Rational Invariants of certain Orthogonal and Unitary Groups

Carlisle, David; Kropholler, Peter H.

doi:10.1112/blms/24.1.57

Cited by 21 publications

(32 citation statements)

References 2 publications

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“…. , y n ) G ; the latter is well known for many classical groups (see [1,2,4,5]). [3] Invariants field of some finite projective linear groups 21…”

Section: A General Resultsmentioning

confidence: 99%

“…Then the element β 1 g 1 + · · · + β n g n in N can be expressed as k 1 e 1 + · · · + k n−1 e n−1 for some integers k i . That is, f = e k 1 1 · · · e k n−1 n−1 . Since each g i is homogeneous, then g i (y 1 , .…”

Section: A General Resultsmentioning

confidence: 99%

“…Using the same strategy, the invariants subfield F q (V) PSL n (F q ) for the special linear group SL n (F q ) was also computed in [3]. On the other hand, it is well known that G acts naturally on the rational function field F q (V) and the invariants field F q (V) G is purely transcendental for many classical groups G, such as the symplectic group Sp 2n (F q ), unitary group U n (F q 2 ) and orthogonal group O n (F q ) (see [1,2,4,5]). …”

Section: Introductionmentioning

confidence: 99%

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The Invariants Field of Some Finite Projective Linear Group Actions

Chen

2011

Bull. Aust. Math. Soc.

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Let F q be a finite field with q elements, V an n-dimensional vector space over F q and V the projective space associated to V. Let G ≤ GL n (F q ) be a classical group and PG be the corresponding projective group. In this note we prove that if F q (V) G is purely transcendental over F q with homogeneous polynomial generators, then F q (V) PG is also purely transcendental over F q . We compute explicitly the generators of F q (V) PG when G is the symplectic, unitary or orthogonal group.2010 Mathematics subject classification: primary 12F20; secondary 20G40.

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“…. , y n ) G ; the latter is well known for many classical groups (see [1,2,4,5]). [3] Invariants field of some finite projective linear groups 21…”

Section: A General Resultsmentioning

confidence: 99%

Section: A General Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Invariants Field of Some Finite Projective Linear Group Actions

Chen

2011

Bull. Aust. Math. Soc.

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“…[7,Section 199 [6,5,2,4]. The purpose of this note is to prove the following theorem of S. D. Cohen.…”

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confidence: 98%

The Ring of Invariants of O(3, Fq)

Smith¹

1999

Finite Fields and Their Applications

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Let p3-be an odd prime integer and % O be the Galois field with q"pJ elements. Let Q"y!xz3% O [x, y, z], which is a nondegenerate quadratic form, and denote by O(3, % O ) the corresponding orthogonal group. The purpose of this note is to give a new proof of a theorem of S. D. Cohen on the structure ofThe novelty of our proof lies in the description of the generators in terms of the geometry of ovals in % O /(2) and Steenrod operations. Academic PressKey ¼ords: polynomial invariants of finite groups; invariants of finite orthogonal groups; Steenrod operations; ovals; Segre's theorem on ovals.Let p3-be an add prime integer and % O be the Galois field with q"pJ] is a nondegenerate quadratic form then, up to linear change of coordinates, Q is given by one of the following standard forms

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“…4 In any case, both groups have the same orbit structure when acting on the Grassmannians of V . Since the invariants of Sp(2l, F) form a complete intersection ( [6], Section 8.2 in [3], or [11]) they are Cohen-Macaulay. Hence so are the invariants of any group between Sp(2l, F) and Γ (Sp(2l, F)).…”

Section: Proposition 24 If F[v ] G Is Cohen-macaulay Then So Is F[v ]mentioning

confidence: 99%

Unstable Cohen–Macaulay Algebras

Neusel

2001

Mathematical Research Letters

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Abstract. We characterize Cohen-Macaulay algebras in the category K fg of unstable Noetherian algebras over the Steenrod algebra via the depth of the P * -invariant ideals. This allows us to transfer the Cohen-Macaulay property to suitable subalgebras. We apply this to rings of invariants of finite groups and to the P * -inseparable closure.For over 50 years the Steenrod algebra P * and algebras over it have proven to be decisive tools in algebraic topology. Applications occur in cohomology theory, invariant theory of finite groups, and most recently in algebraic geometry. This makes it desirable to have a completely algebraic (and not topological) approach to the subject.In this paper we follow the path begun in [12], [13], [14] and [16] of building a "complete" P * -invariant commutative algebra. We find a P * -invariant version of the classical characterization of Cohen-Macaulay algebras. This allows us to show that Cohen-Macaulayness is inherited by subalgebras provided that the P *

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Rational Invariants of certain Orthogonal and Unitary Groups

Cited by 21 publications

References 2 publications

The Invariants Field of Some Finite Projective Linear Group Actions

The Invariants Field of Some Finite Projective Linear Group Actions

The Ring of Invariants of O(3, Fq)

Unstable Cohen–Macaulay Algebras

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