence also on the quotient field [Formula: see text]. A well-known variant of the no-name lemma asserts that the invariant sub-field [Formula: see text] is a purely transcendental extension of [Formula: see text]. In other words, there exist [Formula: see text] which are algebraically independent over [Formula: see text] such that [Formula: see text]. In this paper, we give an explicit construction of suitable elements [Formula: see text].

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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